Lie groups aFrom Wikipedia, the free encyclopedia

Contents

1 2E6 (mathematics) 11.1 Over nite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Over the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 3D4 22.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Over nite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 3D4(23) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 ADE classication 43.1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Binary polyhedral groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Labeled graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Other classications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Trinities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Adjoint representation of a Lie algebra 94.1 Adjoint representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Relation to Ad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Ane group 125.1 Relation to general linear group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.1.1 Construction from general linear group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.1.2 Stabilizer of a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.2 Matrix representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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ii CONTENTS

5.3 Planar ane group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.4 Other ane groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.4.1 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.4.2 Special ane group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4.3 Projective subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4.4 Poincar group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6 Ahlfors niteness theorem 156.1 Bers area inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7 An Exceptionally Simple Theory of Everything 167.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7.1.1 Non-technical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.1.2 Algebraic breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.1.3 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.1.4 Q and A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.2 Chronology and reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8 Automorphic form 238.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3 Automorphic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.4 Poincar on discovery and his work on automorphic functions . . . . . . . . . . . . . . . . . . . . 248.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

9 Glossary of semisimple groups 269.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.10 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.11 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

CONTENTS iii

9.12 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.13 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.14 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.15 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.16 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.17 Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.18 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.19 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.20 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.21 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.22 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.23 W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10 Lie theory 3210.1 Elementary Lie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.2 History and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.3 Aspects of Lie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

11 List of Lie groups topics 3411.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.2 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.3 Foundational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.4 Semisimple theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.5 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

11.6.1 Physical theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.6.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.6.3 Discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.6.4 Algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11.7 Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.7.1 Automorphic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11.8 People . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 39

11.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Chapter 1

2E6 (mathematics)

In mathematics, 2E6 is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form ofE6, depending on a quadratic extension of elds KL. Unfortunately the notation for the group is not standardized,as some authors write it as 2E6(K) (thinking of 2E6 as an algebraic group taking values in K) and some as 2E6(L)(thinking of the group as a subgroup of E(L) xed by an outer involution).Over nite elds these groups form one of the 18 innite families of nite simple groups, and were introducedindependently by Tits (1958) and Steinberg (1959).

1.1 Over nite eldsThe group 2E6(q2) has order q36 (q12 1) (q9 + 1) (q8 1) (q6 1) (q5 + 1) (q2 1) /(3,q + 1). This is similar tothe order q36 (q12 1) (q9 1) (q8 1) (q6 1) (q5 1) (q2 1) /(3,q 1) of E6(q).Its Schur multiplier has order (3, q + 1) except for 2E6(22), when it has order 12 and is a product of cyclic groupsof orders 2,2,3. One of the exceptional double covers of 2E6(22) is a subgroup of the baby monster group, and theexceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.The outer automorphism group has order (3, q + 1) f where q2 = pf .

1.2 Over the real numbersOver the real numbers, 2E6 is the quasisplit form of E6, and is one of the ve real forms of E6 classied by E. Cartan.Its maximal compact subgroup is of type F4.

1.3 References Carter, Roger W. (1989) [1972], Simple groups of Lie type, Wiley Classics Library, New York: John Wiley &Sons, ISBN 978-0-471-50683-6, MR 0407163

Steinberg, Robert (1959), Variations on a theme of Chevalley, Pacic Journal of Mathematics 9: 875891,doi:10.2140/pjm.1959.9.875, ISSN 0030-8730, MR 0109191

Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335 Tits, Jacques (1958), Les formes relles des groupes de type E6, Sminaire Bourbaki; 10e anne: 1957/1958.Textes des confrences; Exposs 152 168; 2e d. corrige, Expos 162 15, Paris: Secrtariat math'ematique,MR 0106247

1

Chapter 2

3D4

Inmathematics, the Steinberg triality groups of type 3D4 form a family of Steinberg or twisted Chevalley groups. Theyare quasi-split forms of D4, depending on a cubic Galois extension of eldsK L, and using the triality automorphismof the Dynkin diagram D4. Unfortunately the notation for the group is not standardized, as some authors write it as3D4(K) (thinking of 3D4 as an algebraic group taking values in K) and some as 3D4(L) (thinking of the group as asubgroup of D4(L) xed by an outer automorphism of order 3). The group 3D4 is very similar to an orthogonal orspin group in dimension 8.Over nite elds these groups form one of the 18 innite families of nite simple groups, and were introduced bySteinberg (1959).

2.1 ConstructionThe simply connected split algebraic group of type D4 has a triality automorphism of order 3 coming from an order3 automorphism of its Dynkin diagram. If L is a eld with an automorphism of order 3, then this induced an order3 automorphism of the group D4(L). The group 3D4(L) is the subgroup of D4(L) of points xed by . It has three8-dimensional representations over the eld L, permuted by the outer automorphism of order 3.

2.2 Over nite eldsThe group 3D4(q3) has order q12 (q8 + q4 + 1) (q6 1) (q2 1). For comparison, the split spin group D4(q) indimension 8 has order q12 (q8 2q4 + 1) (q6 1) (q2 1) and the quasisplit spin group 2D4(q2) in dimension 8 hasorder q12 (q8 1) (q6 1) (q2 1).The group 3D4(q3) is always simple. The Schur multiplier is always trivial. The outer automorphism group is cyclicof order f where q3 = pf and p is prime.This group is also sometimes called 3D4(q), D42(q3), or a twisted Chevalley group.

2.3 3D4(23)The smallest member of this family of groups has several exceptional properties not shared by other members of thefamily. It has order 211341312 = 212347213 and outer automorphism group of order 3.The automorphism group of 3D4(23) is a maximal subgroup of the Thompson sporadic group, and is also a subgroupof the compact Lie group of type F4 of dimension 52. In particular it acts on the 26-dimensional representation of F4.In this representation it xes a 26-dimensional lattice that is the unique 26-dimensional even lattice of determinant 3with no norm 2 vectors, studied by Gross & Elkies (1996). The dual of this lattice has 819 pairs of vectors of norm8/3, on which 3D4(23) acts as a rank 4 permutation group.The group 3D4(23) has 9 classes of maximal subgroups, of structure

2

2.4. SEE ALSO 3

21+8:L2(8) xing a point of the rank 4 permutation representation on 819 points.[211]:(7 S3)U3(3):2S3 L2(8)(7 L2(7)):231+2.2S472:2A432:2A413:4

2.4 See also List of nite simple groups 2E6

2.5 References Carter, Roger W. (1989) [1972], Simple groups of Lie type, Wiley Classics Library, New York: John Wiley &Sons, ISBN 978-0-471-50683-6, MR 0407163

Elkies, NoamD.; Gross, Benedict H. (1996), The exceptional cone and the Leech lattice, International Math-ematics Research Notices (14): 665698, doi:10.1155/S1073792896000426, ISSN 1073-7928, MR 1411589

Steinberg, Robert (1959), Variations on a theme of Chevalley, Pacic Journal of Mathematics 9: 875891,doi:10.2140/pjm.1959.9.875, ISSN 0030-8730, MR 0109191

Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335

2.6 External links D(2) at the atlas of nite groups D(3) at the atlas of nite groups

Chapter 3

ADE classication

Inmathematics, theADEclassication (originallyA-D-E classications) is the complete list of simply lacedDynkindiagrams or other mathematical objects satisfying analogous axioms; simply laced means that there are no multipleedges, which corresponds to all simple roots in the root system forming angles of /2 = 90 (no edge between thevertices) or 2/3 = 120 (single edge between the vertices). The list comprises

An; Dn; E6; E7; E8:

These comprise two of the four families of Dynkin diagrams (omittingBn andCn ), and three of the ve exceptionalDynkin diagrams (omitting F4 and G2 ).This list is non-redundant if one takes n 4 for Dn: If one extends the families to include redundant terms, oneobtains the exceptional isomorphisms

D3 = A3; E4 = A4; E5 = D5;

and corresponding isomorphisms of classied objects.The question of giving a common origin to these classications, rather than a posteriori verication of a parallelism,was posed in (Arnold 1976).The A, D, E nomenclature also yields the simply laced nite Coxeter groups, by the same diagrams: in this case theDynkin diagrams exactly coincide with the Coxeter diagrams, as there are no multiple edges.

3.1 Lie algebrasIn terms of complex semisimple Lie algebras:

An corresponds to sln+1(C); the special linear Lie algebra of traceless operators, Dn corresponds to so2n(C); the even special orthogonal Lie algebra of even-dimensional skew-symmetricoperators, and

E6; E7; E8 are three of the ve exceptional Lie algebras.

In terms of compact Lie algebras and corresponding simply laced Lie groups:

An corresponds to sun+1; the algebra of the special unitary group SU(n+ 1); Dn corresponds to so2n(R); the algebra of the even projective special orthogonal group PSO(2n) , while E6; E7; E8 are three of ve exceptional compact Lie algebras.

4

3.2. BINARY POLYHEDRAL GROUPS 5

An

Dn

E6

E7

E8The simply laced Dynkin diagrams classify diverse mathematical objects.

3.2 Binary polyhedral groupsThe same classication applies to discrete subgroups of SU(2) , the binary polyhedral groups; properly, binarypolyhedral groups correspond to the simply laced ane Dynkin diagrams ~An; ~Dn; ~Ek; and the representations ofthese groups can be understood in terms of these diagrams. This connection is known as theMcKay correspondenceafter John McKay. The connection to Platonic solids is described in (Dickson 1959). The correspondence uses theconstruction of McKay graph.Note that theADE correspondence is not the correspondence of Platonic solids to their reection group of symmetries:for instance, in the ADE correspondence the tetrahedron, cube/octahedron, and dodecahedron/icosahedron corre-spond toE6; E7; E8;while the reection groups of the tetrahedron, cube/octahedron, and dodecahedron/icosahedronare instead representations of the Coxeter groups A3; BC3; and H3:

6 CHAPTER 3. ADE CLASSIFICATION

The orbifold of C2 constructed using each discrete subgroup leads to an ADE-type singularity at the origin, termeda du Val singularity.The McKay correspondence can be extended to multiply laced Dynkin diagrams, by using a pair of binary polyhedralgroups. This is known as the Slodowy correspondence, named after Peter Slodowy see (Stekolshchik 2008).

3.3 Labeled graphsThe ADE graphs and the extended (ane) ADE graphs can also be characterized in terms of labellings with certainproperties,[1] which can be stated in terms of the discrete Laplace operators[2] or Cartan matrices. Proofs in terms ofCartan matrices may be found in (Kac 1990, pp. 4754).The ane ADE graphs are the only graphs that admit a positive labeling (labeling of the nodes by positive realnumbers) with the following property:

Twice any label is the sum of the labels on adjacent vertices.

That is, they are the only positive functions with eigenvalue 1 for the discrete Laplacian (sum of adjacent verticesminus value of vertex) the positive solutions to the homogeneous equation:

= :

Equivalently, the positive functions in the kernel of I: The resulting numbering is unique up to scale, and ifnormalized such that the smallest number is 1, consists of small integers 1 through 6, depending on the graph.The ordinary ADE graphs are the only graphs that admit a positive labeling with the following property:

Twice any label minus two is the sum of the labels on adjacent vertices.

In terms of the Laplacian, the positive solutions to the inhomogeneous equation:

= 2:

The resulting numbering is unique (scale is specied by the 2) and consists of integers; for E8 they range from 58to 270, and have been observed as early as (Bourbaki 1968).

3.4 Other classicationsThe elementary catastrophes are also classied by the ADE classication.The ADE diagrams are exactly the quivers of nite type, via Gabriels theorem.There is also a link with generalized quadrangles, as the three non-degenerate GQs with three points on each linecorrespond to the three exceptional root systems E6, E7 and E8.[3] The classes A and D correspond degenerate caseswhere the line set is empty or we have all lines passing through a xed point, respectively.[4]

There are deep connections between these objects, hinted at by the classication; some of these connections can beunderstood via string theory and quantum mechanics.

3.5 TrinitiesArnold has subsequently proposedmany further connections in this vein, under the rubric of mathematical trinities,[5][6]and McKay has extended his correspondence along parallel and sometimes overlapping lines. Arnold terms these"trinities" to evoke religion, and suggest that (currently) these parallels rely more on faith than on rigorous proof,

3.6. SEE ALSO 7

though some parallels are elaborated. Further trinities have been suggested by other authors.[7][8][9] Arnolds trinitiesbegin with R/C/H (the real numbers, complex numbers, and quaternions), which he remarks everyone knows, andproceeds to imagine the other trinities as complexications and quaternionications of classical (real) mathemat-ics, by analogy with nding symplectic analogs of classic Riemannian geometry, which he had previously proposedin the 1970s. In addition to examples from dierential topology (such as characteristic classes), Arnold considersthe three Platonic symmetries (tetrahedral, octahedral, icosahedral) as corresponding to the reals, complexes, andquaternions, which then connects with McKays more algebraic correspondences, below.McKays correspondences are easier to describe. Firstly, the extended Dynkin diagrams ~E6; ~E7; ~E8 (correspondingto tetrahedral, octahedral, and icosahedral symmetry) have symmetry groups S3; S2; S1; respectively, and the asso-ciated foldings are the diagrams ~G2; ~F4; ~E8 (note that in less careful writing, the extended (tilde) qualier is oftenomitted). More signicantly, McKay suggests a correspondence between the nodes of the ~E8 diagram and certainconjugacy classes of the monster group, which is known asMcKays E8 observation;[10][11] see also monstrous moon-shine. McKay further relates the nodes of ~E7 to conjugacy classes in 2.B (an order 2 extension of the baby monstergroup), and the nodes of ~E6 to conjugacy classes in 3.Fi24' (an order 3 extension of the Fischer group)[11] note thatthese are the three largest sporadic groups, and that the order of the extension corresponds to the symmetries of thediagram.Turning from large simple groups to small ones, the corresponding Platonic groupsA4; S4; A5 have connections withthe projective special linear groups PSL(2,5), PSL(2,7), and PSL(2,11) (orders 60, 168, and 660),[12][13] which isdeemed a McKay correspondence.[14] These groups are the only (simple) values for p such that PSL(2,p) acts non-trivially on p points, a fact dating back to variste Galois in the 1830s. In fact, the groups decompose as productsof sets (not as products of groups) as: A4 Z5; S4 Z7; and A5 Z11: These groups also are related to variousgeometries, which dates to Felix Klein in the 1870s; see icosahedral symmetry: related geometries for historicaldiscussion and (Kostant 1995) for more recent exposition. Associated geometries (tilings on Riemann surfaces) inwhich the action on p points can be seen are as follows: PSL(2,5) is the symmetries of the icosahedron (genus 0)with the compound of ve tetrahedra as a 5-element set, PSL(2,7) of the Klein quartic (genus 3) with an embedded(complementary) Fano plane as a 7-element set (order 2 biplane), and PSL(2,11) the buckminsterfullerene surface(genus 70) with embedded Paley biplane as an 11-element set (order 3 biplane).[15] Of these, the icosahedron datesto antiquity, the Klein quartic to Klein in the 1870s, and the buckyball surface to Pablo Martin and David Singermanin 2008.Algebro-geometrically, McKay also associates E6, E7, E8 respectively with: the 27 lines on a cubic surface, the 28bitangents of a plane quartic curve, and the 120 tritangent planes of a canonic sextic curve of genus 4.[16][17] Therst of these is well-known, while the second is connected as follows: projecting the cubic from any point not ona line yields a double cover of the plane, branched along a quartic curve, with the 27 lines mapping to 27 of the28 bitangents, and the 28th line is the image of the exceptional curve of the blowup. Note that the fundamentalrepresentations of E6, E7, E8 have dimensions 27, 56 (282), and 248 (120+128), while the number of roots is 27+45= 72, 56+70 = 126, and 112+128 = 240.

3.6 See also Elliptic surface

3.7 References[1] (Proctor 1993)

[2] (Proctor 1993, p. 940)

[3] Cameron P.J.; Goethals, J.M.; Seidel, J.J; Shult, E. E. Line graphs, root systems and elliptic geometry

[4] Godsil Chris; Gordon Royle. Algebraic Graph Theory, Chapter 12

[5] Arnold, Vladimir, 1997, Toronto Lectures, Lecture 2: Symplectization, Complexication and Mathematical Trinities, June1997 (last updated August, 1998). TeX, PostScript, PDF

[6] Polymathematics: is mathematics a single science or a set of arts? On the server since 10-Mar-99, Abstract, TeX, PostScript,PDF; see table on page 8

8 CHAPTER 3. ADE CLASSIFICATION

[7] Les trinits remarquables, Frdric Chapoton (French)[8] le Bruyn, Lieven (17 June 2008), Arnolds trinities[9] le Bruyn, Lieven (20 June 2008), Arnolds trinities version 2.0[10] Arithmetic groups and the ane E8 Dynkin diagram, by John F. Duncan, in Groups and symmetries: from Neolithic Scots

to John McKay[11] le Bruyn, Lieven (22 April 2009), the monster graph and McKays observation[12] Kostant, Bertram (1995), The Graph of the Truncated Icosahedron and the Last Letter of Galois (PDF), Notices Amer.

Math. Soc. 42 (4): 959968, see: The Embedding of PSl(2, 5) into PSl(2, 11) and Galois Letter to Chevalier.[13] le Bruyn, Lieven (12 June 2008), Galois last letter[14] (Kostant 1995, p. 964)[15] Martin, Pablo; Singerman, David (April 17, 2008), From Biplanes to the Klein quartic and the Buckyball (PDF)[16] Arnold 1997, p. 13[17] (McKay, John & Sebbar, Abdellah 2007, p. 11)

Bourbaki, Nicolas (1968), Chapters 46, Groupes et algebres de Lie, Paris: Hermann Arnold, Vladimir (1976), Problems in present day mathematics, in Felix E. Browder,Mathematical develop-

ments arising fromHilbert problems, Proceedings of symposia in puremathematics 28, AmericanMathematicalSociety, p. 46, Problem VIII. The A-D-E classications (V. Arnold).

Dickson, Leonard E. (1959), XIII: Groups of the Regular Solids; Quintic Equations, Algebraic Theories,New York: Dover Publications

Hazewinkel, Michiel; Hesseling; Siersma, JD.; Veldkamp, F. (1977), The ubiquity of Coxeter Dynkin dia-grams. (An introduction of the A-D-E problem)" (PDF), Nieuw Archief v. Wiskunde 35 (3): 257307

McKay, John (1980), Graphs, singularities and nite groups, Proc. Symp. Pure Math. (Amer. Math. Soc.)37: 183 and 265

McKay, John (1982), Representations and Coxeter Graphs, The Geometric Vein, Coxeter Festschrift, Berlin:Springer-Verlag, pp. 549

Kac, Victor G. (1990), Innite-Dimensional Lie Algebras (3rd edition ed.), Cambridge: Cambridge UniversityPress, ISBN 0-521-46693-8

McKay, John (January 1, 2001), A Rapid Introduction to ADE Theory Proctor, R. A. (December 1993), Two Amusing Dynkin Diagram Graph Classications, The American

Mathematical Monthly 100 (10): 937941, doi:10.2307/2324217, ISSN 0002-9890, JSTOR 2324217 McKay, J.; Sebbar, Abdellah (2007). Replicable Functions: An introduction. Frontiers in Number Theory,

Physics, and Geometry, II. Springer. pp. 373386. doi:10.1007/978-3-540-30308-4_10. Stekolshchik, R. (2008), Notes on Coxeter Transformations and the McKay Correspondence, Springer Mono-graphs in Mathematics, doi:10.1007/978-3-540-77398-3, ISBN 978-3-540-77398-6

van Hoboken, Joris (2002), Platonic solids, binary polyhedral groups, Kleinian singularities and Lie algebras oftype A,D,E (PDF), Masters Thesis, University of Amsterdam

3.8 External links John Baez, This Weeks Finds in Mathematical Physics: Week 62, Week 63, Week 64, Week 65, August 28,1995 through October 3, 1995, and Week 230, May 4, 2006

The McKay Correspondence, Tony Smith ADE classication, McKay correspondence, and string theory, LuboMotl, The Reference Frame,May 7, 2006

Chapter 4

Adjoint representation of a Lie algebra

In mathematics, the adjoint endomorphism or adjoint action is a homomorphism of Lie algebras that plays afundamental role in the development of the theory of Lie algebras.Given an element x of a Lie algebra g , one denes the adjoint action of x on g as the map

adx : g! g with adx(y) = [x; y]for all y in g .The concept generates the adjoint representation of a Lie group Ad. In fact, ad is the dierential of Ad at the identityelement of the group.

4.1 Adjoint representationLet g be a Lie algebra over a eld k. Then the linear mapping

ad : g! End(g)given by x adx is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Itsimage actually lies in Der (g) . See below.)Within End (g) , the Lie bracket is, by denition, given by the commutator of the two operators:

[adx; ady] = adx ady ady adxwhere denotes composition of linear maps.If g is nite-dimensional, then End (g) is isomorphic to gl(g) , the Lie algebra of the general linear group over thevector space g and if a basis for it is chosen, the composition corresponds to matrix multiplication.Using the above denition of the Lie bracket, the Jacobi identity

[x; [y; z]] + [y; [z; x]] + [z; [x; y]] = 0

takes the form

([adx; ady]) (z) =ad[x;y]

(z)

where x, y, and z are arbitrary elements of g .

9

10 CHAPTER 4. ADJOINT REPRESENTATION OF A LIE ALGEBRA

This last identity says that ad really is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets tobrackets.In a more module-theoretic language, the construction simply says that g is a module over itself.The kernel of ad is, by denition, the center of g . Next, we consider the image of ad. Recall that a derivation on aLie algebra is a linear map : g! g that obeys the Leibniz' law, that is,

([x; y]) = [(x); y] + [x; (y)]

for all x and y in the algebra.That adx is a derivation is a consequence of the Jacobi identity. This implies that the image of g under ad is asubalgebra of Der (g) , the space of all derivations of g .

4.2 Structure constantsThe explicit matrix elements of the adjoint representation are given by the structure constants of the algebra. That is,let {ei} be a set of basis vectors for the algebra, with

[ei; ej ] =Xk

cijkek:

Then the matrix elements for ad are given by

[adei ]kj= cijk :

Thus, for example, the adjoint representation of su(2) is the dening rep of so(3).

4.3 Relation to AdAd and ad are related through the exponential map: crudely, Ad = exp ad, where Ad is the adjoint representation fora Lie group.To be more precise, let G be a Lie group, and let : G Aut(G) be the mapping g g, with g: G G givenby the inner automorphism

g(h) = ghg1 :

It is an example of a Lie group map. Dene Adg to be the derivative of g at the origin:

Adg = (dg)e : TeG! TeGwhere d is the dierential and TeG is the tangent space at the origin e (e being the identity element of the group G).The Lie algebra of G is g = Te G. Since Adg Aut (g) , Ad: gAdg is a map from G to Aut(TG) which will havea derivative from TG to End(TG) (the Lie algebra of Aut(V) being End(V)).Then we have

ad = d(Ad)e : TeG! End(TeG):The use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector x inthe algebra g generates a vector eld X in the group G. Similarly, the adjoint map ady = [x,y] of vectors in g ishomomorphic to the Lie derivative LXY = [X,Y] of vector elds on the group G considered as a manifold.Further see the derivative of the exponential map.

4.4. REFERENCES 11

4.4 References Fulton, William; Harris, Joe (1991), Representation theory. A rst course, Graduate Texts in Mathematics,Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN978-0-387-97527-6

Chapter 5

Ane group

In mathematics, the ane group or general ane group of any ane space over a eld K is the group of allinvertible ane transformations from the space into itself.It is a Lie group if K is the real or complex eld or quaternions.

5.1 Relation to general linear group

5.1.1 Construction from general linear groupConcretely, given a vector space V, it has an underlying ane space A obtained by forgetting the origin, with Vacting by translations, and the ane group of A can be described concretely as the semidirect product of V by GL(V),the general linear group of V:

Aff(A) = V o GL(V )

The action of GL(V) on V is the natural one (linear transformations are automorphisms), so this denes a semidirectproduct.In terms of matrices, one writes:

Aff(n;K) = Kn o GL(n;K)

where here the natural action of GL(n,K) on Kn is matrix multiplication of a vector.

5.1.2 Stabilizer of a pointGiven the ane group of an ane space A, the stabilizer of a point p is isomorphic to the general linear group of thesame dimension (so the stabilizer of a point in A(2,R) is isomorphic to GL(2,R)); formally, it is the general lineargroup of the vector space (A; p) : recall that if one xes a point, an ane space becomes a vector space.All these subgroups are conjugate, where conjugation is given by translation from p to q (which is uniquely dened),however, no particular subgroup is a natural choice, since no point is special this corresponds to the multiple choicesof transverse subgroup, or splitting of the short exact sequence

1! V ! V o GL(V )! GL(V )! 1

In the case that the ane group was constructed by starting with a vector space, the subgroup that stabilizes the origin(of the vector space) is the original GL(V).

12

5.2. MATRIX REPRESENTATION 13

5.2 Matrix representationRepresenting the ane group as a semidirect product of V by GL(V), then by construction of the semidirect product,the elements are pairs (M, v), where v is a vector in V and M is a linear transform in GL(V), and multiplication isgiven by:

(M; v) (N;w) = (MN; v +Mw):

This can be represented as the (n + 1)(n + 1) block matrix:

M v0 1

whereM is an nnmatrix over K, v an n 1 column vector, 0 is a 1 n row of zeros, and 1 is the 1 1 identity blockmatrix.Formally, A(V) is naturally isomorphic to a subgroup of GL(V K) , with V embedded as the ane planef(v; 1)jv 2 V g , namely the stabilizer of this ane plane; the above matrix formulation is the (transpose of) therealization of this, with the (n n and 1 1) blocks corresponding to the direct sum decomposition V K .A similar representation is any (n + 1)(n + 1) matrix in which the entries in each column sum to 1.[1] The similarityP for passing from the above kind to this kind is the (n + 1)(n + 1) identity matrix with the bottom row replaced bya row of all ones.Each of these two classes of matrices is closed under matrix multiplication.

5.3 Planar ane groupAccording to Artzy,[2] The linear part of each anity [of the real ane plane] can be brought into one of thefollowing standard forms by a coordinate transformation followed by a dilation from the origin:

1. x 7! ax+ by; y 7! bx+ ay; a; b 6= 0; a2 + b2 = 1;2. x 7! x+ by; y 7! y; b 6= 0;3. x 7! ax; y 7! y/a; a 6= 0; where the coecients a, b, c, and d are real numbers.

Case (1) corresponds to similarity transformations which generate a subgroup of similarities. Euclidean geometrycorresponds to the subgroup of congruencies. It is characterized by Euclidean distance or angle, which are invariantunder the subgroup of rotations.Case (2) corresponds to shear mappings. An important application is absolute time and space where Galilean trans-formations relate frames of reference. They generate the Galilean group.Case (3) corresponds to squeeze mapping. These transformations generate a subgroup, of the planar ane group,called the Lorentz group of the plane. The geometry associated with this group is characterized by hyperbolic angle,which is a measure that is invariant under the subgroup of squeeze mappings.Using the above matrix representation of the ane group on the plane, the matrix M is a 2 2 real matrix. Accord-ingly, a non-singular M must have one of three forms that correspond to the trichotomy of Artzy.

5.4 Other ane groups

5.4.1 General caseGiven any subgroup G < GL(V ) of the general linear group, one can produce an ane group, sometimes denotedAff(G) analogously as Aff(G) := V oG .

14 CHAPTER 5. AFFINE GROUP

More generally and abstractly, given any group G and a representation of G on a vector space V, : G ! GL(V )one gets[3] an associated ane group V o G : one can say that the ane group obtained is a group extension by avector representation, and as above, one has the short exact sequence:

1! V ! V o G! G! 1:

5.4.2 Special ane groupMain article: Special ane group

The subset of all invertible ane transformations preserving a xed volume form, or in terms of the semi-directproduct, the set of all elements (M,v) with M of determinant 1, is a subgroup known as the special ane group.

5.4.3 Projective subgroupPresuming knowledge of projectivity and the projective group of projective geometry, the ane group can be easilyspecied. For example, Gnter Ewald wrote:.[4]

The setP of all projective collineations of Pn is a group which we may call the projective group of Pn.If we proceed from Pn to the ane space An by declaring a hyperplane to be a hyperplane at innity,we obtain the ane group A of An as the subgroup of P consisting of all elements of P that leave xed.A P

5.4.4 Poincar groupMain article: Poincar group

The Poincar group is the ane group of the Lorentz group O(1; 3) : R1;3 o O(1; 3)This example is very important in relativity.

5.5 See also Holomorph Related articles on a diagram

5.6 References[1] David G. Poole, The Stochastic Group'", American Mathematical Monthly, volume 102, number 9 (November, 1995),

pages 798801[2] Artzy p 94[3] Since GL(V ) < Aut(V ) . Note that this containment is in general proper, since by automorphisms one means group au-

tomorphisms, i.e., they preserve the group structure onV (the addition and origin), but not necessarily scalar multiplication,and these groups dier if working over R.

[4] Gnter Ewald (1971) Geometry: An Introduction, p. 241, Belmont: Wadsworth ISBN0-534-0034-7

Rafael Artzy (1965) Linear Geometry, Chapter 2-6 Subgroups of the Plane Ane Group over the Real Field,Addison-Wesley.

Roger Lyndon (1985)Groups and Geometry, Section VI.1, Cambridge University Press, ISBN 0-521-31694-4.

Chapter 6

Ahlfors niteness theorem

In the mathematical theory of Kleinian groups, the Ahlfors niteness theorem describes the quotient of the domainof discontinuity by a nitely generated Kleinian group. The theorem was proved by Lars Ahlfors (1964, 1965), apartfrom a gap that was lled by Greenberg (1967).The Ahlfors niteness theorem states that if is a nitely-generated Kleinian group with region of discontinuity ,then / has a nite number of components, each of which is a compact Riemann surface with a nite number ofpoints removed.

6.1 Bers area inequalityThe Bers area inequality is a quantitative renement of the Ahlfors niteness theorem proved by Lipman Bers(1967a). It states that if is a non-elementary nitely-generated Kleinian group with N generators and with regionof discontinuity , then

Area(/) 4(N 1)

with equality only for Schottky groups. (The area is given by the Poincar metric in each component.) Moreover, if1 is an invariant component then

Area(/) 2Area(1/)

with equality only for Fuchsian groups of the rst kind (so in particular there can be atmost two invariant components).

6.2 References Ahlfors, Lars V. (1964), Finitely generated Kleinian groups, American Journal of Mathematics 86: 413429,ISSN 0002-9327, JSTOR 2373173, MR 0167618

Ahlfors, Lars (1965), Correction to Finitely generated Kleinian groups"", American Journal of Mathematics87: 759, ISSN 0002-9327, JSTOR 2373073, MR 0180675

Bers, Lipman (1967a), Inequalities for nitely generated Kleinian groups, Journal d'Analyse Mathmatique18: 2341, doi:10.1007/BF02798032, ISSN 0021-7670, MR 0229817

Bers, Lipman (1967b), On Ahlfors niteness theorem, American Journal of Mathematics 89: 10781082,ISSN 0002-9327, JSTOR 2373419, MR 0222282

Greenberg, L. (1967), On a theorem of Ahlfors and conjugate subgroups of Kleinian groups, AmericanJournal of Mathematics 89: 5668, ISSN 0002-9327, JSTOR 2373096, MR 0209471

15

Chapter 7

An Exceptionally Simple Theory ofEverything

"AnExceptionally Simple Theory of Everything"[1] is a physics preprint proposing a basis for a unied eld theory,very often referred to as "E8 Theory",[2] which attempts to describe all known fundamental interactions in physicsand to stand as a possible theory of everything. The paper was posted to the physics arXiv by Antony Garrett Lisi onNovember 6, 2007, and was not submitted to a peer-reviewed scientic journal.[3] The title is a pun on the algebraused, the Lie algebra of the largest "simple", "exceptional" Lie group, E8. The papers goal is to describe how thecombined structure and dynamics of all gravitational and Standard Model particle elds, including fermions, are partof the E8 Lie algebra.[2] In the paper, Lisi states that all three generations of fermions do not directly embed in E8with correct quantum numbers and spins, but that they might be described via a triality transformation, noting thatthe theory is incomplete and that a correct description of the relationship between triality and generations, if it exists,awaits a better understanding.The theory received accolades from a few physicists amid a urry of media coverage, but also met with widespreadskepticism.[4] Scientic American reported in March 2008 that the theory was being largely but not entirely ignoredby the mainstream physics community, with a few physicists picking up the work to develop it further.[5]

In a follow-up paper, Lee Smolin proposed a spontaneous symmetry breaking mechanism for obtaining the classicalaction in Lisis model, and speculated on the path to its quantization.[6]

In July 2009, Jacques Distler and Skip Garibaldi published a critical paper inCommunications inMathematical Physicscalled There is no 'Theory of Everything' inside E8,[7] arguing that Lisis theory, and a large class of related models,cannot work. They oer a direct proof that it is impossible to embed all three generations of fermions in E8, or toobtain even the one-generation Standard Model without the presence of an antigeneration. In response to Distler andGaribaldis paper, Lisi argued, in a new paper An Explicit Embedding of Gravity and the Standard Model in E8,[8]peer reviewed and published in a conference proceedings, that some assumptions about fermion embeddings areunnecessary and that the antigeneration is not by itself a problem sucient to rule out the one-generation StandardModel.[8][9] In December 2010 and May 2011, Lisi wrote in the popular magazine Scientic American a featurearticle on the E8 Theory of Everything and an entry in the blog section of the magazine addressing some of thecriticism of his theory and how it has progressed, noting that the theory is still incomplete and makes only tenuouspredictions, with the three generation issue remaining as a signicant problem.[9]

7.1 OverviewLisis model is a variant and extension of a Grand Unication Theory (a GUT, describing electromagnetism, theweak interaction and the strong interaction) to include gravitation, a Higgs boson and fermions in an attempt to de-scribe all elds of the Standard Model and gravity as dierent parts of one eld over four-dimensional spacetime.More specically, Lisi combines the left-right symmetric PatiSalam GUT with a MacDowellMansouri descrip-tion of gravity, using the spin connection and gravitational frame combined with a Higgs boson, necessitating acosmological constant. The model is formulated as a gauge theory, using a modied BF action, with E8 as the Liegroup. Mathematically, this is an E8 principal bundle, with connection, over a four-dimensional base manifold. Lisisembedding of the Standard Model gauge group in E8 leads him to predict the existence of 22 new bosonic particles

16

7.1. OVERVIEW 17

at an undetermined mass scale.Lisi states that the fermions enter via an unconventional use of the BRST technique, as Grassmann number eldsvalued in part of the E8 Lie algebra. The bosons are combined with these fermions as one-form and Grassmannnumber parts of a kind of superconnection, each valued in separate parts of the E8 Lie algebra. The curvatureof this superconnection is calculated, producing the Riemann curvature, gauge eld curvature, gravitational torsion,covariant derivative of the Higgs, and the covariant Dirac derivative of the fermions. This curvature is used to buildthe modied BF action by hand, in an attempt to match the dynamics of the Standard Model and gravity.In the paper, Lisi describes several deciencies in this model. The most important deciency is noted as an incorrect,or poorly understood, inclusion of the second and third generations of fermions in E8, relying on triality. Thisdeciency, and the incomplete nature of the model, precludes the prediction of masses for new or existing particles.Also, Lisi notes the use of explicit symmetry breaking in building his action, rather than oering a more desirablespontaneous symmetry breaking mechanism. And, no attempt is made to provide a quantum description of thetheorythis being left for future work. About it, Lee Smolin proposed a spontaneous symmetry breaking mechanismfor obtaining the action in Lisis model, and speculates on the path to its quantization as a spin foam.[6]

7.1.1 Non-technical overview

The ber bundle of electromagnetism, for example, describes the electromagnetic eld as being made up of circlesattached to every point of spacetime.[10] Crucially, each circle can rotate a little relative to its spacetime neigh-bors. The so-called connection eld of a ber bundle describes how neighboring bers are related by these sym-metry rotations. The electric and magnetic force elds lling spacetime correspond to the curvature of this berbundlegeometrically, the electric and magnetic elds are how the circular bers twist over time and space. Anelectromagnetic wave is the undulation of circles over spacetime. One quantum of an electromagnetic waveaphotonis a propagating particle of light. Each kind of elementary particle corresponds to a dierent ber overspacetime. All the electrons of the world result from the twisting of a single kind of berexplaining, among otherthings, why all electrons are identical. The bers of electrically charged particles, such as electrons, wrap aroundthe circular bers of electromagnetism like threads around a screw. How fast a particles ber twists around thecircle is equal to its electric charge, determining how the particle responds to the force of electromagnetism. Be-cause twists must meet around the circle, these charges are integer multiples of some standard unit of electric charge.Of the elementary matter particles, called fermions, electrons have electric charge 1 (three twists), up quarks haveelectric charge +23 (two opposite twists), down quarks have electric charge 13 (one twist), and neutrinos have 0.The antimatter particles, such as positrons and antiquarks, have twists in the opposite direction around the electro-magnetic circle, giving them the opposite electric charges. When particles collide, they may be converted into newtypes, but the outgoing particles have exactly the same total charge as the incoming ones did. This crucial fact isa consequence of ber geometry: When any two particles meet, their twists add. In this way, the ber-bundle pic-ture explains what we know about electromagnetism. The electric charges describe the geometric structure of thecombined electromagnetic and matter ber bundle, determining what interactions are possible between electricallycharged particles.[10]

In Lisis model, the Lie group used is E8, a group with 248 parameters.[11]

In general, in each gauge theory based on a YangMills action, the symmetries of the Lie group are associatedto a specic kind of particles known as gauge bosons (like photons, W and Z bosons, and gluons in the StandardModel; in models involving supersymmetry this is a little more complicated). These gauge bosons can interact witheach other and with fermions according to the geometry of the group and its fundamental representations. One ofLisis challenges is that his theory identies fermions with the symmetries that are usually associated only with gaugebosons. Generally this is considered not possible and this aspect of the theory still needs to be completed. Anotheraspect dierent from the common approaches is that Lisis theory includes also gravity in the Lie Group E8. Whilethis aspect has been proven to be possible in supersymmetric theories and impossible in a large class of theories(Coleman-Mandula theorem), it is sometimes attempted in other theories and models that do not strictly belong tothose classes. It is still not clear if this feature is achievable in Lisis theory.[12]

In general, a unied theory has a Lie group large enough to contain the Standard Model symmetries. There are manysuch theories, some of which have used E8 for a long time (like string theory). In Lisis specic model, as introducedabove, each of the 248 symmetries of E8 corresponds to a dierent elementary particle: Standard Model gaugebosons, gravitons and Standard Model fermions, which can all interact (as usual in this kind of theories) according tothe geometry of the group, in this case E8. Lisi states that: E8 reproduc[es] all known elds and dynamics throughpure geometry.[1]

18 CHAPTER 7. AN EXCEPTIONALLY SIMPLE THEORY OF EVERYTHING

The complicated geometry of Lie groups, E8 amongst them, is described graphically using group representationtheory. Using this mathematical description, each symmetry of a groupand so each kind of elementary particlecan be associated with a point in a weight diagram. The coordinates of these points are the quantum numbersthechargesof elementary particles, which are conserved in interactions.In order to form a theory of everything, Lisis model must eventually predict the exact number of fundamental par-ticles, all of their properties, masses, forces between them, the nature of spacetime, and the cosmological constant.Much of this work is still in the conceptual stagein particular, quantization and predictions of particle masses havenot been done and the model at the moment cannot reproduce all the known particles.Lisi himself acknowledges it as a work-in-progress: The theory is very young, and still in development,[13] and thethree generation issue remains the most signicant problem, and until it is solved the theory is not complete andcannot be considered much more than a speculative proposal. Without fully describing how the three generations offermions work, the theory and all predictions from it remain tenuous.[9]

7.1.2 Algebraic breakdownLisi proposes a decomposition of e8, the 248 dimensional Lie algebra of E8, into parts accommodating the gravita-tional and standard model elds according to the following schema:[1][14][15][16]

These two generations are only formally identied as second and third generations, this being a problematic aspectof the theory,[9] as explained above.

7.1.3 PredictionsBy matching 226 known standard model particles to some of the 248 symmetries of E8, the theory is able to predictthe existence and quantum numbers of 22 new particles.[1] Three of these are the same new su(2)R and u(1)BLgauge bosons as predicted in the PatiSalammodel, theW' and Z' bosons. Another is a new u(1) gauge boson, with acorresponding new quantum number. And the remaining 18 new bosons predicted are new colored elds, interactingwith the strong force. Lisi states that some of these 22 particles might be seen at the Large Hadron Collider.[17]

Since Lisi does not specify masses for these particles their prediction is not falsiable by non-discovery in any givenexperiment, because the masses could exceed the experiments reach. However, the discovery of new particles thatdo not t in Lisis classication, such as superpartners, would fall outside the model, and falsify Lisis match to E8.Also, because the theory at the moment fails to predict all the known particles and the matching of the three fermiongenerations is tentative and problematic in the model, Lisi places a low condence in these predictions.

7.1.4 Q and AQ: How does time arise from symmetry breaking in E8?Lisi: Its not precisely clear yet, but there are four time-like directions in E8 that mix to give the direction of time inspacetime.

7.2 Chronology and reactionThree previous arXiv preprints by Lisi deal withmathematical physics related to the theory. CliordGeometrodynamics,[18]in 2002, endeavors to describe fermions geometrically as BRST ghosts. Cliord bundle formulation of BF gravitygeneralized to the standard model,[19] in 2005, describes the algebra of gravitational and Standard Model elds act-ing on a generation of fermions, but does not mention E8. Quantummechanics from a universal action reservoir,[20]in 2006, attempts to derive quantum mechanics using information theory.Before writing his 2007 paper, Lisi discussed his work on an Foundational Questions Institute (FQXi) forum,[21] atan FQXi conference,[22] and for an FQXi article.[23] Lisi gave his rst talk on E8 Theory at the Loops '07 conferencein Morelia, Mexico,[24] soon followed by a talk at the Perimeter Institute.[25] John Baez commented on Lisis work

7.2. CHRONOLOGY AND REACTION 19

in This Weeks Finds in Mathematical Physics (Week 253)",[26] and Lisi was interviewed on Sabine HossenfeldersBackreaction blog.[27] Lisis arXiv preprint, An Exceptionally Simple Theory of Everything, appeared on Novem-ber 6, 2007, and immediately attracted a great deal of attention. Lisi made a further presentation for the InternationalLoop Quantum Gravity Seminar on November 13, 2007,[28] and responded to press inquiries on an FQXi forum.[29]He presented his work at the TED Conference on February 28, 2008.[30]

Numerous news sites from all over the world reported on the new theory in 2007 and 2008, noting Lisis personalhistory and the controversy in the physics community. The rst mainstream and scientic press coverage began witharticles in The Daily Telegraph[13] and New Scientist,[31] with articles soon following in many other newspapers andmagazines.Lisis paper spawned a variety of reactions and debates across various physics blogs and online discussion groups.The rst to comment was Sabine Hossenfelder, summarizing the paper and noting the lack of a dynamical symmetrybreaking mechanism.[32] Lubo Motl oered a colorful critique, objecting to the addition of bosons and fermionsin Lisis superconnection, and to the violation of the Coleman-Mandula theorem.[33] In the presentation Whatsnew at the arXiv?" on May 20, 2008, Simeon Warner stated that Lisis paper is the most downloaded article on thearXiv.[34][35] Among the physicists early to comment on E8 Theory, Sabine Hossenfelder, Peter Woit and Lee Smolinwere generally supportive, while Lubo Motl and Jacques Distler were critical.On his blog, Musings, Jacques Distler oered one of the strongest criticisms of Lisis approach, claiming to demon-strate that, unlike in the StandardModel, Lisis model is nonchiral consisting of a generation and an anti-generation and to prove that any alternative embedding in E8 must be similarly nonchiral.[16][36][37] These arguments weredistilled in a paper written jointly with Skip Garibaldi, There is no 'Theory of Everything' inside E8,[7] publishedin Communications in Mathematical Physics. In this paper, Distler and Garibaldi oer a proof that it is impossibleto embed all three generations of fermions in E8, or to obtain even the one-generation Standard Model. In a pressrelease from his university, Rock climber takes on surfers theory,[38][39] Garibaldi states that his article with Distleris a rebuttal of Lisis theory. In response, Lisi argues that Distler and Garibaldi made unnecessary assumptions abouthow the embedding needs to happen.[9] Addressing the one generation case, in June 2010 Lisi posted a new paperon E8 Theory, An Explicit Embedding of Gravity and the Standard Model in E8,[8] peer reviewed and publishedin a conference proceedings, describing how the algebra of gravity and the Standard Model with one generation offermions embeds in the E8 Lie algebra explicitly using matrix representations. When this embedding is done, Lisiagrees that there is an antigeneration of fermions (also known as mirror fermions) remaining in E8; but while Dis-tler and Garibaldi state that these mirror fermions make the theory nonchiral, Lisi states that these mirror fermionsmight have high masses, making the theory chiral, or that they might be related to the other generations. Addressingthe three generation case, Lisi agrees that three generations of fermions cannot be directly embedded in E8, butsuggests that a gauge transformation related to triality might be used to relate the 64 mirror fermions and 64 otherE8 generators to two other generations of 64 fermions.[9]

The group blog, The n-Category Cafe, provides some of the more technical discussions, with posts by Lisi, UrsSchreiber,[14] Kea,[40] and Jacques Distler.[40]

Sixteen arXiv preprints have cited Lisis work. Lee Smolin's The Plebanski action extended to a unication ofgravity and YangMills theory, December 6, 2007, proposes a symmetry breaking mechanism to go from an E8symmetric action to Lisis action for the Standard Model and gravity.[6] Roberto Percaccis Mixing internal andspacetime transformations: some examples and counterexamples[12] addresses a general loophole in the Coleman-Mandula theorem also thought to work in E8 Theory.[9] Percacci and Fabrizio Nestis Chirality in unied theories ofgravity[41] conrms the embedding of the algebra of gravitational and Standard Model forces acting on a generationof fermions in so(3; 11) 64 , mentioning that Lisis ambitious attempt to unify all known elds into a singlerepresentation of E8 stumbled into chirality issues.[41] Mathematician Bertram Kostant discussed Lisis work in acolloquium presentation at UC Riverside.[42] In a joint paper with Lee Smolin and Simone Speziale,[43] published inJournal of Physics A, Lisi proposes a new action and symmetry breaking mechanism. In An Explicit Embedding ofGravity and the Standard Model in E8,[8] Lisi describes E8 Theory using explicit matrix representations.On August 4, 2008, FQXi awarded Lisi a grant for further development of E8 Theory.[44][45]

In September 2010, Scientic American reported on a conference inspired by Lisis work.[46]

In October 2010, Lisi, Smolin and Simone Speziale published a partially related paper on unication, in a peer-reviewed journal, proposing an action and symmetry breaking mechanism, and using an alternative treatment offermions.[43] In December 2010 Scientic American published a feature article on E8 Theory, A Geometric Theoryof Everything,[2] written by Lisi and James Owen Weatherall.In December 2011, in his paper, String and M-theory: answering the critics,[47] for a Special Issue of Foundations

20 CHAPTER 7. AN EXCEPTIONALLY SIMPLE THEORY OF EVERYTHING

of Physics: Forty Years Of String Theory: Reecting On the Foundations, Michael Du argues against Lisis theoryand the attention it has received in the popular press.[48] Du states that Lisis paper was incorrect, citing Distler andGaribaldis proof, and criticizes the press for giving too much positive attention to an outsider scientist and theory.

7.3 References[1] A. G. Lisi (2007). An Exceptionally Simple Theory of Everything. arXiv:0711.0770 [hep-th].

[2] A.G. Lisi; J. O.Weatherall (2010). AGeometric Theory of Everything. Scientic American 303 (6): 5461. doi:10.1038/scienticamerican1210-54. PMID 21141358.

[3] Greg Boustead (2008-11-17). Garrett Lisis Exceptional Approach to Everything. SEED Magazine.

[4] Amber Dance (2008-04-01). Outsider Science. Symmetry Magazine. Archived from the original on 5 July 2008. Re-trieved 2008-06-15.

[5] Collins, Graham P. (March 2008). Wipeout?". Scientic American: 3032. Retrieved 2008-06-18.

[6] Lee Smolin (2007). The Plebanski action extended to a unication of gravity and Yang-Mills theory. arXiv:0712.0977[hep-th].

[7] Jacques Distler; Skip Garibaldi (2009). There is no 'Theory of Everything' inside E8. arXiv:0905.2658 [math.RT].

[8] A. G. Lisi (2010). An Explicit Embedding of Gravity and the Standard Model in E8. arXiv:1006.4908 [gr-qc].

[9] A G Lisi (2011-05-11). Garrett Lisi Responds to Criticism of his Proposed Unied Theory of Physics. Scientic Amer-ican. Archived from the original on 2011-07-02. Retrieved 2011-07-30.

[10] A Geometric Theory of Everything (PDF).

[11] Mathematicians Map E8". AIM. Archived from the original on 2007-12-29. Retrieved 2007-12-30.

[12] Roberto Percacci (2008). Mixing internal and spacetime transformations: some examples and counterexamples. arXiv:0803.0303[hep-th].

[13] Roger Higheld (2007-11-14). Surfer dude stuns physicists with theory of everything. The Daily Telegraph. Archivedfrom the original on 2008-06-19. Retrieved 2008-06-15.

[14] Urs Schreiber (2008-05-10). E8 Quillen Superconnection. The n-Category Cafe. Archived from the original on 2008-06-19. Retrieved 2008-06-15.

[15] Jacques Distler (2007-12-09). A Little More Group Theory. Musings. Retrieved 2008-08-30.

[16] Jacques Distler (2007-11-21). A Little Group Theory. Musings. Archived from the original on 12 May 2008. Retrieved2008-06-15.

[17] The Big Bang: what will we nd?". The Daily Telegraph. 2008-03-25. Retrieved 2008-06-15.

[18] A. G. Lisi (2002). Cliord Geometrodynamics. arXiv:gr-qc/0212041.

[19] A. G. Lisi (2005). Cliord bundle formulation of BF gravity generalized to the standard model. arXiv:gr-qc/0511120.

[20] A. G. Lisi (2006). Quantum mechanics from a universal action reservoir. arXiv:physics/0605068.

[21] A. G. Lisi (2007-06-09). Pieces of E8. FQXi forum. Archived from the original on 2 June 2008. Retrieved 2008-06-15.

[22] A. G. Lisi (2007-07-21). Standard model and gravity. inaugural FQXi conference. Retrieved 2008-06-15.

[23] Scott Dodd (2007-10-26). Surng the Folds of Spacetime (PDF). FQXi article. Retrieved 2008-06-15.

[24] A. G. Lisi (2007-06-25). Deferential Geometry. Loops '07 conference. Retrieved 2008-06-15.

[25] A. G. Lisi (2007-10-04). An Exceptionally Simple Theory of Everything. Perimeter Institute talk. Retrieved 2008-06-15.

[26] John Baez (2007-06-27). This Weeks Finds in Mathematical Physics (Week 253)". Archived from the original on 30June 2008. Retrieved 2008-06-15.

[27] Sabine Hossenfelder (2007-08-06). Garrett Lisis Inspiration. Backreaction. Retrieved 2008-06-15.

7.4. EXTERNAL LINKS 21

[28] A. G. Lisi (2007-11-13). A Connection With Everything. International Loop Quantum Gravity Seminar. Archived fromthe original on 22 May 2008. Retrieved 2008-06-15.

[29] A. G. Lisi (2007-11-20). An Exceptionally Simple FAQ. FQXi forum. Archived from the original on 2 June 2008.Retrieved 2008-06-15.

[30] A. G. Lisi (2008-02-28). Garrett Lisi: A beautiful new theory of everything. TED talks. Archived from the original on18 October 2008. Retrieved 2008-10-17.

[31] ZeeyaMerali (2007-11-15). Is mathematical pattern the theory of everything?". New Scientist. Archived from the originalon 12 May 2008. Retrieved 2008-06-15.

[32] Sabine Hossenfelder (2007-11-06). A Theoretically Simple Exception of Everything. Backreaction. Archived from theoriginal on 26 May 2008. Retrieved 2008-06-15.

[33] Lubo Motl (2007-11-07). Garrett Lisi: An exceptionally simple theory of everything. The Reference Frame. Retrieved2008-06-15.

[34] Peter Woit (2008-05-28). INSPIRE. Not Even Wrong. Retrieved 2008-08-05.

[35] Simeon Warner (2008-05-20). Whats new at the arXiv?". HEP Information Resource Summit. Retrieved 2008-07-22.(The slide containing this statement was subsequently removed from the presentation le.)

[36] Jacques Distler (2007-12-09). A Little More Group Theory. Musings. Retrieved 2008-11-15.

[37] Jacques Distler (2008-09-14). My Dinner with Garrett. Musings. Archived from the original on 2008-11-19. Retrieved2008-11-15.

[38] Carol Clark (2010-03-18). Rock climber takes on surfers theory. esciencecommons. Retrieved 2011-07-30.

[39] No 'Simple Theory of Everything' Inside the Enigmatic E8, Researcher Says. ScienceDaily. 2010-03-26. Retrieved2011-07-30.

[40] http://golem.ph.utexas.edu/category/2008/05/e8_quillen_superconnection.html#c016877

[41] R. Percacci; F. Nesti (2009). Chirality in unied theories of gravity. arXiv:0909.4537 [hep-th].

[42] Bertram Kostant (2008-02-12). On Some Mathematics in Garrett Lisis 'E8 Theory of Everything'". UC Riverside math-ematics colloquium. Archived from the original on 28 June 2008. Retrieved 2008-06-15.

[43] A. G. Lisi; Lee Smolin; Simone Speziale (2010). Unication of gravity, gauge elds, and Higgs bosons. arXiv:1004.4866[gr-qc].

[44] E8 Theory. FQXi. 2008-08-04. Archived from the original on 2008-08-09. Retrieved 2008-08-05.

[45] FQXi Grants. FQXi. Archived from the original on 3 July 2008. Retrieved 2008-08-08.

[46] Merali, Zeeya (September 2010). Rummaging for a Final Theory. Scientic American. Retrieved 2010-08-25.

[47] M. J. Du (2011). String and M-theory: answering the critics. arXiv:1112.0788v1.

[48] Peter Woit (2011-12-07). String and M-theory: answering the critics. Not Even Wrong. Retrieved 2011-12-21.

7.4 External links Deferential Geometry - Lisis wiki, containing detailed mathematical background Animation of E8 - a New Scientist video describing the theory using a visual representation The Elementary Particle Explorer - an online E8 investigation tool for rotating and examining the particleassignments, charges, and interactions in the standard model and Lisis E8 Theory

A beautiful new theory of everything - Lisi presents his theory at TED

22 CHAPTER 7. AN EXCEPTIONALLY SIMPLE THEORY OF EVERYTHING

Levels of magnication:1. Macroscopic level - Matter2. Molecular level3. Atomic level - Protons, neutrons, and electrons4. Subatomic level - Electron5. Subatomic level - Quarks6. Lie group geometrical representation level

Chapter 8

Automorphic form

The Dedekind eta-function is an automorphic form in the complex plane.

In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological groupG to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup Gof the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean spaceto general topological groups.Modular forms are automorphic forms dened over the groups SL(2, R) or PSL(2, R) with the discrete subgroupbeing the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is anextension of the theory of modular forms. More generally, one can use the adelic approach is a way of dealing withthe whole family of congruence subgroups at once. From this point of view, an automorphic form over the groupG(AF) for an algebraic group G and an algebraic number eld F, is a complex-valued function on G(AF) that is leftinvariant under G(F) and satises certain smoothness and growth conditions.Poincar rst discovered automorphic forms as generalizations of trigonometric and elliptic functions. Through theLanglands conjectures automorphic forms play an important role in modern number theory.[1]

8.1 FormulationAn automorphic form is a function F on G (with values in some xed nite-dimensional vector space V, in thevector-valued case), subject to three kinds of conditions:

1. to transform under translation by elements 2 according to the given factor of automorphy j;2. to be an eigenfunction of certain Casimir operators on G; and3. to satisfy some conditions on growth at innity.

It is the rst of these that makes F automorphic, that is, satisfy an interesting functional equation relating F(g) withF(g) for 2 . In the vector-valued case the specication can involve a nite-dimensional group representation acting on the components to 'twist' them. The Casimir operator condition says that some Laplacians have F aseigenfunction; this ensures that F has excellent analytic properties, but whether it is actually a complex-analytic

23

24 CHAPTER 8. AUTOMORPHIC FORM

function depends on the particular case. The third condition is to handle the case where G/ is not compact but hascusps.The formulation requires the general notion of factor of automorphy j for , which is a type of 1-cocycle in thelanguage of group cohomology. The values of j may be complex numbers, or in fact complex square matrices,corresponding to the possibility of vector-valued automorphic forms. The cocycle condition imposed on the factorof automorphy is something that can be routinely checked, when j is derived from a Jacobian matrix, by means ofthe chain rule.

8.2 HistoryBefore this very general setting was proposed (around 1960), there had already been substantial developments ofautomorphic forms other than modular forms. The case of a Fuchsian group had already received attention before1900 (see below). The Hilbert modular forms (also called Hilbert-Blumenthal forms) were proposed not long afterthat, though a full theory was long in coming. The Siegel modular forms, for which G is a symplectic group, arosenaturally from consideringmoduli spaces and theta functions. The post-war interest in several complex variables madeit natural to pursue the idea of automorphic form in the cases where the forms are indeed complex-analytic. Muchwork was done, in particular by Ilya Piatetski-Shapiro, in the years around 1960, in creating such a theory. The theoryof the Selberg trace formula, as applied by others, showed the considerable depth of the theory. Robert Langlandsshowed how (in generality, many particular cases being known) the Riemann-Roch theorem could be applied to thecalculation of dimensions of automorphic forms; this is a kind of post hoc check on the validity of the notion. Healso produced the general theory of Eisenstein series, which corresponds to what in spectral theory terms would bethe 'continuous spectrum' for this problem, leaving the cusp form or discrete part to investigate. From the point ofview of number theory, the cusp forms had been recognised, since Srinivasa Ramanujan, as the heart of the matter.

8.3 Automorphic representationsThe subsequent notion of automorphic representation has proved of great technical value for dealing with G analgebraic group, treated as an adelic algebraic group. It does not completely include the automorphic form ideaintroduced above, in that the adele approach is a way of dealing with the whole family of congruence subgroups atonce. Inside an L2 space for a quotient of the adelic form of G, an automorphic representation is a representationthat is an innite tensor product of representations of p-adic groups, with specic enveloping algebra representationsfor the innite prime(s). One way to express the shift in emphasis is that the Hecke operators are here in eect puton the same level as the Casimir operators; which is natural from the point of view of functional analysis, though notso obviously for the number theory. It is this concept that is basic to the formulation of the Langlands philosophy.

8.4 Poincar on discovery and his work on automorphic functionsOne of Poincar's rst discoveries in mathematics, dating to the 1880s, was automorphic forms. He named themFuchsian functions, after the mathematician Lazarus Fuchs, because Fuchs was known for being a good teacher andhad researched on dierential equations and the theory of functions. Poincar actually developed the concept of thesefunctions as part of his doctoral thesis. Under Poincar's denition, an automorphic function is one which is analytic inits domain and is invariant under a discrete innite group of linear fractional transformations. Automorphic functionsthen generalize both trigonometric and elliptic functions.Poincar explains how he discovered Fuchsian functions:

For fteen days I strove to prove that there could not be any functions like those I have since called Fuchsianfunctions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two,tried a great number of combinations and reached no results. One evening, contrary to my custom, I drankblack coee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak,making a stable combination. By the next morning I had established the existence of a class of Fuchsianfunctions, those which come from the hypergeometric series; I had only to write out the results, which tookbut a few hours.

8.5. SEE ALSO 25

8.5 See also Automorphic factor Factor of automorphy

8.6 References[1] Friedberg, Solomon. _Solomon_Friedberg,_Boston_College.pdf Automorphic Forms: A Brief Introduction"] (PDF).

Retrieved 10 February 2014.

A.N. Parshin (2001), Automorphic Form, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

Henryk Iwaniec, Spectral Methods of Automorphic Forms, Second Edition, (2002) (Volume 53 in GraduateStudies in Mathematics), American Mathematical Society, Providence, RI ISBN 0-8218-3160-7

This article incorporates material from Jules Henri Poincar on PlanetMath, which is licensed under the Creative Com-mons Attribution/Share-Alike License.

Chapter 9

Glossary of semisimple groups

This is a glossary for the terminology applied in the mathematical theories of semisimple Lie groups. It also coversterms related to their Lie algebras, their representation theory, and various geometric, algebraic and combinatorialstructures that occur in connection with the development of what is a central theory of contemporary mathematics.Contents :

Top 09 A B C D E F G H I J K L M N O P Q R S T

26

9.1. A 27

U V W X Y Z

9.1 A Adjoint representation

The adjoint representation of any Lie group is its action on its Lie algebra, derived from the conjugation action of thegroup on itself.

Ane Lie algebra

An ane Lie algebra is a particular type of KacMoody algebra.

Algebraic group

9.2 B (B, N) pair Borel subgroup Borel-Bott-Weil theorem Bruhat decomposition

9.3 C Cartan decomposition Cartan matrix Cartan subalgebra Cartan subgroup Casimir invariant ClebschGordan coecients Compact Lie group Complex reection group coroot Coxeter group Coxeter number Cuspidal representation

28 CHAPTER 9. GLOSSARY OF SEMISIMPLE GROUPS

9.4 D Discrete series Dominant weight

The irreducible representations of a simply-connected compact Lie group are indexed by their highest weight. Thesedominant weights form the lattice points in an orthant in the weight lattice of the Lie group.

Dynkin diagram

9.5 E E6 (mathematics) E7 (mathematics) E7 (Lie algebra) E8 (mathematics) En (Lie algebra) Exceptional Lie algebra

9.6 F F4 (mathematics) Flag manifold Fundamental representation

For the irreducible representations of a simply-connected compact Lie group there exists a set of fundamental weights,indexed by the vertices of the Dynkin diagram of G, such that dominant weights are simply non-negative integer linearcombinations of the fundamental weights.The corresponding irreducible representations are the fundamental representations of the Lie group. In particular,from the expansion of a dominant weight in terms of the fundamental weights, one can take a corresponding tensorproduct of the fundamental representations and extract one copy of the irreducible representation corresponding tothat dominant weight.In the case of the special unitary group SU(n), the n 1 fundamental representations are the wedge products

Altk Cn

consisting of alternating tensors, for k=1,2,...,n-1.

Fundamental Weyl chamber

9.7 G G2 (mathematics) Generalized Cartan matrix Generalized KacMoody algebra Generalized Verma module

9.8. H 29

9.8 H

Harish-Chandra homomorphism

Highest weight

Highest weight module

9.9 I

Iwasawa decomposition

9.10 J

9.11 K

KacMoody algebra

Killing form

Kirillov character formula

9.12 L

Langlands decomposition

Langlands dual

Levi decomposition

Lie algebra

Lowest weight

9.13 M

Maximal compact subgroup

Maximal torus

9.14 N

Nilpotent cone

Elements in a semisimple Lie algebra that are represented in every linear representation by a nilpotent endomorphism.

30 CHAPTER 9. GLOSSARY OF SEMISIMPLE GROUPS

9.15 O

9.16 P Parabolic subgroup PeterWeyl theorem Positive root

9.17 Q

9.18 R Real form Reductive group Reection group Root datum Root system

9.19 S Schur polynomial

A Schur polynomial is a symmetric function, of a type occurring in the Weyl character formula applied to unitarygroups.

Semisimple Lie algebra Semisimple Lie group Simple Lie algebra Simple Lie group Simple root Simply laced group

A simple Lie group is simply laced when its Dynkin diagram is without multiple edges

Steinberg representation

9.20 T

9.21 U Unitarian trick

9.22. V 31

9.22 V Verma module

9.23 W Weight (representation theory) Weight module Weight space Weyl chamber

A Weyl chamber is one of the connected components of the complement in V, a real vector space on which a rootsystem is dened, when the hyperplanes orthogonal to the root vectors are removed.

Weyl character formula

The Weyl character formula gives in closed form the characters of the irreducible complex representations of thesimple Lie groups.

Weyl group

Chapter 10

Lie theory

Lie theory (/li/ LEE) is one of the areas of mathematics, developed initially by Sophus Lie and worked out byWilhelm Killing and lie Cartan. The foundation of Lie theory is the exponential map relating Lie algebras to Liegroups which is called the Lie groupLie algebra correspondence. The subject is part of dierential geometry sinceLie groups are dierentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structureof the Lie algebra is expressed by root systems and root data.Lie theory has been particularly useful in mathematical physics since it describes important physical groups such asthe Galilean group, the Lorentz group and the Poincar group.

10.1 Elementary Lie theoryThe one-parameter groups are the rst instance of Lie theory. The compact case arises through Eulers formula inthe complex plane. Other one-parameter groups occur in the split-complex number plane as the unit hyperbola

fexp(jt) = cosh(t) + j sinh(t) : t 2 Rg;

and in the dual number plane as the line fexp(t) = 1 + t : t 2 Rg: In these cases the Lie algebra parameters havenames: angle, hyperbolic angle, and slope. Using the appropriate angle, and a radial vector, any one of these planescan be given a polar decomposition. Any one of these decompositions, or Lie algebra renderings, may be necessaryfor rendering the Lie subalgebra of a 2 2 real matrix.There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identiedwith the 3-sphere. Its Lie algebra is the subspace of quaternion vectors. Since the commutator ij ji = 2k, the Liebracket in this algebra is twice the cross product of ordinary vector analysis.Another elementary 3-parameter example is given by the Heisenberg group and its Lie algebra. Standard treatmentsof Lie theory often begin with the Classical groups.

10.2 History and scopeEarly expressions of Lie theory are found in books composed by Sophus Lie with Friedrich Engel and Georg Scheersfrom 1888 to 1896.In Lies early work, the idea was to construct a theory of continuous groups, to complement the theory of discretegroups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincar. Theinitial application that Lie had in mind was to the theory of dierential equations. On the model of Galois theoryand polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, thewhole area of ordinary dierential equations.According to historian Thomas W. Hawkins, it was Elie Cartan that made Lie theory what it is:

32

10.3. ASPECTS OF LIE THEORY 33

While Lie had many fertile ideas, Cartan was primarily responsible for the extensions and applications ofhis theory that have made it a basic component of modern mathematics. It was he who, with some helpfrom Weyl, developed the seminal, essentially algebraic ideas of Killing into the theory of the structureand representation of semisimple Lie algebras that plays such a fundamental role in present-day Lietheory. And although Lie envisioned applications of his theory to geometry, it was Cartan who actuallyreated them, for example through his theories of symmetric and generalized spaces, including all theattendant apparatus (moving frames, exterior dierential forms, etc.)[1]

10.3 Aspects of Lie theoryLie theory is frequently built upon a study of the classical linear algebraic groups. Special branches include Weylgroups, Coxeter groups, and buildings. The classical subject has been extended to Groups of Lie type.In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress ofMathematicians in Paris.

10.4 See also List of Lie group topics

10.5 Notes and references[1] Thomas Hawkins (1996) Historia Mathematica 23(1):925

John A. Coleman (1989) The Greatest Mathematical Paper of All Time, The Mathematical Intelligencer11(3): 2938.

10.6 Further reading M.A. Akivis & B.A. Rosenfeld (1993) lie Cartan (18691951), translated from Russian original by V.V.Goldberg, chapter 2: Lie groups and Lie algebras, American Mathematical Society ISBN 0-8218-4587-X .

P. M. Cohn (1957) Lie Groups, Cambridge Tracts in Mathematical Physics. Nijenhuis, Albert (1959). Review: Lie groups, by P. M. Cohn. Bulletin of the American Mathematical

Society 65 (6): 338341. doi:10.1090/s0002-9904-1959-10358-x.

J. L. Coolidge (1940) A History of Geometrical Methods, pp 30417, Oxford University Press (Dover Publi-cations 2003).

Robert Gilmore (2008) Lie groups, physics, and geometry: an introduction for physicists, engineers and chemists,Cambridge University Press ISBN 9780521884006 .

F. Reese Harvey (1990) Spinors and calibrations, Academic Press, ISBN 0-12-329650-1 . Hawkins, Thomas (2000). Emergence of the Theory of Lie Groups: an essay in the history of mathematics,

18691926. Springer. ISBN 0-387-98963-3. Sattinger, David H.; Weaver, O. L. (1986). Lie groups and algebras with applications to physics, geometry, and

mechanics. Springer-Verlag. ISBN 3-540-96240-9.

Stillwell, John (2008). Naive Lie Theory. Springer. ISBN 0-387-98289-2. Heldermann Verlag Journal of Lie Theory

Chapter 11

List of Lie groups topics

This is a list of Lie group topics, by Wikipedia page.

11.1 ExamplesSee Table of Lie groups for a list

General linear group, special linear group SL2(R) SL2(C)

Unitary group, special unitary group SU(2) SU(3)

Orthogonal group, special orthogonal group Rotation group SO(3) SO(8) Generalized orthogonal group, generalized special orthogonal group

The special unitary group SU(1,1) is the unit sphere in the ring of coquaternions. It is the group ofhyperbolic motions of the Poincar disk model of the Hyperbolic plane.

Lorentz group Spinor group

Symplectic group Exceptional groups

G2 F4 E6 E7 E8

Ane group Euclidean group Poincar group Heisenberg group

34

11.2. LIE ALGEBRAS 35

11.2 Lie algebras Commutator Jacobi identity Universal enveloping algebra Campbell-Hausdor formula Casimir invariant Killing form KacMoody algebra Ane Lie algebra Loop algebra Graded Lie algebra

11.3 Foundational results One-parameter group, One-parameter subgroup Matrix exponential Innitesimal transformation Lies third theorem MaurerCartan form Cartans theorem Cartans criterion Local Lie group Formal group law Hilberts fth problem Hilbert-Smith conjecture Lie group decompositions Real form (Lie theory) Complex Lie group Complexication (Lie group)

11.4 Semisimple theory Simple Lie group Compact Lie group, Compact real form Semisimple Lie algebra Root system Simply laced group

36 CHAPTER 11. LIST OF LIE GROUPS TOPICS

ADE classication Maximal torus Weyl group Dynkin diagram Weyl character formula

11.5 Representation theorySee also: List of representation theory topics

Representation of a Lie group Representation of a Lie algebra Adjoint representation of a Lie group Adjoint representation of a Lie algebra Unitary representation Weight (representation theory) PeterWeyl theorem BorelWeil theorem Kirillov character formula Representation theory of SU(2) Representation theory of SL2(R)

11.6 Applications

11.6.1 Physical theories Pauli matrices Gell-Mann matrices Poisson bracket Noethers theorem Wigners classication Gauge theory Grand unication theory Supergroup Lie superalgebra Twistor theory Anyon Witt algebra Virasoro algebra

11.7. SPECIAL FUNCTIONS 37

11.6.2 Geometry Erlangen programme Homogeneous space

Principal homogeneous space Invariant theory Lie derivative Darboux derivative Lie groupoid Lie algebroid

11.6.3 Discrete groups Lattice (group) Lattice (discrete subgroup) Frieze group Wallpaper group Space group Crystallographic group Fuchsian group Modular group Congruence subgroup Kleinian group Discrete Heisenberg group CliordKlein form

11.6.4 Algebraic groups Borel subgroup Parabolic subgroup Arithmetic group

11.7 Special functions Dunkl operator

11.7.1 Automorphic forms Modular form Langlands program

38 CHAPTER 11. LIST OF LIE GROUPS TOPICS

11.8 People Sophus Lie (1842 1899) Wilhelm Killing (1847 1923) lie Cartan (1869 1951) Hermann Weyl (1885 1955) Harish-Chandra (1923 1983) Lajos Puknszky(1928 1996) Bertram Kostant

11.9. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 39

11.9 Text and image sources, contributors, and licenses11.9.1 Text

2E6 (mathematics) Source: https://en.wikipedia.org/wiki/2E6_(mathematics)?oldid=608540893 Contributors: Michael Hardy, Tom-ruen, Rjwilmsi and