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A GENERIC HALL ALGEBRA OF THEKRONECKER ALGEBRACsaba Szántó aa Faculty of Mathematics and Computer Science , “Babeş-Bolyai” University Cluj-Napoca , Chair of Algebra , Cluj-Napoca , RomaniaPublished online: 16 Aug 2006.

To cite this article: Csaba Szántó (2005) A GENERIC HALL ALGEBRA OF THE KRONECKERALGEBRA, Communications in Algebra, 33:8, 2519-2540, DOI: 10.1081/AGB-200065132

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Comlllllllications in Algebra®. 33: 25 19- 2540. 2005 Cop,Tigh t © Ta ,·lo r & Francis. Inc. ISS N: 0092-7872 print / I 532-4 125 o nline DOl : I 0.108 1/ AG B-200065 132

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A GENERIC HALL ALGEBRA OF THE KRONECKER ALGEBRA#

Csaba Szanto Faculty of Mathematics and Computer Science. "Babe~-Bolyai" Un iversity Cluj-Napoca, Chair of Algebra. Cft(j-Napoca. Romania

We constmct a generic Hall algebra of tlze Kronecker algebra ami pr01•e that its twisted ••ersion is a polynomial algebm in infinitely llltliiJ' variables m•er tlze twisted generic composition algebm. The variables are explicitly gh•eu as some ce11tral elements in the xeneric Hall algebra.

Thus, H'e obtain generic versions in t!te Kronecker case of theoreJIIS by Hua-Xiao and Se••enlumt- Van den BerJ:h.

Key Words: Compositio n a lgebra : Generic composition algebra: Generic Ha ll a lgebra; Hall a lgebra; Hall po lynomials: Kronecker a lgebra; Quantum a ffin e a lgebra.

2000 Mathematics Subject Classification: 16G20 ( l 7B3 7).

INTRODUCTION

Let A be a finite-dimen sional associative algebra with I over a finite field k and consider the catego ry mod-A of finite ly generated (hence finite) right modules over A. Denote by [M] the isomorphism class of a modu le M E mod-A.

The Ha ll a lgebra :lt(A) associated to the a lgebra A is the <Q-space having as basis the isomorphism classes in mod-A together with a multiplication defined by:

[NI][N~ ] = L F~:N~ [M] , fMJ

where the structure constant F~N~ is the number of submodules U of M such that U ~ N1 a nd M/ U ~ N1 • These structure con stants are a lso called Hal/numbers . One can see that ~e(A) is an associative <Q-algebra with identity [0] (see for example Ringe l, 1990a).

The composition algebra ce (A) of A is the unital subalgebra of :7t(A) generated by the isomorphism classes of the simple modules from mod-A.

Received March 15. 2004; Accepted May I 0, 2004 "Comm un icated by D. Happel. Address correspondence to Csaba Szan to. Faculty o r M athematics and Computer Science,

'' Babe~- Bo l ya i ·· University Cluj -Napoca , C ha ir o f Algebra , Str. Miha il Koga lniceanu r. I, R0-400084 C luj-Napoca. Roman ia; Fax: (40) 264-59 1906; E-mail : szanto@math.ubbcluj.ro

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2520 SZANTO

The Hall algebras of hereditary a lgebras made it possible to study quantum groups by using the machinery of representation theory of algebras. Ringel and Green have shown that a twisted generic version of the composition algebra of a hereditary a lgebra with generalized Cartan matrix 6. is in fact the positive part of the Drinfeld-Jimbo quantized enveloping a lgebra of the Kac-Moody a lgebra having type 6. (see for example Green. I 995 ; Ringel , I 990b, I 996).

A natural question related to Hall algebras is to measure the difference (if there is any) between the Hall a lgebra and its composition a lgebra . We know that in the case of a representation-fin ite a lgeb ra A, the Hall algebra lf(A) coincides with its composit ion subalgebra rf6 (A) . The contrary will happen if A is representation­infini te . More precisely Hua and Xiao (2002) showed that for A hereditary of the tame type, JC (A) is a polynomial algebra in infinitely many va riables over cli? (A) . This is in accordance with the res ul t of Sevenhant and Van den Bergh (2001) which proves that the Hall a lgebra associated to the path a lgebra of a quiver is the positive part of a quantized enveloping a lgebra of a generalized Kac-Mood y a lgebra. We should mention here that none of these results is generic. More precisely, on the one hand, the Hall and composition algebra and the variables used by Hua and Xiao depend on the base-field k (so they are non-generic); on the other hand, the parameter used by Seven hant a nd Van den Bergh is also non-generic.

The main purpose of this article is to present generic ve rsions for the above mentioned results in the Kronecker case.

Using some recent results of the author on the Hall numbers and the structure of the composit ion algebra for the Kronecker quiver (see Szanto, 2004), as well as some results on so-called regular modules (presented in section 2). in section 4 we construct a generic Hall a lgebra 7f(K. <Q (q)) , wh ich does not depend on the base fie ld k and contain as a unital suba lgebra the generic composition a lgebra rg (K. <Q(q)) (introduced by Ringel and Green). We also provide a basis of PEW­type for ':lt (K. <Q(q)) and consider the twisted versions lf* (K, <Q( v)) and 76*(K , <Q( v)) of the generic Hall and composition a lgebras.

In section 5, we will explicitly construct a system of central elements in ':!f* (K . <Q (v)), using the PBW-bases (and their Hall polynomials) for 16*(K , <Q (v)) obtained by the a uthor in Szanto (2004) and a lso the new PBW-basis of r€* (K, <Q( v)) constructed in section 3.

In the last section. we achieve our main goal by showing that ':!f* (K , <Q( v)) is a polynomial algebra over C*(K. <Q( v)) in infinitely many variables, which in fact the central elements constructed in the previous section.

1. THE CATEGORY OF FINITE RIGHT MODULES OVER THE KRONECKER ALGEBRA. THE HALL ALGEBRA OF THE KRONECKER ALGEBRA

Let K be the Kronecker-quiver, k a finite field with lkl = qk and kK the correspond ing path-algebra over k , ca lled Kronecker algebra. We will consider the category mod-kK of fi nitely generated (hence finite) right modules over kK , wh ich wi ll be identified with the category rep-k K of the finite dimensional k-representat ions of the Kronecker qui ver. For general notions concern ing the representation theory of quivers , we refer to A uslander et al. (1995) or Ringel ( 1984).

Up to isomorphism, we will have two simple objects in mode-kK corresponding to the two vert ices . We shall denote them by kS1 a nd kS2 . For a

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A GENERIC HALL A LGEBRA OF THE KRONECKER ALGEBRA 2521

m od ule M E mod-kK, [M] will denote the isomorphism class of M. The number of a utomo rphisms of M will be denoted by 'Y.M a nd the dimension vector of M by d im M = (mcks,)(M), mcks,1(M)), where m1, 5,1(M) is the number of compositio n factors of M isomorphic to kS;. For a module M let 1M:= M ® · · · ® M (Himes).

The indecomposables in mod-kK are divided into the three fa milies: the preprojectives, the regulars, and the preinjectives.

The preprojective (respectively preinjective) indecomposa ble modules are up to isomorphism uniquely determined by their dimensio n vecto rs. For n E IN we will denote by k P" (respect ively with k I,J the indecomposable preprojective module of dimension (n + 1, 11) (respectively the indeco mposable prei njecti ve module of dimensio n (11, 11 + 1)). So k P11 , k P1 are the projective indecomposable modules (kP0 = 1S1 begin simple) and k / 0 = kS2, k/ 1 the inject ive indecomposa ble modules (k /0 = kS2 being simple). A preprojective (respective ly a preinjective) mod ule, i.e., a module with a ll its indecomposable direct summand s preprojective (respectively preinjective). will be usua ll y denoted by P (respect ively by ! ). Note tha t the preprojective (preinject ive) modules form a full, extension-closed , additive su b­ca tegory of mod-kK .

Viewed as finit e dimensional k-representations of the Kronecker quiver, the regu lar indecomposables up to isomorphism are (see fo r exam ple in Zhang et a l. , 2001):

X id

k R'( (t) := k[X] j(X') != k[X] j (X'), k R~' (t) := k[ X] / ((X - !-l) ') != k[ X] / (( X - ;t)'), id X

where 1 :=:: I and 11 E k;

id

k Rj' (t) := k[ X] / ( rp1( X)') != k[ X] /( <p1(X) ') . X

where 1 :=:: 1.1 :=:: 2 a nd <p1(X) is a m o nic irreducible polynomial of degree lover k. Let N(qk, I) = + Ldi llt( ~ )qf, where l :=:: I , and 11 is the Mobius fu nction. It

is well-k nown that N(qk, l) is the number of monic. irreducible po lynomials of degree I over a field with qk elements. M oreover, for l fi xed N(q, I) is strictly monotono us increasing in q :=:: I , i.e .. N(q 1, l ) < N(q2 , I) for 1 _::: q1 < (/2. N(q, I):=:: I a nd N(q, I)= I if q =I= 2 (see for example Ruskey et a l. , 2001 for details).

Let M(q , I):= N(q, l) when l :::: 2 a nd M(q, I):= N(q, I)+ I = q + l. To somewhat simplify the nota tions, we sha ll arbitrari ly fix bijections

f 1 : {11 l it E k} U { o} ---7 {I, . .. , qk + I} and j~ : { <p1 I <p1 monic irreducible polynomia l of degree I over k} ---7 {I , . .. , N(qk.l)) (where l :=:: 2) and then let kR{'(o)( l) := kRT( t) , kR{'(1

'1(r) := k R~'( t) , k R{' <'~'' \t) := kR'j' (l) . So using the notatio ns a bove, o ur

regular indecomposables a re kR'J'( t) , where l :::: I , a= I , M(qk, 1), L :=:: I. Using the terminology of the Auslander-Reiten theory (see Ausla nder et a l. ,

1995 or Ringe l, 1984) we say that a seq uence of the form [kR;'( l )], .. . , [kR;' (t)], . .. is the ve rtex-seq uence of a homogeneo us tube kTf'. In thi s termino logy the regula r indecomposable kR;'( l ) is called quasi-simple and kR;'( I) is called of quasi-length 1

a nd quasi-socle kR;'( l ). A module with a ll its indecomposable direct summands in the tube kT/' wil l be deno ted by k R;' .

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2522 SZANTO

Simila rl y to the preprojective (p reinjecti ve) case, a regula r m odule, i.e. , a m od ule with a ll its indecom posable direc t summa nds regula r, will be usua lly denoted by R. Note tha t the regul a r modules a lso fo rm a full , exact, ex tensio n-closed a belia n subca tego ry of mod-k K , with simp le o bjects a ll the quasi-simples. We will usua ll y deno te by k R" a regul a r m odule o f the form k R;',' (t 1) ~ · · · ~ k R;',· (t ,), where (a

1, / 1) • • • • • (a., !J a re pa irw ise diffe rent (i.e .. a ll the indecomposable components

belo ng to pa irwise diffe rent tubes). a nd / 1 t 1 + · · · + l J s = n (i.e. , the dimensio n o f k R

11 is (n, n)).

W e refe r to A usla nder et a l. ( 199 5), Ringel ( 1984) (see a lso Szanto, 2004, Lemma 1.1.) fo r a ll the fac ts o n mo rphisms a nd extensio ns in mod-k K .

Nex t. we present some fac ts o n H a ll a lgebras. For M. N1 , •• • • N, E mod-k K we defin e

Let A be a commuta ti ve rin g with the unit eleme nt 11,. Then the H a ll a lgebra YC (k K , A) associated to the Kro necke r algebra kK a nd A is the free A-module having as basis the isom o rphi sm classes in mod-kK together wi th a multiplica ti o n defin ed by:

[NI ][ N2] = L F~~N, [M]. [M[

The structure consta nts Ff!,N, = I{M 2 U IU ~ N2 , M / U ~ N1 }I are call ed Hall numbers.

It is easy to see tha t the Ha ll -a lgebra is a we ll -defin ed , associative, usua ll y no ncommuta ti ve algebra with unit e lement the isom o rphi sm class of the ze ro m odule. W e have YC(k K , A)= YC (kK , ll ) ~~A and 'JC (k K , A)= @dE~ ' 'lC (kK , A)" is a /l 2 grading fo r YC (k K , A), where 7C (k K , A)" is the A-s ubm od ule o f 7C (k K , A) genera ted by [M] with dim M = d.

The uni ta l suba lgebra of YC(k K. A) gene ra ted by the two simple isomorphism classes [StJ, [S2 ] is ca lled t he compos iti o n a lgebra o f the Kro necker a lgebra and it is deno ted by 7i (kK, A).

Let us now assume tha t v is a n invertible elemen t in A suc h that u2 = q k I K ·

Then we can twi st the m ultiplica ti o n in 7C(kK , A) such that [N1] * [N2] := u(dim N, .d im N, )[N1J[ N2] . whe re (dim N1, dim N2) = dimkH o m (N1, N2 ) - dim kExt 1(N1, Nz)

is the Euler-fo rm . We obta in thus the twisted H a ll a lgebra 7C* (k K , A), which is a n associa tive A-a lgebra '?6*(kK , A) is the unital suba lgebra o f 7C* (kK, A) genera ted by [kS1] a nd [kS2].

H a ll a lgebras a nd their compositio n suba lgebras were used by C. M. Rin gel a nd J. A . G reen to co nnect the representa ti o n theo ry of fi nite dimensio na l a lgebras with the theo ry of qua ntum g ro ups (see G reen . 1995 ; Ringel. 1990b, 1996). In t his sense. it is impo rtant to o bta in fo rmulas fo r the Ha ll numbers.

Since we have tha t [N1][N2] = [N1 ~ N2 ] fo r N1, N2 E mod-kK with Ex t 1(N1, N2) =Ho m (N2 , N1)=0, it fo llows tha t [ P][R][l] =[P ~ R ~ l], so fo r [M] = [P ~ R ~ !] and [M ' ] = [P' ~ R' ~ ! ' ]

[M][M' ] = [P][R][!][P'][R' ][!'] .

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A GENERIC HALL ALGEBRA OF TH E KRONECKER ALGEBRA 2523

This shows that, in order to obtain a ll the Hall numbers, we should be able to describe the Hall products of the form

(!][ /'] , [ !][P] , [ /][ R], [ R][ R' ], [ R][ P] , [ P][ P' ].

Baumann and Kassel (200 I) obta ined some formulas for the cases [!][!'] , [P][P'], [!][R] , [R][P]. However they work over the category of coherent shea ves on the projective line (derived eq uiva lent with mod-kK ), where for example the case [!][P] doesn ' t exists. Szanto (2004) extended the formula li st in Baumann and Kassel (200 I) and also described the missing case [!][ P], using a representation­theo retical approach.

The formulas read as follows. For the cases [!][/'] and duall y [P][P' ] (see Baumann a nd Kassel, 2001 or

Szanto, 2004):

for i < j

I u + !l] k [ u k P;] [ v k P;] = v ( qk) [ ( u + v) PJ, where

1 ~ 1

[' P.][kP.] = i- J+I[kp . .o.k p .]+(qi- J+ I -qi- J- 1) ~[k P 10.k p . ] I .J qk I '<Y J k k ~ 1- /1 '<Y }+II

and dually

/t= l

fori > j

[ukl;] [vk !;] =I u: v ]<qk)[(u + v)*t;]

19 1

(1)

(2)

for i > j

(3)

(4)

(5)

[k I;][* l j] = cfi- i+l [k I;® k ! ) ] + (qri I - q{- i- 1) t [k 1;+, ® k l j_,J for i < j. (6) II = I

For the case [!][P] (see Szanto, 2004):

[k ][kP] k ,.., , _I[*P][k! ] ' ll - 1- i i = .)r_ , + qk i 11 - 1- i ' (7)

where

the summation going over all finite sets {[k R;li (t 1) ] , •. • , [k R;;· (t J]} of isomorphism classes of regular indecomposables satisfying the conditions: (a 1 , 11 ) , ••• , (a,, l ,) are

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2524 SZANTO

pairwise different (so they are taken from pairwise different tubes) and / 1 t 1 + · · · + l,t,. = 11.

For the cases [I][R] and dually [R][P]: on the one hand (see Baumann and Kassel , 2001 or Szanto. 2004)

and dually

on the other hand (see Szanto, 2004)

1- l

[kR;I( t)WPnJ = q~l [ k P,WR;' (t)] + [kpll+ll ] + L(q~(l- i) - q~{l - i- l)) [k P,+IiWR;I (t- i)] i=l

(I I)

and dually

I - I

[k ][kR"( )] 11 [k "( )][k ] [k ] "'\'( 1(1- iJ 1(1-i-l l )[k "( .) ][k ] 1, I t = qk Rl t 1, + 1,+11 + L.. qk - qk Rl t- I l ,+li . i=l

( 12)

The case [ R][ R'] can be described using resu lt s o n the classical Hall algebra . We will present some of these results in the next section.

2 . THE REGULAR MODULES IN THE HALL ALGEBRA

We know that for two regulars k R;1 , k R;': taken from different tubes (i.e. , with (a. I) i= (a', /') ) , we have Ext 1(kR;', kRn =HomeR;' . k R ;~' ) = 0. so we obtain that [k R;'W R;q = [k R/ W R;1] = [k R;' EB k R;n . This implies that in order to describe [ R][ R' ] it is eno ugh to describe [k R;1][k R;"] .

The isomorphism classes [k R;' ] (and [0]) form a A-basis of a unital A­subalgebra 'lt (k Ti' . A) of 'lt (kK , A) , ca lled the Hall algebra of the tube kTf'. The quasi-length gives a natural ~-grading of this a lgebra , so we have 'Jt (k Tf', A) = ffi ,a 'lt (k Tf1

, A) ,, with 'Jf (k Tf'. A) , having as A-basis the isomorphism classes of regulars with quasi-length 11.

Since ;;ee Tf' , A) coincides with the classical Hall algebra studied by Ph. Hall , we can apply all the results due to Hall , Macdonald and Zelevinsky (see MacDonald, 1995. II + Appendix for a ll the genera l notions and notations).

Let i . = u l , .. . , JcJ E lP a partit ion. We introduce the fo ll owing notations (where a ll the products are Hall products):

M(q,.l)

k {/ ( ) [ k R(l (I)] k II (0) [0] I k (I ( 1 ) k . (/ ( 1 ) k (I ( 1 ) k ( ) ) r1 t := t 1 : r1 = = ; r1 1. := r1 ~c 1 • • • r1 1.,. ; r1 , = L kr;'(i,); (I= I

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A G ENERI C HALL ALG EBRA OF THE KRO NEC KER ALG EBRA 2525

M(q.,l)

k P/' (t) := [k R;' (t)]; k p;' (O) =[OJ= I ; k p;'(i.) := A p;' (), I) .. .k p;'(}c.); kp,(J,) = L k p;' (i,). a= l

Observe that fo r /.,pE IP we have krf' (ic)krf'(p)=k rf'(i.U p) and k P/'(i. ) k P/'(p) = k P/'(i, U J..L) .

Theorem 2.1 ( Hall ; M acD ona ld , 1995) . a) If Jc (l ) < 1, • · • < 1, Jc (p (n)) are the e/emenls of lP (n) in increasing lexicographical order, we have

where (a !.(i).;IJ) (q)) is a stricllv lower unitriangular mmrix over ll [q] not depending on k, a and I.

b) We have thai

Y{ e T/', ll ) = 7l [k r;' ( I), . .. , k rf' ( n) , .. . ] and a /so

Y{(k T/', <Q ) = <Q[kp;'( l ), .... kp;'(n), . . . ].

In parlicular 7C(T/', ll ) is commulative.

c) kp/' U) = I:1,, l=k·l b;1,(qi) krf'(Jl). where b!1,(q) E ?l[q] is not depending on k, a and l. In particular k p1(},) = I:

1,,

1=

1; _

1 b;

1, ( q[)k 'AJL).

Corollary 2.2. a) The elemenls of rhe set {k rl'( t) I I :=::: I , a= I , M(q, , l), t :=::: I ) ~ YC (k K, ll ) are algebraically independent over ll.

b) The elements of the set {' p/' (1) 11 :=::: l ,a= l ,M(qk, /), 1 :=::: l) ~'JC(kK , <Q ) are algebraically independent over <Q.

Proof We will proof a), the proof of b) being simila r. Suppose some no n-tri via l ?l- li nea r combina tion of mo nomia ls in the give n elements is zero. Denote by ( / 1• a 1), •• • , (Ill, all) the pa irwise diffe rent " tube- indexing" pa irs which a re presen t in these mo nomials. This mea ns tha t a non-tri vial ?l-linear combina tio n of mo nomia ls o f the fo rm li (ic 1

, • •• ,i.ll) := kr;' 1 (J, 1)- • .kr;'"(i.ll) (where i.1, ... ,i.ll E IP) is zero. F o r

I " two such mono mials we defi ne the to tal o rdering .!ll (J. 1

, • •• , i.ll ) :=::: J1!l (J-t 1, ••• , pll)

if eithe r (li.1l, . .. , li.lll) > 1, (lp1l, ... -IPil l) (using the lex icogra phica l ordering) . I (I 'll l'lll)-(111 I Ill ) d • I _ I ., _ I '1+ 1 ~ 1+1 0 1 e se 1. , . .. , 1. - Jl , . . . , Jl a n A - p , .. . , 1. - JL , 1, 1 p ,

(p '+1, ).'+1) E L l;•+ll (o r equ iva lentl y 1-1'+1' < 1, },'+1' ) fo r a 1 E {0, . .. , n) (see M acDona ld. 1995, 1.9 fo r deta ils). T his usin g Theorem 2. 1. a) a nd (1.9) in M acDona ld ( 1995) we can see that o nl y the biggest mo nomia l .!ll (:J. 1

, • • • ,7.1l) present in ou t ?l-linea r com bina tio n will con ta in as term the isomo rphi sm class ku;' 1 (7. 1' ) •• . •u;'" (cxll' ). So, we have reached a contradictio n . 0

I "

Consider now the po lyno mia l suba lgebra kR1 := <Q[k rf' (l) I a= I , M(qk, l ), t :=::: I] :s 'JC (k K, <Q) , where I is identi fied with [0] , the unit element of 'JC (kK , <Q).

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2526 SZANTO

The symmet ric group SM(q,.t) acts on this a lgebra by permuting the indices a in the va riables. A polynomial in kR1 will be ca lled symmet ric if it is in va riant under this acti on. The symmetric polynomials fo rm a subalgebra kS1 = k R~M1 '"·' 1 of kR1•

Let now A= (i.1, • . . , i. 1

) be a lexicographica ll y decreasing fi nite sequence of nonzero pa rtitions. The number /(A) := t will be ca lled the length of A and IAI := 1Jc 11 + · · · + li,'l is the weight of !\. We consider the fo ll owing elements:

fo rt = /(A) ::S M(qk, /) let J.. n1(A) := 'I:, kr/(/)···k r

1M(q, .t) (y'H(q, .tl ),

r

where r = (/, . . . , ~,M(q,.l)) runs over a ll di stinct permutations of A = {ic 1

, • • • , i,1, (0) . ... , (0));

fo rt = /(A) > M(qk, I) let k n1(!\) := 0;

Using the previous resul ts, one can easily see that the elements kn1(!\) with /(A) :::=: M(q~.., I) fo rm a natural <Q-basis of kS1.

One can also observe that for A= (i,) (i.e., for /(A)= 1). kn1(A) = kr1(/.) .

For A = (1,1, ••• , ),') .let krJA) := kr1(i,

1) • • • kr1(i.1

) and kr1(0) := [0] =I. Just changing the va ri ables krf'( t) with the variables kp;'( t) . simila rl y, we can

consider the polynomial algebra kR; = <Q [k p;' ( t) I a = I , M(qk, /), t ~ I] :::=: '7/C (k K. <Q ),

the symmetric polynomial algebra ks; = kR/ ''11q' 11 and the elements k m1(A) and kPt(A).

Proposition 2.3. The elements k r1(A) (respectively k p1(!\)) with !(}.) :::=: M(qk, /)form a <Q-basis in kS, (respecti vely ks; ).

Proof When A= 0 (i.e., l (i.) = 0) then k nJ0) = k r1(0) = [OJ = I. When A= (}c) (i. e. /(A) = I) aga in kn1(A) = kr1(A) = k1AI.).

Consider now A = (i_~ , . . . , ;.1). where I :::=: t = /(A) :::=: M(qk, ! ). Then one can

eas ily see that we have the formula

kr1(A) = kc1.A/n1(A) + L kct. M kn1(f) , I S /( 1") < /(A)

(13)

where kc/..1 ,\ E IN* and fo r /(f) < /(A) we have kcu r E IN. Moreover. notice that kcu r = 0 whenever lfl # IAI .

Converse ly. we obtain tha t

kn1(A) = kd1 A/ r1(A) + L kd1_,\/ r,(f). ( 14) I S i(f/ < 1(1\ )

where kd/..1 .\ E <Q* and for /(f) < /(A) we have kc/1. \I" E <Q. Here a lso kdur = 0 whenever lfl # IAI .

Now the assertion easily fo llows. 0

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A GENERIC f-IALL ALGEBRA OF Tf-I E KRONECKER ALGEBRA 2527

We can see from formulas (13) and (14), that if, k, k' are finite fields and

l, I'~ I such that 1 ::::= /(f) < /(A)::::= M(qk, /), M(qk' • /' ),then

k I k' d . d G i.i\i\ = I' . AA =. i\i\ '

Consider now the case /(A) > M(qk, !). Then, since M(q, l) is m onotonous increasing in q, there is a big enough field k' such that /(A) ::::= M(qk' . l). But this means k'n,(A) = d 11 1 k'r,(A) + Ll.:<:l(ll < I(A) d 11 1k 'r1(f), so identify ing the variab les k'rf'(t) with

the variables kr;'(t) for a= 1, M(qk , /)and with 0 for a= M(qk, l) + I , M(qk'• I) we

get. that 0 = d!li\ kr1(A) + LI "' I( I) < I(A J d 1rkr1(f). We can conclude, that for every A, k and l

k n 1(A) = d!li\ kt"t(A) + I: d,11-' r1(f) , I <: l(r) < l( i\)

(15)

where d M E <Q* and d i\1 E <Q are independent from k and /, moreover d 11 r = 0

whenever fr! -I- [A[. In the same way we obtain that for every A, k and l

kmAA) = e!li\ k p1(A) + I: e,1/ p 1(f), 1<: /( Jl <i(i\)

( 16)

where eM E <Q* and e11 r E <Q are independent from k and /, moreover eAr= 0,

whenever fr! -I- [A[ . For a nonzero partition }_= (A. 1 , ••• , ..1.,) denote by),:= ((/. 1), • •• , (i.,)) and let

(0) = 0. Using the notions above and formula (8) we can observe, that

lee q~ ( I ) '<i.'J ( I )'(i2J ( I )'ii"l .'51.

11 = L -- I - - I - ? . . . I - -

qk - I qk q-;: q~

k ("l)k (12) k (" II ) X m 1 A 111 2 / , · · · 11111

l c , ( 17)

where the summation goes over i. 1, },2 , ... , },11

E IP with the property [Jc 1[ + 2[},2 [ + · · · + nfi,11

[ = n.

3. BASES OF PBW TYPE IN THE COMPOSITION ALGEBRA

Szanto (2004) proved that the elements

[k p ] ... [k p ] k 91L [k I ] ... [k I ] 1 1 11· /. it )J '

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2528 SZANTO

with i - (i i )E IP kfi. = k?A - ... kJ/1. - . t , s.i1 . . .. , i, .. J·1, .... J·,E IN. i1 _< . - ·I · ... , ·d , ' · ,_, ' ·d .

. . . ::: i" j 1 ::: • • • ::: j ,, form a PBW-basis in re (kK. <Q) with structure constants given by formulas ( I )-(8) and

n- 1

k -' [kp ] _ " [kP ] ' , h ( 211 - 1 2"-2 )[' p ] "'( , +; _ 11+i- 2)[k p ]k?/1 .51, 111 - Cfk 111 •7' , + qk + qk 111+11 + L Cfk qk m+ l · 11 - i

i= l

( 18)

and dually

Il - l

[k/ ]krJfl _ 11 k']"fl [' J ] + ( 211 - l + 211 - 2)[kf ] + "'( 11 + i _ ll + i-2 ) k j/ [k/ ] m ,J t , - q k ·JL, 111 CfJ..: CfJ..: m + n L CfJ..: Cfk · n - i m+ l ·

i=l

It is also proven in Szanto (2004) that the elements

with i. E IP. il ::: .. . ::: j ,.

[k ] [k ] k r [k ] [k / ] P; * · · · * P; * .'/i; * 11- * ·-- * 1- , l I . I I

where vk = .j(j;. k 1b • _ II ~-=-~-_k cA .-'n, . - II Vk 7 .Y/ 11 q'('- I

j n- 1 k,1 ._ kr;q _ -( ' _I)"'( _ -) k , b k ,l/ ·~ II .- .J/ II Lfk -'--' /1 ( .J I i . . 11-i

17 i= l

( 19)

and

form a PBW -basis in r(i' * (k K , @2 [ vk]) . The structure constants are given by the twisted versions of formulas ( I )-(8)

k r· [k ] [' ] k r: [k J .J:J , * P,, = P111 * .J:J, + P,+111 (20)

and dually

[k ] k r k r [k ] [ ' ] /Ill * .J}, , = .J:J, * /,, + ' "+"' . (2 1)

Besides the bases above. we will need one more. We define inductively

I I n- 1 . k'::t (f) ·= (- l )"+l_k?fl + _ "'(-l) '+lk'?ek r-1 -(f)

u ll • n/ L- 1f 11 - 1

17 n i= l (22)

Proposition 3 .1. We have in '(l* (kK. <Q[vk]):

(23)

and dually

(24)

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A GENERIC HALL ALGEBR A OF TI-lE KRONECKER ALGEBRA 2529

Proof Induction on n using formulas (20) , (2 1 ), (22) repeatedly. D

Using a ll the previous res ults we can conclude that

Corollary 3.2. Th e elements

[k p ] * .. . * [k p ] * k r.!!J . ( 1 ) * [k I ] * ... * [k I ] I] '., / , Jf }] '

. I . ( ' . ) IP k ~ ( I) kr ( I) kr ( I) · . . . . lN H'/{ 11. = 1. 1 , •• • ,/cd E , //;_ = .:V;_1

• • • J J;., , I , S, t l , . .. , t 5 , j l . . .. , j 1 E ,

i 1 S · · · S i,. j 1 S · · · S j"form a PBW-basis in ,.W (kK. <Q[vk]).

4. A GENERIC AND TWISTED GENERIC HALL ALGEBRA. HALL POLYNOMIALS

Let '3f. be a set of finite field s and assume that the order of these fields is not bounded. Consider the product

n = n 7/e (k K, <Q) . kE .J!.

which ca n be viewed as <Q[q] -a lgebra, where the va ri a ble q is identified with (qk . 1 fl (k K ()) ) ) in n. The fact that q is indeed a va riab le follows easil y using the conditio n on X .

We define the fo llowing elements in n P" := ([k P

11]), 1

11 := ([k 1

11]) . n1(i\ ) = (kn1(i\)) , m1(1\ ) = (km1(i\ )) . ?It"= (19i,J

r1(i.) := ('r1(i. )), r1(i\) := (k r 1(i\)) = rli,1) · · · t"t( Jc'). p1(J,) := ( k p 1(i.)),

p 1(A) := (1p 1(A)) = p 1(}.

1) · · · p 1(A.').

where n E IN , Jc E IP and A= (i,1, • ••• i,' ) is a lexicogra phically decreasing sequence

of nonzero pa rtition s of length /(A) = t .

For i\ = 0 (i.e. , /(A)= 0) we have r 1(0) = p 1(0) = m 1(0) = n 1(0) =((OJ)= I, where I is the unit element in n. Remark 4.1. We have r1((0)) = p 1((0)) = M(q, ! ) · I.

Conside r now the unit a l <Q[ q] -subalgebra of n generated by S1• S2 and the elements r1(i.), with l :::: 1, i. E IP. We will ca ll this a lgebra generic Hall algebra of the Kronecker a lgebra and denote it by 'Jf (K , <Q[q]) . F o ll owing Ringel and Green , the unital subalgebra of n generated by the elements S1• S2 is ca lled f:eneric compos iLion algebra of the Kronecker a lgebra and denoted by ,.IZ(K, <Q[ q ]). Taking scala r extensions let 'JC (K , <Q (q)) = 'lf (K , <Q[q]) Q9 <Q(q) and ,.t (K , <Q (q)) = re(K, <Q[q]) Q9 <Q(q) . Of course ,.IZ(K , <Q (q)) s "lf (K, <Q (q)) .

Using Co roll ary 2.2, one can easily see that the elements r1(1.), with I :::: I fixed a nd i.E IP are <Q (q)-linea rly independent , so we can define that <Q(q)-spaces V1(n) s 'Jf (K, <Q(q)) having as basis the elements r1(i.), with Jc E IP (n) and V1 =

L /1 v,(n) = EB/1 v,(n) E 'Jf (K , <Q (q)) . For 11 E lN denote by <Q(q)[xl, X2, . . . ], the <Q (q)-subspace of <Q(q)[x1, x2, . .. J having as basis the elements X;. = X;

1 • • • xi., • be denoted

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2530 SZANTO

by Pw Using the graduations above, take now the graded <Q(q)- linear isomorphism .f~ : <Q (q)[x 1 , x2 , .. . ]......,. V1 given by f(x;) = r1(}. ) , so in particular f 1(x<O> = I)= r1((0)) = M(q, 1) · I. Let .1; 11 be the projection of f 1 on V1(n), so we have f 1p 11 = ftx

lemma 4.2.

a) For i,= (/ 1, ... , i.,) E IP* we have

1!1 r,(l)P111 = L Pm+it.f; li l-i((q; '1x;, + X;

1_ 1) · · · (q!'

1X ;, + X; ,- 1)) (25)

i=O

and dualfr

Ii i

1111 r1().) = Lf~. l i l -i((q;' 1 x;, + X;1

_ 1) · · · (q '•' x;, + X;_,_ 1)) 1111 +it· (26) i=O

b) Fori\ a fexicographica ffy decreasing finite sequence of nonzero partitions we have

m 1(i\) = eA ,\ p,(i\) + L e.\rp,(f). (27) 1::; / (1/<i(A)

where eAA E <Q* and eM E <Q, moreover eAr= 0 whenever lfl-=/= IAI. c) For). E IP we have

p1(A) = L b;Jl(q 1)r1(Jl). (28) It• I= Iii

d) We have

(29)

where

II ( I )/(i 1

) ( I ) 'V-11 ( I ) l(i.") - - -

711." = L __!J___l I -- I --:;- . . . I -- m 1 (i.1)m2().

2) · · · m

11(i. 11

),

q- q q- q" (30)

the summation going over /,1, Jc 2

, .. . , i," E lP with the propertY li, 11 + 21/21 + · · · + nli."l = n.

e) We have

19 1 pp = qi-J+ 1 p p + (qi-J+ I - qi-J- 1) """'P P fior i > ;·, i- ;· odd

I ) ) I ~ J+lf t~lf

u= l

(3 1)

l !f l- 1 pipj = qi- j+1 p j pi + (qi- j - qi-j-1)P~ + (qi - J+I - qi- j-1 ) t pj+upi- u

u= l

fori > j. i- j even (32)

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A GENERIC HALL ALGEBRA OF THE KRONECKER ALG EBRA 253 1

and dually

I s-' I II - j~i+ I I I + ( j~ i+ I j~ i~ I) '"""' I I

i j - q j i q - q L.., j ~ ll i+ll for i < j , j - i odd II = I

(33)

liy ] ~ l

IJ ; = qj-i+IIJ; + (q j-i - qj-i~ l)/~ + (qj-i+l - qJ~i~ l ) L I;~Ji+ll u= l

fo ri < j , j - i even. (34)

Proof a) We consider the <Q-spaces kV," (n) (respectively kV1(n)) having a n <Q­basis the elements k r;' ()_), wit h X E IP(n) (respectively k r1(J,), with X E IP(n) ), and let kV," = L 11 kV,"(n) = EB" kV," (n) ( respect ively kV1 = L 11 kV1(n) = ffi" kV1(n)) . For n E IN denote by <Q[x 1, x 2 , . .. ], the <Q-subspace of <Q[x 1, x 2 , .. . ] having as basis the elements X; = x;, · · · x;,, wi th }, = (}. 1, •.• , i,J E IP (n) . The canonica l p rojection of <Q[x 1 , x1 • . . . ] on <Q[x 1 , x1 , .. . ], will be denoted by nw Using the gradua tions a bove we also define the graded <Q-Iinear isomo rphism kf 1": <Q[x 1,x1 .... ] ~k V1", k.ff'(x;_) = k rf'(i,) (respectively kf1 : <Q[x 1 , x2 , ... ] ~ k V1, kf 1(x;_) = k r1(1.)) and its

Pro iection kfa and k V" (n) (respecti vely kj· o n k V (n)) so we have ' f""n = kj ·" J 1.11 I 1.11 I ' . I II 1.11

(respect ively kf1n" = k.f1 ,).

0 "] t] kf "( . . ) _ kf " ( )kj""( ) k/"( ) _ '\' M(qk .i) kfa( ) ne can eas1 y see 1a t . 1 .x 1_.1 1, - 1 X ;. 1 x1, , . 1 X; - L..a=l 1 X;_ a nd .f1(qx;_) = q.f1(x;_) = (q/f~ (x;_)) = (kj~ (qkx;_) ) , so to show (25) it is enough to show for X E IP* that

l/.1

k r!' V.W P,,] = L[k Pm+il ]k .f(' lil~ ;( (q:· 'l.r ;, +X;, ~ I) · ·· (q/x;, + X;, ~ l )). (3 5) i= O

We proceed by induction on I(J,) = t. For t = I formu la (35) is just fo rmula (9). Suppose (35) is true for t and prove it for t + I. Let Jl = (11 1 , ••• , JLr+ 1) of length t + I. Then using the induction hypothesis a nd fo rmula (9) we get

i=O

b) It fo llows fro m (16).

c) It fo llows fro m Theorem 2. 1.c.

d) It fo llows from (7) and ( 17).

e) It fo llows from (3) and (6). 0

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2532 SZANTO

Theorem 4.3. The elements

P · · · P r1 (i\ 1) · · · r1 (i\ d)) ! .. . f

IJ 1, I d Jt }] '

with 11 < · · · < ld E N *. i\ *, . .. , i\d lexicographically decreasing .finit e sequences of nonzero partitions, s, d, I , i 1, •. . , i, j 1, ••• , j , E IN, i 1 :S · · · :S i,. j 1 :S · · · :S j,. form a PBW-basis in 'JC (K, <Q (q)).

Th e structure constants are given by formulas (25)- (34 ).

Proof Deno te by 2ll the set of the given elements. It is easy to see tha t 2ll ~ 'JC (K , <Q (q)). Indeed . we have that .31- 1 = 525 1 - 5 151 E rrz (K , <Q (q)). On the o ther hand, using the generic ve rsion of form ula ( 18), we get tha t P,+1 = q~ l (9'1. 1 P, -qP, 1t 1) , so it follows inductively that P, E ?6(K , <Q (q)) ~ 'JC (K . <Q (q)). In the same way. we have that/, E 'fZ (K , <Q (q)) ~ lC (K , <Q (q)) and tri via ll y r1(i\) E 7e(K , <Q (q)) .

A pplyi ng formulas (25)-(34) fro m Lemma 4.2. it fo llows that 93 is <Q (q) ­linea rl y generating 'JC (K , <Q (q)) .

It remains to show that the elements of .03 are linearly independent over <Q (q). Since we have [P][R][I] = [P EB R EB /] , one can easil y see that it is enough to prove the linea r independency over <Q (q) of the sets { P;

1 • • • P;, I s. i 1, . . . , i , E N , i 1 ::=:

··· :S i , ) , {fj, ·· ·Iitl t , j 1, •.• , j, E lN,j1 :S ···:S.i,} and h 1(i\ 1) ···r1"(i\d))ldEJN, 11 < · · · < ld EN*, A 1, ••• , i\ ci lex icogra phicall y decreasing finite sequences of no nzero part itions}. For linear independency of the first two sets just use formulas ( I )- (6). T o prove the linea r inde pendency of the las t set, due to Coro llary 2.2 a), it is eno ugh to show, that the elements r1(i\) , where I is fixed and i. E IP, a re a lgebra ica ll y independent ove r <Q[q] , or equi va lently the elements r1(i\) are <Q[q]­linea rl y independent for I fixed (where i\ denotes a lex icographicall y decreasing finit e sequence of partitions) .

Suppose L AE J f\(q) r1(i\) = 0, where .".f is some finite set of lexicographica lly dec reas ing finite (possibly empty) sequences of partitions. Let k0 E ."J{ such that l(i\) ::=: M(qko, I) for all A E . ..f. Such a field exists due to the condition on :f{ and the fact that M(q , I) is strict ly m o notono us increasing in q :=::: I. M o reover, there a re infinitely ma ny k E :f{ such that qk :=::: qko and in thi s way I :S M(qko• I ) :S M(qk, 1). It follows then (using Proposit io n 2.3. ) that for all these k E 'J{ a nd i\ E J' we have f\ (q,) = 0, so f 11 = 0 for a ll i\ E .'!' . 0

Remark 4.4. One ca n easily see tha t a lso the elements

p · · · P n1 (A 1) • .. n1 (i\d)l · .. J

I J 11 J tf ) 1 )] '

with 11 < · · · < ld EN*. i\ 1, ••.• i\ rl lex icographicall y decreas ing finite sequences of

no nzero pa rtitions, s. d , I, i 1, ... , i ,, j 1 • ••• ,j, E JN , i 1 ::=: · · · ::=: i " j 1 ::=: · · · ::=: j l' form a PBW-basis in Je(K, <Q (q)).

Szanto (2004) proved. that the elements

P ···P fi l·· · f I ] I, I . ) 1 }J '

with i . = (/. 1 , •• . , ).d ) E IP, ?It;_ = '!A ;. 1 · · · ?It ;.", t , s, i 1 , ..• , i" j 1 , •.• , j , E N , i 1 ::=: · · · ::=: i" j 1 :S · · · :S j l' form a PBW-bas is in rfi (K , P<Q (q)). the structure constants being given

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A GENERIC H ALL ALGEBRA OF THE KRO NECKER ALGEBRA 2533

by formulas (29), (3 1 )- (34)

n- 1

"" p _ "P '!D + ( 211- 1 + 211 - 2)? + """'( 11 + i 11 +i-2)P ;n JL/1 111 - q m.JL" Cf Cf m+ 11 L q - q m+ I-AII - i (36) i= l

a nd dua ll y

n- 1 I ;D - " "" I ( 211- 1 211 - 2 ) 1 """'( ll + i ll +i-2 ) "" I m ·.JL" - Cf ~JL" 111 + q + Cf m+ 11 + L Cf - Cf ~n11- i m+ l · (37)

i = l

We can see in fo rmulas (25)-(34), (36). (37) that a ll the structure constants (re latively to the p rese nted bases) are polynomia ls in <Q[q] . We will ca ll these polynom ia ls Hall polynomials. We can see tha t we can define directly 'Jt (K , <Q (q)) (a nd a lso '~ ( K, <Q(q)) , see Szanto, 2004) via these H a ll po lynomials. Indeed , we can take the <Q (q)-space having as basis the "sym bols"

P ... p r (i\ 1) .. ·r (A") I ... 1

I] 1 .~ /1 /d .IJ ./1 '

with /1 < · · · < /" E * , i\ 1, ••• , i\'1 lex icographica ll y decreasing fi nite sequences of

nonzero partitions, s, d, t , i 1 .... , is, j 1, . .. , .ir E IN, i 1 ::: · · · ::: i" j 1 ::: · · · ::: .if' and define the m ultipl icat ion using fo rmulas (25)- (34), i. e., using the Ha ll polynom ia ls.

We sha ll now consider the twisted case. Let v, = ~· Consider the product

n· = n 'lt* (k K. <Q[vd) , ke X

which can be viewed as a <Q[ v, v- 1]-algeb ra , where the vari able v (respective ly v- 1

) is iden ti fied with (vk · l H' (k K.<Q[v,J) ) (respectively with (v; 1 · l 1c' (kK. <Qi vk ll) = ~vk ·

l ll'(k K<Qi vk[) )) in n·. Taking sca la r extensions. we have tha t n· ® <Q(v) is a <Q (v) a lgebra . Let q := v2.

T he specia l elements listed in n wi ll be e lemen ts a lso in n·. We furt her consider the fo llowing elements in n·:

(38)

(39)

q - 1 vD ._ (koo ) _ "--- :// .:~un . - .-:.~rJ n - nv ., . ,, q-"- I

(40)

(4 1)

where ). = (i.1, • • . , i.") E IP .JJ ), JJ:= i. ~ + · · · + ),~ and A = (1,1, ••• , i,r) is a

lexicographica ll y decreasing seq uence of no nzero pa rtitio ns of length /(A) = t .

For A= 0 (i.e. , /(A)= 0) we have rt (0) = rt(O) =([OJ) = l the unit element inn· .

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2534 SZANTO

The unital <Q (v)-s ubalgebra of n• generated by S 1, S2 and the elements rt ().), with I ~ I , i. E IP will be called twisted generic Hall algebra of the Kronecker algebra and denoted by 'Jt* (K , <Q (v)) . Fo llowing Ringel and Green, the unital <Q ( v) -suba lgebra of n· generated by the elements sl ' s2 is called twisted generic composition algebra of the Kro necker a lgebra and deno ted by ?i!*(K, <Q (v)). Trivia ll y C*(K, <Q (v)) _:::: 7/C* (K, <Q (v)).

Consider now the <Q (v) -spaces Vt' (n) _:::: H*(K , <Q (v)), having as basis the

elements rt' (i.) with 11.1 = n , a nd let Vt' = .L" Vt' (n) = EB" V1*(n) E le* (K, <Q ( v)). For n E IN denote by <Q (v)[x 1,x2 . .. . ], the <Q (v) -subspace of <Q (v)[x 1,x2 •... ]

ha ving as basis the elements x i. = x i. , ·· -xi.,• with i. = (i.1, . •. , ), ,) E IP(n ). The canonical projection of <Q (v) [x 1 , x2 , .•. ] o n <Q (v)[x 1, x2, ... ], will be denoted by p~ .

Using the grad uatio ns above take now the graded <Q (v) -linea r isom orphism f 1* : <Q (v)[x 1,x2 ... . ] ~ V1* given by f ,* (x;) = r1*(i.), so in particular f(x (OJ = I) = I. Let f ,*" be the projection of f 1* o n V1*(n), so we have .f;* p ~ = f ,*".

The twisted version of Lemma 4 .2 a nd Theorem 4.3 is

Theorem 4.5. The elements

with 11 < · · · < I" E IN *, A 1, ... , A" lexicographically decreasing finite sequences of nonzero partitions, s, d, t , i 1, •.. , i ,, ) 1 •••• , j , E IN, i 1 _:::: · · • _:::: i '"' j 1 _:::: • • · _:::: j, form a PBW-basis in 7C* (K. <Q (v)).

The structure constants are given by the fo rmulas (27) , (28) , (30), (38)

ISL I P P 2p p ( 2 -2) "' 2up p

i * j = V j * i + V - V L V j+u * i- u fo r i > j , i - j odd. u= l

P P 2 p , p + i - j-2 ( 2 I ) p , p + ( 2 - 2) ;* j =v j "' ; v v- Y "' 9 v -v

i !.y!H "' 2up p X L V j+u * i - u fo r i > j , i - j even u=l

and dually

for i < j. j - i odd. u= l

IS' H "' 2u I I X L- V j- u * i+u for i < j. j - i even. II = I

[i. [

rt' (i.) * P," = L pm+il * f,~li. [ -i((xi., + xi,- 1) . .. (xi., + x),_ l)) i= O

(42)

(43)

(44)

(45)

(46)

(47)

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A GENERIC HALL ALGEBRA OF THE KRONECKER ALGEBRA 2535

and dua lly

Iii

1111 * r7(i.) = Lf~~!i! -;((x; , + X;1

_ 1) · ··(xi,,+ xlc, _1)) * l m+it· (48) i=O

Using the results from section 3. we immediate ly have

Corollary 4.6. The elements

with ). E IP, 2R;, = 9(;1

• • • .CJt;" = 2k;1 * · · · * .Off_;" ' I, s, i 1, . . . , i, , ) 1 , ... , ) 1 E lN,

i1

:S · · · :S i,, ) 1 :S · · · :S j , form a PBW-basis in ~g* (K , <Q(v)), the structure constants being given by formu las (42)-(46)

n- 1 ;~ p _ p . rm + -n ( 2n- 1 + 2n-2 )P +" ;( ; _ i-2 ) P 0~

•7 ' 11 * 111 - "' * en" V q q m+ n L.- V q q m+ i * ·7 Ln- i (49) i= l

and dua lly

n- 1

I tY> _ 0 b I + - n( 2n - 1 + 2n - 2 ) 1 +" i( i _ i- 2) 0"~ 1 Ill * CIL/1 - CIL/1 * Ill V q q 111 + 11 L,_ V q q CJLII - i * m+i · (50)

i= l

Corollary 4.7. The elements

P * · ·· * P * 03· * 1· * ·· ·* I · IJ I < A J1 ) I '

with ). E IP. 03;, = '13;, · · · 03;" = 03;_1 * · · · * JA;" ' I , s, i 1, ... , i5 , ) 1 , • •• , j , E IN ,

i; :S · · · :S i , j1

:S · · · :S j , form a PBW-basis in rg* (K , <Q(v)) , the structure constants being given by fo rmulas (39), (40), (42)- (46)

(5 1)

and dua lly

(52)

Corollary 4.8. The elements

P *···* P * 'd!- ( 1) * 1 *· .. * ! , I J 1_, I . }, }1

with ),E IP, '?b;.( I)= '.!D; , (1) · · · 2ll;.,( l), t, s, i 1 , . .. , i,. 11 , ... , j ,E lN, 11 :S ··· :S i,, j

1 :S · · · :S j ,. form a PBW-basis in Cfi* (K, <Q(v)), the structure constants being given by

formulas (39 )- ( 46)

'?b"(l) * Pill = Pill* S'l"( l ) + Pm+l * 20 11 - l ( I ) (53)

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2536 SZANTO

and dual/\·

(54)

5. CENTRAl ElEMENTS IN THE TWISTED GENERIC HAll AlGEBRA

Consider the <Q (v)-spaces W1*(n) ::: 'Jf*(K , <Q( v)) , having as basis the elements r1*(i.)r,*(Jt) with i . ~ lo JL (using the lexicographical ordering) and li,l + IPI = n. Let w,· = L ll W,*(n) = EBII Wt (n ) E 7f* (K, <Q( v)). For n E IN , denote by Q( v)[x1, x 2 . . . . , y 1, y2 , .. . L the <Q( v)-subspace of Q( v) [x1, x2 , . . . • y1, -"" · ... ]. having as basis the elements x~_ v,, , with li.l + IPI = n. When we speak about homogeneous polynomials of some degree. we will use the graduation above. Take now the graded <Q( v)-linear epimorphism g; : <Q(v)[.\ 1, x1 , ... . y 1, y2 , • •. ]---+ W1* given by g; (x ;Y,,) = r1' ( i.) rt (p) . where .r; = X;_

1 • • • x ;_, (x0 being I), and let g7.

11 be its projection

on W1*(n). Observe that g;- 1(rt (J.)rt (!l)) = {x;_v,, x1,Y; }.

Applying fo rmulas (47) . (48), we get

lemma 5.1. Let h 11 (x 1 ••• . • X 11 , v 1 , • • • • .' '11 ) a homogeneous polmomial of degree n in Q( v)[.r 1, x2 • ...• .''1, r 2 ••• . ]. Th en

II

= LP;, *8~ ~~ - ; (h ll (xl + I .... ,.rll + xii- I•YI + l , .. . . r ll +rll - 1)) (55) i=O

and dually

= L g~~~- ; (h ll (x l + I, ... , .rll + x11 - 1· Y1 + I ..... Y11 + -"11 - 1)) * lit . (56) i= O

Next. H'e defin e inductil'e ly in Q( v )[x 1• x 2 • . ..• r 1, y 2 , .. . ]the polYnomials

n- 1

T1 = x 1, T11 = 11X11 - LX11 _;T;. i= l

Of course, T11 = T11 (x 1, .. . , X 11 ) and its homogeneous of degree n. For a partition i . = (l 1 , ••• , }, ,) E lP let

(57)

T; := (T;1 (.r1, .... x;J - T;_

1 (y1, . . .. r ;J ) · · · (T;, (x1, ... , x;)- T;,(_r1, ... , Y;) ).

(58)

One can see that T;_ = T;_ (x 1, ... , X;_1

, y 1, ... , r ;_1

) and it is homogenous of degree li.l-

lemma 5.2.

a) Til ( x 1 + I , x 2 + x 1 . . . . , x II + x 11 _ 1 ) = T11

( x 1 , . . . , x,) + (- I ) II+ 1 .

b) T;_(x 1 + I, . . . , X;_1

+ X;1

_ 1. r 1 + I .... , y1_1 + r ;_

1_ 1) = T;_(x

1, • ••• X;_

1, v1, . . . , _\' ;_

1).

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r\ GENERIC HALL ALGEBRA OF THE KRONECKER ALGEBRA 2537

c) If i. = (}. 1 .. . . , ; ,, ) E IP and 1 = l(i.) is odd. then g7(T;) = 0. d) Ifi. = (i.1, ... , ),) E IP and I=/().) is even, then

g~ (T;) = (2i.1 • • • i,, )rt (}c ) + L cL,r1*(X) r1*(p' ), (59) i.> ,,,;.·~,.,, ~,,( 0). 1/.' l+l li l=li-1

where i"_J E 71.. / .' Jl '

Proof a) Use induction on n, for 11 = l the statement being trivia ll y true. By (57) and the induct ion hypo thesis we have

Il - l

= n(.rll +XII_ ,)- :LCxll - i + xll - i- I) T;(x , + I ' x2 + x,, .. . , X;+ X;_,) i= l

n- t

= n(.rll + X11 _ 1) - L(X 11 _ ; + X11 _ ; _ 1 )(T;(..r1, x2 , . .. , x;) + (-I r+') i= l

11 - !.

- L xii - 1- ;T;(x ,, . .. , x;) i=l

= Tll(.r,, Xl, .. . , x,J + (- Til - l (.r ,, . . . , XII- I) + ( - I )"+I + Til- l (x, , ... , XII-I)

= T,lr1, x2 , . .. , xJ + ( - I )"+ '.

b) Fo llows from (58) and a).

c) -g;( T;) .

d) Follows from (57), (58) and the definition of g;. Now we are ready to define our central elements in '!!C* (K. <Q( v)). Fo r }, = (i.1 , ••• , i. ,) E IP wit h 1 = /(}.) even , let:

c1().) := g7(T;) = (2/c 1 · • • l. ,)r1*(}c) + L cL,t/ (A'h *(f.l' ) ,

where cl-,,, E 71.. For I. = (n) , let:

i.> ,,;:~ ,,, ,,_,, (o). l i.'l +l ti I=IJ. I

For i. = (}. 1 • •. . , },, ) wit h t = /().) :::=: 3 odd we define inductively:

i.,

0

(60)

(61)

cJ(/,1, ... . i.,_ 1, A.,)):= g;(x;, T0 , ;,_,))- L c't((}c 1, ... , ),,_1, i.,- i)) '!t; (l). (62) i= l

· i

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2538 SZA t TO

Theorem 5.3 . The elements c1(/c) defined above are in the center of 'J(;* (K , <Q (v)).

Proof Since regulars are commuting with regu lars (see Theorem 2.1) it is enough to show, that the specified elements a re commuting with S1 (a nd dua ll y with S2).

Consider the case l(i.) even. Then using Lemma 5. 1. and Lemma 5.2. c) we have that c1(i.) * S1 = g;(T;_) * S1 = S1 * g;( TJ = S1 * C1(/,).

When i. = (n). then by Lemma 5.1 , Lemma 5.2. a) . and formula (5 1) we obta in c1(( n)) * S1 = (g; (T,) + (- I )":1J, 1) * S1 = S1 * g; (T,) + ( - 1)"+1 P,1 + (- I )"(SI * '33 ,t + P, 1) =51 * c/((n)).

Now let ;, = Ut .... , i.,) with 1 = 1(1.) :::: 3 odd. Then, using (23), we have

1.,

= g; (x;J ci, .i,_,) ) * S1- l::>tW'I• ... , ;.,_!· i.,- i) )':'iJ;(/) * S1 i= l

- (s1 * -gc1((}. 1, ... , J., _1, i., - i)) D;(/) + P1 * -gc1((). 1, ... . i. ,_1, ;_,- i)):-2!;_ 1(/) )

= sl * cl((i.l, . .. . i ., _l , )., ))-PI* ctC(i· l· . . .. i. ,_l, i.,- I))+ PI * (g; (xi,- 1 Tv, i, _,) )

i.l - !

- L c1((i. 1 , ••• , i. ,_ 1 , 1.,- I - j))D1(/)) }= I

6. STRUCTURE THEOREM FOR THE TWISTED GENERIC HALL ALGEBRA

D

Theorem 6.1. 'JC* (k , <Q (v)) is a polrnornial algebra in infinite/\· monr variables over ?i:* (K, <Q (v)). More preciselY,

7{* ( K , <Q (v)) = rrz* ( K , <Q (v))[c1(i.)ll :::': I I.E IP\ {(0) , (1)} ].

The degree of c1().) being 111.1, rhere will be - I + L Jin p( ~) variables of deg ree n E *. where p(m) is the number of parfilions of m.

Proof Denote by J{~ (K , <Q (v)) the unita l subalgebra of 7(* (K , <Q (v)) generated by the elements r/ (/.) with I :::: I and ). E IP*. In fact. th is is the "regular part" of lC* (K , <Q (v)). Due to Theorem 4.5, we have that 7t;(K. <Q (v)) is a polynomial a lgebra in infin itely man y variables. mo re precisely

'/lt;(K . <Q (v)) = <Q (v )[rt' (i.) ll :::: I.;, E IP*],

so the degree of r1*(i.) being 11/_l, we wi ll have L dln p( ~ ) varia bles of degree n E IN *. Consider now the total ordering on IN * x IP* given by (!. i.) 2::: (/' , p) if

either I > I' or I = I' and i. ::':to 1-1. Then using fo rmulas (27), (28), (30), (38). one

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A GENERIC HALL ALGEBRA OF THE KRONECKER ALGEBRA

ca n see that

M(q , n)(q"- I ) .CJ/" = 1( ) r,7((l)) +a <Q (v)-polynomial in va riables

v q- I

rt' (J,) with n > l, i.e., (n, ( I)) >- (L, Jc) ,

so for each n E JN* we can change the va riable <( (I) ) with f!!i.11

•

Rearranging the variables, it follows that

7/CZ(K. <Q (v)) = <Q (u)[f!!l. 1, ••• , Uli 11 , •• • ][r1*(/c) I L ::0: I , Jc E IP \ {(0) , ( 1)}].

2539

(63)

Due to formulas (39), (40), (4 1) we have that 9ll11

, t:}£11

( / ) E <Q (v)[f!!l. 1, ... ,

f!!!. ", ... ] , so using (60), (6 1), (62) and the opposite of (63) (for express ing r1*((l)) as some <Q (u)[9t 1, ... , f!!!." , .. . )-polynom ial in variables r,7,(p) with l > m, Jl E

IP \ {(0) , ( !)}),we deduce for I ::0: I and i.E lP\ {(0) , (1)} that

c1U) = cxrt'(J.) +some <Q (u)[2!1. 1 , • . • , 9'1. 11 , • •• )-po lynomial in variables

r,7,(1l) with (1, Jc) >- (m, 11),

where a E IN *, p E IP\ {(0), ( I)} . But this means that

'JCZ( K, <Q (v)) = <Q (v)[.oA 1, .. . , :!lt 11 , •• • )[c1().) II ::0: I , A E IP\ {(0), ( I) }],

therefo re, using Theorem 4.5, Corollary 4.6, and T heorem 5.3, the assertion fo ll ows. D

ACKNOWLEDGMENTS

I wou ld like to thank Professor Claus Michael Ringel and Professor Steffen Konig fo r their help and encouragement. I am also grateful to Istvan Agoston and Andre i Marcus fo r supporting me throughout my studies and my work.

REFERENCES

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Hua . J. , Xiao. J. (2002). On Ringci-Ha ll algebras of tame hereditary algebras. Algebr. Repr. Th eory 5:527- 550.

MacDonald , I. G. ( 1995). Symmerric Funclions and Hall Polynomials. Oxford : C la rend on Press .

Ringel, C. M. (1984). Tame algebra and lllfegral Quadraric Forms . Lect. Notes Mat h. 1099. Berlin-New York: Sp ringer-Verlag.

Ringel, C. M. ( 1990a). Hall algebras. Topics in a lgebra. Banach Cer11er Pub!. 26:433-44 7.

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