Degree of Parabolic Quantum Groups

Transcription

1 Unverstà degl stud Roma Tre Dottorato d rcerca n Matematca XVI cclo Tes d Dottorato Degree of Parabolc Quantum Groups Dottorando Rccardo Pulcn Drettore d tes Corrado De Concn September 26, 2005 Coordnatrce del dottorato Luca Caporaso

2 CONTENTS Introducton Part I Useful algebrac and geometrc noton 1 1. Posson algebrac Groups Posson manfolds Posson algebras and Posson manfolds Symplectc leaves Le balgebras and Mann trples Le balgebras Mann trples Posson groups Posson affne algebrac group Posson Le group The correspondence between Posson Le groups and Le balgebras Symplectc leaves n Posson groups Symplectc leaves n Posson Le groups and dressng acton Symplectc leaves n smple complex Posson Le groups Algebras wth trace Defnton and propertes Representatons Twsted polynomal algebras Useful notaton and frst propertes Defnton Representaton theory of twsted dervaton algebras Representaton theory of twsted polynomal algebras Maxmal order

3 Contents Part II Quantum Groups General Theory Quantum unversal envelopng Algebras P.B.W. bass Degeneratons of quantum groups Quantum groups at root of unty The center of U ǫ Parametrzaton of rreducble representaton of U ǫ Degree of U ǫ Degree of U ǫ (b) Quantum unversal envelopng algebras for parabolc Le algebras Parabolc quantum unversal envelopng algebras Defnton of U(p) Defnton of U ǫ (p) The center of U ǫ (p) Parametrzaton of rreducble representatons Character of a representaton A deformaton to a quas polynomal algebra The case p = g general case The degree of U ǫ (p) A famly of U ǫ (p) algebras Genercally semsmplcty The center of Uǫ χ (p) Bblography Index

4 INTRODUCTION Quantum groups frst arose n the physcs lterature, partcularly n the work of L.D. Faddeev and the Lengrad school, from the nverse scatterng method, whch had been developed to construct and solve ntegrable quantum systems. They have generated a great nterest n the past few years because of ther unexpected connectons wth, what are at frst glance unrelated parts of mathematcs, the constructon of knot nvarants and the representaton theory of algebrac groups n characterstc p. In ther orgnal form, quantum groups are assocatve algebras whose defnng relatons are expressed n terms of a matrx of constants (dependng on the ntegrable system under consderaton) called quantum R matrx. It was realzed ndependently by V. G. Drnfel d and M. Jmbo around 1985 that these algebras are Hopf algebras, whch, n many cases, are deformatons of unversal envelopng algebras of Le algebras. Indeed Drnfel d and Jmbo gve a general defnton of quantum unversal algebra of any semsmple complex Le algebra. On a somewhat dfferent case, Yu. I. Mann and S. L. Woronowcz ndependently studes non commutatve deformatons of the algebra of functons on the groups SL 2 (C) and SU 2, respectvely, and showed that many of the classcal results about algebrac and topologcal groups admt analogues n the non commutatve case. The am of ths thess s to calculate the degree of some quantum unversal envelopng algebras. Let g be a semsmple Le algebra, fxed a Cartan subalgebra h g and a Borel subalgebra h b g, we denote wth the correspondent set of smple roots. Gven, we assocate to parabolc subalgebra p b. Followng Drfel d, the consdered stuaton can be quantzed. We obtan Hopf algebras over C [ q, q 1], U q (b) U q (p) U q (g). When we specalze the parameter q to a prmtve l th root ǫ of 1 (wth some restrctons on l). The resultng algebras are fnte modules over ther centers, they are a fntely generated C algebra. In partcular, every rreducble representatons has fnte dmenson. Let us denote by V the set of rreducble representatons, Shur lemma gves us a surjectve applcaton π : V Spec(Z). To determne the pull back of a pont n Spec(Z) s very dffcult work. But genercally the problem becomes easer. Snce our algebras are doman,

5 Contents v there exsts a non empty open Zarsk set V Spec(Z), such that π π 1 (V ) s bjectve and moreover every rreducble representaton n π 1 (V ) has the same dmenson d, the degree of our algebra. The problem s to dentfy d. Note that, a natural canddate for d exsts. We wll see that n the case of U ǫ (p), we can fnd a natural subalgebra Z 0 Z, such that as U ǫ (p) subalgebra s a Hopf subalgebra. Therefore t s the coordnate rng of an algebrac group H. The deformaton structure of U ǫ (p) mples that H has a Posson structure. Let δ be the maxmal dmenson of the symplectc leaves, then a natural conjecture s d = l 2. δ Ths s well know n several cases, for example, p = g and p = b (cf [DCK90] and [KW76]). Our job has been to prove ths fact and to supply one explct formula for δ. Before descrbng the strategy of the proof, we explan the formula for δ. Set l the Lev factor of p. Let W be the Weyl group of the root system of g, and W l W that one of subsystem generated by. Denote by ω 0 the longest element of W and ω0 l the longest element of Wl. Recall that W acts on h and set s as the rank of the lnear transformaton w 0 w0 l of h. Then δ = l(w 0 ) + l(w l 0) + s, where l s the length functon wth respect to the smple reflecton. We descrbes now the strategy of the thess. In order to make ths, the man nstrument has been the theory of quas polynomal algebras, or skew polynomal, S. In ths case we know, followng a result of De Concn, Kaç and Proces ([DCKP92] and [DCKP95]), that to calculate the degree of S corresponds to calculate the rank of a matrx (wth some restrcton on l). In order to take advantage of ths result we have constructed a deformaton of U ǫ (p) to S and one famly, Uǫ t,χ, of fntely generated algebras parameterzed by (t, χ) C Spec(Z 0 ). Then the fact that on C[t, t 1 ] our deformatons have to be trval, together wth the rgdty of semsmple algebras can be used to perform the degree computaton at t = 0. In ts the algebra we obtan s a quas dervaton algebra, so that the theorem of De Concn, Kaç and Proces ([DCKP95]) can be appled and one s reduced to the soluton of a combnatoral problem, the computaton of the rank of an nteger skew symmetrc matrx. The actual determnaton of the center of the algebra U ǫ (p) remans n general an open and potentally trcky problem. However we wll propose a method, nspred by work of Premet and Skryabn ([PS99]), to left elements of the center of the degenerate algebra at t = 0 to elements of the center at least over an open set of Spec Z 0. We close ths ntroducton wth the descrpton of the chapters that compose the thess. In the frst chapter, we recall the noton of Posson group, ths s the nstrument necessary n order to descrbe the geometrc property of the Hopf subalgebra Z 0.

6 Contents v In the second and thrd chapter, we descrbe the slght knowledge of algebra wth trace and quas polynomal algebra. These are the nstruments necessary n order to descrbe the theory of the representatons of U ǫ and U ǫ (p). Moreover we descrbe how to calculate the degree of quas polynomal algebra. Ths concludes the frst part of the thess. In the second part, we wll be studyng the theory of quantum groups. In chapter 4 we recall the man defntons and the man propertes of quantum unversal envelopng algebras assocated to a semsmple complex Le algebra g, and we consder how to calculate the degree of U ǫ (g) and U ǫ (b). The last chapter s dedcated to the defnton of quantum envelopng algebra assocated to a parabolc subalgebra p of g and the proof of the formula for the degree.

7 Part I USEFUL ALGEBRAIC AND GEOMETRIC NOTION

8 1. POISSON ALGEBRAIC GROUPS In ths chapter we recall some basc facts about Posson groups that wll prove useful n the study of quantum groups. The nterested reader can fnd more detals n a vast varety of artcles and monographs, for example n [CP95], [KS98] or [CG97]. 1.1 Posson manfolds Posson algebras and Posson manfolds Defnton A commutatve assocatve algebra A over a feld k s called a Posson algebra f t s equpped wth a k-blnear operaton {, } : A A A such that the followng condtons are satsfed: 1. A s a Le algebra wth the bracket {, }; 2. the Lebnz rule are satsfed,.e. for any a, b, c A, we have {ab, c} = a {b, c} + {a, c}b. If these condtons are satsfed, the operaton {, } s called Posson bracket, and ξ a = {a, } s called Hamltonan dervaton. Defnton Let A and B be a Posson algebra over k. An algebra homomorphsm f : A B s called Posson homomorphsm f f ({a, b}) = {f(a), f(b)}. Posson algebras form a category, wth morphsm beng Posson homomorphsm. Defnton A smooth manfold M s called a smooth Posson manfold f the algebra A = C (M) of smooth complex value functon on M s equpped wth a structure of Posson algebra over C Defnton An affne algebrac k-varety M s called an affne algebrac Posson k-varety f the algebra A = k[m] of regular functon on M s equpped wth a structure of Posson algebra over k

9 1. Posson algebrac Groups 3 Defnton Let M and N be smooth Posson manfolds. A smooth map f : M N s called a Posson map f the nduced map f : C (N) C (M) s a Posson homomorphsm. 2. Let M and N be algebrac Posson k-varety. An algebrac map f : M N s called a Posson map f the nduced map f : k[n] k[m] s a Posson homomorphsm. It s clear that smooth, and algebrac, Posson manfolds form a category, wth morphsms beng Posson map. Defnton Suppose that M s a smooth Posson manfold, let A = C (M). 1. Gven φ A, the vector feld ξ φ assocated to the Hamltonan dervaton {φ, } of A s called Hamltonan vector feld. 2. A submanfold (not necessarly closed) N M s called Posson submanfold f the vector ψ φ (n) s tangent to N for any n N and φ A. Let M a smooth Posson manfold, then there s an equvalent defnton of Posson manfold n term of bvector felds. Recall that an n-vector feld s a secton of the bundle n TM where TM s the tangent vector bundle of M. In partcular, we call 2-vector felds bvector felds. Recall also the defnton of the Schouten bracket of n-vector felds whch generalze the usual Le bracket on vector felds. The Schouten bracket of an m-vector feld wth an n vector feld s an (m + n 1)-vector feld whch s locally defned by [u 1... u m, v 1... v m ] =,j ( 1) +j [u, v j ] u 1... û u m v 1... ˆv j... v n where u 1,...,u m, v 1,...,v n T m M, m M, and [, ] denote the Le bracket of vector felds. Denote by T M the cotangent bundle to M. Gven a bvector feld π on M, we defne a blnear operaton {, } on C (M) by {φ, ψ} = π, dφ dψ (1.1) where, s the natural parng between the sectons of the bundle T M T M and TM TM. Proposton The bracket 1.1 defnes a Posson manfold structure on M f and only f [π, π] = 0

10 1. Posson algebrac Groups 4 Proof. Easy verfcaton of the defnton. Let (M, π) be a Posson manfold. Consder the morphsm of vector bundle ˇπ : T M TM nduced by π, we have Defnton A Posson manfold s called symplectc manfold f the map ˇπ s an somorphsm Symplectc leaves One of the most fundamental facts n the theory of Posson manfolds s that for any Posson manfold M there s a stratfcaton of M by symplectc submanfolds whch are called symplectc leaves n M. In a certan sense, symplectc manfolds are smple objects n the category of Posson manfolds. In what follows we assume that M s a smooth Posson manfold. Defnton A Hamltonan curve on a smooth Posson manfold M s a smooth curve γ : [0, 1] M such that there exst f C (M) wth the property that γ(t) = ξ f (γ(t)) for any t (0, 1) Defnton Let M be a smooth Posson manfold. 1. We say that two ponts x, y M are equvalent f they can be connected by a pecewse Hamltonan curve. 2. An equvalence class of ponts of M s called symplectc leaf of M Property. Let S be a symplectc leaf of a smooth Posson manfold M. Then: () S s a Posson submanfold of M; () S s a symplectc manfold; () M s the unon of ts symplectc leaves. We need a tool to determne symplectc leaves. Defnton Let P 1 and P 2 be Posson manfolds, S a symplectc manfold. A dagram P 1 f 1 S f 2 P2 s called a dual par, f f 1 and f 2 are Posson maps and the Posson subalgebras f 1 C (P 1 ) and f 2 C (P 2 ) of C (S) centralze each other wth respect to the Posson bracket. A dual par s called full f f 1 and f 2 are submersons. Theorem Let P 1 f 1 S f 2 P2 be a full dual par. Then the blow-up M x1 = f 2 f 1 1 (x 1) s a symplectc leaf n P 2 for any x 1 P 1. Proof. See [KS98], page 8.

11 1. Posson algebrac Groups Le balgebras and Mann trples Le balgebras Defnton A Le balgebra s a par (g, φ) where g s a fnte dmensonal Le algebra over k, and φ : g g g, called cobracket, satsfes the followng condtons: 1. the dual map φ : g g g makes g nto a Le algebra; 2. the cobracket φ : g g g s a 1-cocycle on g,.e. for any a, b g, φ([a, b]) = a.φ(b) bφ(a) where a.(b c) = [a a, b c] = [a, b] c + b [a, c]. Le balgebras form a category, wth morphsms beng Le algebra homomorphsm whch commute wth the cobracket. Property. Gven a Le balgebra g, the vector space g carres a canoncal structure of Le balgebra, the Le bracket beng the map dual to the cobracket n g and the cobracket beng the map dual to the bracket n g. Defnton The Le balgebra g s called the dual Le balgebra of g. Example Any Le algebra g wth trval cobracket (.e., zero) s a Le balgebra. Example Let g be a Le algebra. Consder the dual vector space g as a commutatve Le algebra. Then the map φ : g g g dual to the Le bracket on g, defne a Le balgebra structure on g, t s the dual Le balgebra to the one n example Example Let g be a complex smple Le algebra wth a fxed non degenerate nvarant symmetrc blnear form, whch s necessarly a scalar multple of the Kllg form. Choose a Cartan subalgebra h g, wth n = dmh, the rank of g. Choose a set of smple roots α 1,...,α n h. Ths gves a decomposton g = n + h n (1.2) where n + (resp. n ) s the nlpotent subalgebra spanned by the postve (resp. negatve) root subspaces. Let X ±, H, = 1,...,n, be the Chevalley generators correspondng to the smple root α, and A = (a,j ) the Cartan matrx whose entres are a,j = 2 (α,α j ) (α,α ), where (, ) s the symmetrc blnear form on h nduced by the blnear form on g. Recall that g s generated by X ±, H and the Serre relatons

12 1. Posson algebrac Groups 6 [ ] [H, H j ] = 0, H, X j ± = ±a,j X j ± [ ], X +, X j = δ,j H, ad 1 a ( ),j X ± X ± j = 0, where ad(a)(b) = [a, b] s the adjont acton of g on tself. Then the followng cobracket φ defnes a Le balgebra structure on g: φ(h ) = 0, φ(x ± ) = d X ± H, where d, = 1,...,n, are postve ratonal numbers that satsfy d a,j = d j a j,. For more detals see [KS98]. Defnton The Le balgebra structure descrbed n example s called the standard Le balgebra structure on g Example Consder the smple complex Le algebra g = sl 2 (C), then the Chevalley generator are ( ) ( ) ( ) X + =, H =, X =, 1 0 then the cobracket s We verfy only that φ s a cocycle. φ(h) = 0, φ(x ± ) = X ± H. φ([h, X ± ]) = φ(±2x ± ) = ±2X ± H H φ(x ± ) X ± φ(h) = H (X ± H) = [H, X ± ] H = ±2X ± H φ([x +, X ]) = φ(h) = 0 X + φ(x ) X φ(x + ) = X + X H X X + H = [X +, X ] H + X [X +, H] [X, X + ] H X + [X, H] = H H + 2X X + = 0 +H H + 2X + 2X The fact that φ defnes a Le bracket s an easy exercse.

13 1. Posson algebrac Groups Mann trples For our proposton t s convenent to thnk of Le algebras n terms of Mann trple. Defnton Let g be a Le algebra equpped wth a nondegenerated nvarant symmetrc blnear form,, and g + and g Le subalgebra of g. The trple (g, g +, g ) s called Mann trple f: 1. g = g + g, 2. g + and g are maxmal sotropc subspaces wth respect to,. Suppose that (g, g +, g ) s a Mann trple. Snce both g + and g are maxmal sotropc subspaces of g, we can dentfy g wth g ± as a vector space. Theorem () Suppose that (g, g +, g ) s a Mann trple. The n (g ±, φ) s a Le balgebra where the cobracket φ : g ± g ± g ± s the dual map to the Le bracket n g. () Suppose that (g, φ) s a Le balgebra. Consder the vector space g equpped wth the dual Le balgebra structure (cf. defnton 1.2.2). Then: (g g, g, g ) s a Mann trple, wth the Le algebra structure on g g gven by [a + α, b + β] = [a, b] + [α, β] + ad α(b) ad β (a) ad b (α) + ad a(β), for any a, b g, α, β g, and the blnear form on g g gven by Note that, n partcular a + α, b + β = β(a) + α(b). [α, b] = ad α(b) ad b (α), [a, β] = ad a(β) ad β (a). Proposton Let g be a Le balgebra, and D(g) = g g the le algebra descrbe n theorem Then there exst a canoncal Le balgebra structure on D(g) gven by the cobracket for any a g and α g φ D(g) (a + α) = φ g (a) + φ g (α), Defnton Le balgebra D(g) s called the double Le balgebra (sometmes t s called the classcal double of g)

14 1. Posson algebrac Groups 8 Remark Mann trple whch corresponds to the double Le balgebra s (D(g) D(g), D(g), g g ) where D(g) s embedded nto D(g) D(g) as the dagonal, and g g (the dual Le balgebra of D(g)) s the drect sun of the Le algebras g = (g, 0) D(g) D(g) and g = (0, g ) D(g) D(g). The blnear form s gven by for any a, b, c, d D(g). (a, b), (c, d) = a, c D(g) b, d D(g), Double Le balgebra has an mportant property. Namely, the cobracket on such Le balgebra s descrbed by a very smple formula, as shown n the followng proposton Proposton Let g be a Le balgebra, and D(g) the double Le balgebra. Suppose that {e α } s a bass of g and {e α } s the dual bass of g. Let us dentfy e α wth (e α, 0) D(g) and e α wth (0, e α ) D(g). Then the cobracket n D(g) s gven by for any a D(g), where φ(a) = [a a, r] r = α e α e α D(g) D(g), s the canoncal element related to,. We gve now some mportant examples of Le balgebra and correspondng Mann trple. Example Let g be a complex smple Le algebra equpped wth the standard Le balgebra structure (cf. defnton 1.2.5) related to a fxed nvarant blnear form,, let h be the correspondng Cartan subalgebra and defne b ± = n ± h a Borel subalgebra of g. Then the correspondng Mann trple s (g g, g, s), where g s embedded dagonally nto g g, and s = {(x, y) b + b : x h + y h = 0}, where x h denote the Cartan part of x b ±. In other words, x = x ± + x h where x ± n ± and x h h. The nvarant blnear form g g s gven by (a, b), (c, d) = a, c g b, d g. Note that as Le algebra D(g) s somorphc to g g. For more detals one can see for example [KS98].

15 1. Posson algebrac Groups 9 Example Let g be as n the prevous example. Consder the Borel subalgebras b ±, note that they are n fact Le balgebras wth respect to the restrcton of the cobracket from g to b ±. Then the correspondng Mann trple s (g h, b +, b ), where the Borel subalgebras are embedded nto g h so that f a b + then a (a, a h ) and f a b then a (a, a h ), where we use the notaton n the prevous example, the blnear form s gven by (a, b), (c, d) = a, c g b, d h. wth a, c g and b, d h, and, h as the restrcton of, g to h. In partcular, we see that the Borel subalgebras b + and b are dual to each other as Le balgebra. We consder, n the next example, an ntermedate case between Borel subalgebra b and Le algebra g. Example Usng the notaton of the prevous example, we call parabolc subalgebra any subalgebra p of g such that b p, note that b ± and g are examples of parabolc subalgebra. Set R as the set of rot assocated to g and X α α R the root vectors, we defne R p := {α R : X α p}. We call l := h α R p: α R p CX α the Lev factor and u = α R p: α/ R p CX α the unpotent part of p. Moreover, we have p = l u, for more detals one can see [Bou98] or [Hum78]. Proposton p s a Le balgebra wth respect to the restrcton of the cobracket from g to p. Proof. Smple verfcaton of the defnton. Proposton The Mann trple correspondng to p s (g l, p, s), where p s embedded nto g l so that a (a, a l ) and { } s = (x, y) b b l + : x h + y h = 0, where b l + = b + l, the blnear form s gven by (a, b), (c, d) = a, c g b, d l. wth a, c g and b, d l, and, l s the restrcton of, g to l.

16 1. Posson algebrac Groups 10 Proof. Observe that, the followng decomposton hold n g g = u l u wth u := α R\R p, and dm(g l) = dm(g) + dm(l) = dm(p) + u + dm(l) = 2 dm(p) where u = dm u (a, b) p s, then = dmu + Let us now check that p s = {0}, take (a, b) p b = a l (a, b) s a b, b b l + and a h + b h = 0 Then b = t h = b + b, and a = u + t wth u u b +, but a b, so t follows that u = 0 and a = b = t. Then the last condton gves us 0 = a + b = 2t t = 0. In concluson we have a = b = 0. Hence p s = {0}. Observe that then we have dms = dmb + + dmb l dmh = dmp, g l = p s. It remans to show that p and s are sotropc subspace wth respect to,. Recall that, g s an nvarant symmetrc blnear form for g, note that for any parabolc subalgebra b + q g, ts Lev factor m we have: x, y g = x h, y h m for every x, y q. Let (a, b) p then b = a l, and we have (a, b), (a, b) = a, a g b, b l = a, a g a l, a l l = 0. Let (a, b) s then a h + b h = 0. Recall that the Lev factor for b ± s h. We get (a, b), (a, b) = a, a g b, b l = a h, a h h b h, b h h = a h, a h h a h, ah h = a h, a h h a h, a h h = 0. Ths fnshed the proof that (g l, p, s) s the Mann trple assocated to p.

17 1. Posson algebrac Groups Posson groups Posson affne algebrac group Let G an affne algebrac groups over an algebrac closed feld k. Let k[g] ts coordnate rng. We know that k[g] s a Hopf algebra wth Comultplcaton: Antpode: : k[g] k[g] k[g] S : k[g] k[g] Count: ǫ : k[g] k gven respectvely by (f)(h 1, h 2 ) = f(h 1 h 2 ), h 1, h 2 G, S(f)(h) = f(h 1 ), h G, ǫ(f) = f(e) where e G s the dentty element. Defnton Suppose that a Hopf algebra A s equpped wth a Posson algebra structure. We say that A s a Posson Hopf algebra f both structure are compatble n the sense that the comultplcaton s a Posson algebra homomorphsm, where the Posson structure on A A s gven by {a b, c d} = {a, c} bd + ac {b, d} Property. Let A be a Posson Hopf algebra, then the count ǫ s a Posson algebra homomorphsm, and the antpode S s a Posson algebra antautomorphsm. Proof. See [KS98], page 18. We know that the algebra k[g] of regular functons on an algebrac group G s a Hopf algebra. Recall that t s also a Posson algebra f G s a Posson algebrac varety. Defnton Suppose G s an algebrac group over the feld k equpped wth a Posson manfold structure. We say that G s a Posson algebrac group f the algebra k[g] of regular functons on G s a Posson Hopf algebra. 2. Equvalently, G s a Posson algebrac group f the multplcaton m : G G G s a Posson map. Property. Let G a Posson algebrac group, then the map s : G G, such that s(g) = g 1 for every g G, s an ant-posson map,.e. the dual map s : k[g] k[g] s an anthomomorphsm of Posson map. Note. Posson algebrac groups form a category, wth morphsms beng homomorphsms whch are the same tme Posson maps.

18 1. Posson algebrac Groups Posson Le group The defnton of Posson algebrac group formulated n the language of ponts can be easly carred over to the case of Posson Le groups. The prncpal dffculty s that the comultplcaton maps goes nto C (G G) but not necessarly nto C (G) C (G) C (G G) Defnton Let G be a Le group and at the same tme, a Posson manfold. We say that G s a Posson Le Group f the multplcaton m : G G G s a Posson map. Note. Posson Le groups form a category, wth morphsms beng homomorphsms whch are the same tme Posson maps. Proposton Let G be both a Le group and a Posson manfold, and let π be the bvector feld correspondng to the Posson manfold structure on G. Then G s a Posson Le group f and only f π satsfes the followng condton: π(g 1 g 2 ) = (l g1 ) π(g 2 ) + (r g2 ) π(g 1 ) for any g 1, g 2 G, where l g s the left multplcaton and r g s the rght multplcaton. Proof. See [KS98], page 19. Corollary The unt element of a Posson Le group s always a zerodmensonal symplectc leaf In ths thess we always used Posson algebrac groups so we shall descrbe Posson brackets on the algebras of regular functons. Example Every Le group G wth trval Posson bracket s a Posson Le group. Example Consder the abelan Le group G = C n. Note that by lnearty t suffces to defne the Posson bracket on the coordnate functon. To get a Posson Le group structure we can take: {x, x j } = n k=1 c k,jx k where x m, m = 0...n, are the coordnate functon, and the structure constant c k,j satsfy the followng condton: n l=1 c k,j ( ) c l,jc m l,k + cl j,k cm l, + cl k, cm l,j = c k j, = 0

19 1. Posson algebrac Groups 13 Example An mportant specal case of the prevous example s The Krllov-Kostant bracket. Let g a le algebra. Consder the dual vector space g equpped wth the structure of abelan Le group. The Krllov-Kostant bracket on g s gven by: {a, b} = [a, b] g where a, b g are regarded as lnear functon on g. Example Consder the Le group G = SL 2 (C) of complex 2 2 matrces wth determnant 1. Then the followng relatons defne a Posson Le group structure on G: {t 11, t 12 } = t 11 t 12, {t 11, t 21 } = t 11 t 21, {t 12, t 22 } = t 12 t 22, {t 21, t 22 } = t 21 t 22, {t 12, t 21 } = 0, {t 11, t 22 } = 2t 12 t 21, where t j, (, j = 1, 2), are the matrx elements The correspondence between Posson Le groups and Le balgebras One of the most mportant facts of the Le theory s the correspondence between Le groups and Le algebras. Recall that gven a Le group G, the tangent space at the unt element has a canoncal Le algebra structure. Conversely, gven a Le algebra g, there exsts a unque connected and smple connected Le group whose tangent space at the unt element s somorphc to g as Le algebra. We establsh now a Posson counterpart of ths result. Let G a Posson Le group, and g ts Le algebra. As usual dentfy g wth the tangent space T e G to G at the unt element of the group. Defne a lnear map φ : g g g as the lnear map whch s dual to the Le bracket n g = T e G gven by for any α, β g, where f, g C (G) are such that [α, β] = d e {f, g} (1.3) d e f = α, d e g = β. Theorem () Let G be a Posson Le group. Then there exsts a canoncal Le balgebra structure on the Le algebra g = T e G wth the cobracket φ : g g g gven by 1.3. () Let G be a Le group, and suppose that the Le algebra g = T e G s equpped wth a Le balgebra structure. Then there exst a unque Posson Le group structure on G such that 1.3 holds.

20 1. Posson algebrac Groups 14 Proof. See [KS98], page 21. Proposton () The correspondence G g = T e G establshed a covarant functor from the category of the Posson Le groups to the category of Le balgebras. () The correspondence establshes an equvalence between the full subcategory of connected and smply connected Posson Le groups and the category of Le balgebras. Example Let G be a Le group wth trval Posson structure then g s a Le balgebra wth trval cobracket Example Let G be the Le group wth Kostant-Krllov structure, defne n example 1.3.8, then g s the Le balgebra gven n the example Defnton Let G be a Posson Le group, g the Le balgebra of G. 1. A connected Posson Le group G whch corresponds to the Le balgebra g s called a dual Posson Le group. 2. A connected Posson Le group D(G) whch corresponds to the double Le balgebra D(g) s called a double Posson Le group of G 1.4 Symplectc leaves n Posson groups Symplectc leaves n Posson Le groups and dressng acton Now we use theorem to descrbe symplectc leaves n a Posson Le group. The double Posson Le group constructon allows us to descrbe the symplectc leaves locally as the orbts of the so-called dressng acton. Throughout the secton the word locally means n a neghborhood of the unt element of the group. Let G be a Posson Le group, and g the Le balgebra of G. Recall that we have a smple descrpton of the dual Le balgebra structure on D(g) by φ(a) = [a a, r] for any a D(g), where r = α e α e α, {e α } beng a bass of g and {e α } the dual bass of g. Proposton The Posson bracket on D(G) s gven by {f 1, f 2 } = (δ α f 1 δ α f 2 δ α f 1 δ α f 2 ) ( δ α f 1 (δ α ) f 2 (δ α ) f 1 δ ) αf 2 α α where δ α (resp. δ α) s the rght (resp. the left) nvarant vector feld on D(G) whch takes the value e α at the unt element of D(G), whle δ α (resp. (δ α ) ) s the rght (resp. the left) nvarant vector feld on D(G) whch takes the value e α at the unt element of D(G).

21 1. Posson algebrac Groups 15 Proof. See [KS98], page 23. Proposton The multplcaton maps m : G G D(G), (1.4) m : G G D(G) (1.5) are local Posson dffeomorphsms n a neghborhood of the unt element. Proof. Ths s an easy consequence of the fact that D(g) = g g as a vector space. It s clear that one can dentfy locally G wth G \D(G) or wth D(G)/G. Consder the natural projecton p 1 : D(G) G \ D(G) and p 2 : D(G) D(G)/G. Then we have the followng property Proposton The Posson manfold structure on the double Le group D(G) nduces Posson manfold structure on G \ D(G) and D(G)/G such that the natural projectons p 1 and p 2 are Posson maps. Both manfolds are somorphc to G as Posson manfolds n a neghborhood of the coset G. Now we ntroduce the noton of left and rght dressng actons. Frst we defne them locally. Gven g G and h G whch le n some neghborhoods of the unt element of D(G), usng proposton there exst unque g h G and h g G such that hg = g h h g (1.6) Ths formula defnes a local left acton of G on G and a local rght acton of G on G gven by h : g g h, g : h h g (1.7) respectvely. Note that we can replace G by G and vce versa, so that we have also a local rght acton of G on G and a local left acton of G on G. Defnton The local left (rsp. rght) acton of G on G defne by 1.6 and 1.7 s called local left (rsp. rght) dressng acton of G on G. If t can be extended to a global acton, the latter s called global left (resp. rght) dressng acton of G on G. Proposton Let G a Posson Le group and G the Posson Le group dual to G. Then the symplectc leaves n G locally concde wth the orbts of the rght (or left) dressng acton of G. In partcular, f the rght (resp. left) dressng acton s defned globally, the symplectc leaves are the orbts of the rght (rsp. left) dressng acton. Proof. See [KS98], page 26.

22 1. Posson algebrac Groups 16 Theorem Let g G belong to a suffcently small neghborhood of the unt element n G. The symplectc leaf n G passng trough g locally (n some neghborhood of g) s the mage of the double coset G gg D(G) under the natural projecton D(G) D(G)/G G Proof. See [KS98], page Symplectc leaves n smple complex Posson Le groups Let G be a fnte dmensonal smple complex Le group, and g the Le algebra of G. Suppose that g s equpped wth the standard balgebra structure descrbed n the example It nduces a Posson Le group structure on G whch s also called standard. Our goal s to descrbe the symplectc leaves n G. Let (g g, g, s) the Mann trple assocated to g, descrbed n the example It s easy to see that D(G) s somorphc to G G as Le group and we can choose G = {(g 1, g 2 ) : g 1 B +, g 2 B and (g 1 ) H (g 2 ) H = e} as dual Posson group. It s also clear that G G s dense and open n D(G). Moreover, the multplcaton map m : G G G G s a coverng space. We conclude that the mage G of the quotent map p : G D(G)/G s dense and open n D(G)/G. By theorem and theorem we get Lemma () Any symplectc leaf n G s a connected component of the fber p 1 (S), where S s a symplectc leaf n G () Any symplectc leaf n G s of the form G G gg /G for some g G D(G). Now we want to descrbe the symplectc leaves. It appears that they are related to the Bruhat decomposton. Let b ± = n ± h be the Borel subalgebra of g, and B ± the correspondng Borel subgroup of G. Recall that the Weyl group W s generated by the smple reflecton s : h h gven by: s (λ) = λ 2(λ, α ) (α, α α where α 1,...,α r are the smple root, and r = dm h s the rank of g. The followng results are well know (cf. [Ste68]) Proposton The followng decomposton of G holds: G = B ± ωb ± (1.8) ω W It s called the Bruhat decomposton of G

23 1. Posson algebrac Groups 17 Let X = G/B ± the flag manfold. By proposton 1.4.8, we have X = X ω where X ω B ± ωb ± /B ± ω W Defnton X ω s the so-called Schubert cell of X correspondng to ω W. It s well know that X ω s naturally somorphc to C l(ω), where l(ω) s the length of ω. Proposton The followng Bruhat decomposton holds: D(G) = P (ω 1, ω 2 ) P, (1.9) (ω 1,ω 2 ) W W where P = HG and H s the dstngushed Cartan subgroup of G assocated to h. Consder the followng sets: C (ω1,ω 2 ) = (G (ω 1, ω 2 ) G )/G, B (ω1,ω 2 ) A (ω1,ω 2 ) = C (ω1,ω 2 ) G, = p 1 ( h H hb (ω1,ω 2 ) Proposton () each symplectc leaf n G s of the form hb (ω1,ω 2 ) for some h H and (ω 1, ω 2 ) W W. () each symplectc leaf n G s of the form ha (ω1,ω 2 ) for some h H and (ω 1, ω 2 ) W W. Proposton Denote s(ω 1, ω 2 ) = codm h ker(ω 1 ω 1 2 1). Then ) C (ω1,ω 2 ) H s(ω 1,ω 2 ) C l(ω 1)+l(ω 2 ) Example The followng s the full lst of the symplectc leaves n SL 2 (C): {( )} t 0 T t = 0 t 1, t 0, {( ) } t b T t A (e,ω0 ) = 0 t 1 : b 0, t 0, {( ) } t 0 T t A (ω0,e) = c t 1 : c 0, t 0, {( ) a b T t A (ω0,ω 0 ) = : b, c 0, b } c d c = t2, t 0.

24 1. Posson algebrac Groups 18 Fnally, consder the map µ : G G gven by µ((b 1, b 2 )) = b 1 1 b 2, so that µ s a coverng of the bg Bruhat cell B B +. De Concn, Kaç and Proces shows n [DCKP92], that as soon as C G s a conjugacy class, untl dmc > 0, µ 1 (C) G s a sngle symplectc leaf of G. If dmc = 0,.e. C = {x} s an element of the center of G, then µ 1 (x) has a fnte number of elements each of whch s a symplectc leaf.

25 2. ALGEBRAS WITH TRACE In ths chapter we wll requre some slght knowledge of the theory of algebras wth trace that wll be useful, n the next chapters, for the study of quantum groups and quas polynomal algebras. More detals and a more general approach n order to study these algebras can be found n [Pro87], [Pro73], [Pro74] or [Pro79]. 2.1 Defnton and propertes Let A be an assocatve algebra wth an unt element 1 over a feld k of characterstc 0 and let us denote the algebrac closure of k by k. Defnton A trace map n an algebra A s a lnear map tr : A A satsfyng the followng axoms: for all pars of element a, b A 1. tr(ab) = tr(ba) 2. tr(a)b = btr(a) 3. tr(tr(a)b) = tr(a)tr(b) An algebra wth a trace map s called algebra wth trace Note. The value of the trace s a subalgebra of the center of A (by condton 2). An deal I of A algebra wth trace s an ordnary deal closed under trace, so that A/I nherts a trace. We are nterested n a partcular famly of algebra wth trace as n [Pro87]. Once we have a trace map we want to defne for all a A the element σ k (a) "the symmetrc functon over the egenvalue of a", by declarng that tr(a k ) should be the sum of the k th power of the egenvalues of a. To do ths recall that n the rng Q[x 1,...,x n ] t defnes the elementary symmetrc functon by the dentty n (t x ) = ( 1) σ t d =0

26 2. Algebras wth trace 20 and the power sums functon ψ k = x k. It s easy to prove that, for every k n, s a polynomal p k (y 1,...,y k ) wth ratonal coeffcent, ndependent of n and such that: σ k = p k (ψ 1,...,ψ k ). We then set σ k (a) = p k (tr(a),...,tr(a k )). Next we can formally defne for every element a A and for every nteger d a d th -characterstc polynomal: χ d,a [t] = d ( 1) σ (a)t d =0 Defnton We say that an algebra A wth trace satsfes the d th -formal Cayley-Hamlton theorem f χ d,a [a] = 0 for all a A. 2. We say that A has degree d f t satsfes the d th -formal Cayley-Hamlton theorem and tr(1) = d. Note that algebra wth trace of degree d form a category wth morphsms beng algebra morphsms compatble wth trace, whch wll be denoted by C d. 2.2 Representatons We are nterested n a representaton theory of algebra wth trace of degree d. Let gve an example Example Let A be a commutatve algebra, then M d (A) wth the ordnary trace s an algebra wth trace of degree d. Defnton A n dmensonal representaton of an algebra wth trace R wth value n a commutatve algebra A s an homomorphsm ρ : R M n (A) compatble wth the trace map. If A = k we thnk of ths representaton as a geometrc pont. Remark We have necessarly n = d, snce d = ρ(tr(1)) = tr(ρ(1)) = tr(i) = n, where I s the dentty matrx of M n (A). Before statng the man theorem of ths secton, we wll gve some examples of algebra wth trace. In order to smplfy the treatment and stck to a geometrc language we assume, from now untl the end of the chapter, that k s algebracally closed and of characterstc 0.

27 2. Algebras wth trace 21 Example Consder A to be an order n a fnte dmensonal central smple algebra D. Ths means that the center of A s a doman, A s torson free over Z and, we have D = A Z Q(Z), where Q(Z) s the feld of fractons of Z. In other words, A embeds naturally n D whch s ts rng of fractons. If Q(Z) s the algebrac closure of Q(Z), we have that A Z Q(Z) s the full rng M d (Q(Z). Hence we have on D, and on A, the usual reduced trace map tr : D Q(Z). It s well known that tr(a) = Z, f A s a fnte Z module, Z s ntegrally closed and the characterstc s 0. So under ths hypotheses A s an algebra of degree d. For more detals cf. [Pro73] or [MR87]. Example The second example s gven by Azumaya algebras (cf [Art69]). Recall that: Defnton An algebra R over a commutatve rng A s called an Azumaya algebra of degree d over A, f there exsts a fathfully flat extenson B of A such that R A B s somorphc to the algebra M d (B). In ths case t s easy to show that the ordnary trace maps R nto A. Let R be a fntely generated algebra, we want to descrbe t s d dmensonal representaton. Theorem Assumes that R C d s a fntely generated algebra. Set T = tr(r). () T s a fntely generated algebra, and R s s fnte module over T. In partcular T s the coordnate rng of an affne algebrac varety X T = Maxspec(T). () The ponts of X T parameterze equvalence classes of d-dmensonal, trace compatble, and semsmple representatons of R () Set Spec(R) equvalences classes of rreducble representatons of R. The canoncal map Spec(R) χ X T, nduced by the central characters, s surjectve and each fber conssts of all those rreducble representatons of R whch are rreducble components of the correspondng semsmple representaton. In partcular each rreducble representaton of R has dmenson at most d. (v) The set Ω R = {a Spec(T),such that the correspondng semsmple representaton s rreducble} s a Zarsk open set. Ths s exactly the part of Spec(T) over whch R s an Azumaya algebra of degree d. Proof. See [DCP93] theorem 4.5, page 48.

28 2. Algebras wth trace 22 Remark (1) If R s an order n a central smple algebra of degree d then T equals the center of R, furthermore snce the central smple algebra splts n a d dmensonal matrx algebra whch one may consder as a generc rreducble representaton, t s easly seen that the open set Ω R s non empty. (2) If T s a fntely generated module over a subalgebra Z 0, we can consder the fnte surjectve morphsm τ : Spec(T) Spec(Z 0 ). Then by the properness of τ we get that the set Ω 0 R := { a Spec(Z 0 ) : τ 1 (a) Ω R } s a Zarsk open dense subset of Spec(Z 0 ). We wll use ths remark n the theory of quantum groups where there s a natural subalgebra Z 0 whch appears n the pcture.

29 3. TWISTED POLYNOMIAL ALGEBRAS In ths chapter we ntroduce the man noton of quas polynomal algebras, or skew polynomal. Note that as the quantum envelopng algebras are the quantum verson of the unversal envelopng algebras of a Le algebra, we can thnk that twsted polynomal algebras are the quantum verson of the symmetrc algebra of a Le algebra. More detals on twsted polynomal algebras can be found, for examples, n [DCP93] or [Man91]. 3.1 Useful notaton and frst propertes Before gvng the defnton of twsted polynomal algebra, we want to ntroduce some notatons, all wll be useful n the sequel. Let fx an nvertble element q C dfferent from 1 and 1 so that the fracton 1 q q 1 s well defned. For all n Z, set [n] = qn q n q q 1 = qn 1 + q n q n+3 + q n+1. We have the followng relaton: [ n] = [n] [n + m] = q n [m] + q m [n] Observe that f q s not a root of unty then n Z, non zero, [n] 0. If q s a prmtve l th root of unty, wth l > 2, defne Now s easy to check that { l f l s odd e = l 2 f l s even.. Property. If q s a prmtve l root of unty then () [n] = 0 n 0 mod e () [n] l = [n]. We can now defne the q analogue of the factorals and of the bnomal coeffcents

30 3. Twsted polynomal algebras 24 Defnton For nteger 0 k n, set [0]! = 1, f k > 0, and [ n k [k]! = [1] [k], ] = [n]! [k]! [n k]!. Proposton If x and y are varables subject to the followng relaton xy = q 2 yx then, for n > 0, (x + y) n = n [ q k(n k) n k k=0 ] x k y n k. (3.1) Proof. We begn by statng the q analogue of the Pascal dentty: [ ] [ ] [ ] q k n + 1 = q n+1 n + 1 n + k k 1 k then by nducton on n the statement follow Corollary If q s a prmtve l root of unty,and xy = q 2 yx then Proof. Observe that [ e k (x + y) e = x e + y e. ] = 0 for all k such that 0 < k < e. Apply ths n the formula 3.1 and the statement fallow. We gve now some notatons that wll be useful n chapter 4 n order to defne the relatons of the quantum groups. Fx d N, for all n Z, set [n] d = qn q n q d q d. We can now extend the defntons of q-factoral and q-bnomal, n the followng way Defnton For nteger 0 k n, set [0]! d = 1, [k]! d = [1] d [k] d, f k > 0, and [ n k ] d = [n]! d [k]! d [n k]! d.

31 3. Twsted polynomal algebras Defnton Let A be an algebra over an algebrac closed feld k, and σ an automorphsm of A. Defnton A twsted dervaton of A relatve to σ s a lnear map D : A A such that: a, b A. D(ab) = D(a)b + σ(a)d(b) Defnton A twsted dervaton D s called nner, f t exsts n an element a A such that: and we denote t ad σ a. D(b) = ab σ(b)a Property. Let a A and σ be an automorphsm such that σ(a) = q 2 a where q s a scalar. Then (ad σ a) m (b) = m [ ( 1) j q j(m 1) m j j=0 ] a m j σ j (b)a j Corollary Under the hypothess of Property 3.2 we have: f q s a prmtve l-th root of 1. (ad σ a) e (x) = a e x σ e (x)a e σ Fx an automorphsm σ of A and a twsted dervaton D of A relatve to Defnton We defne the twsted dervaton algebra A σ,d [x] n the ndetermnate x to be the k-module A k k[x] thought as formal polynomals wth multplcatons defned by the rule: xa = σ(a)x + D(a). When D = 0, we wll call t twsted polynomal algebra and we denote t by A σ [x]. Let us notce that f a, b A and a s nvertble we can perform the change of varables y := ax + b and we see that A σ,d [x] = A σ,d [x],

32 3. Twsted polynomal algebras 26 for a sutable par (σ, D ). In order to see ths, t s better to make the formulas explct separately when b = 0 and when a = 1. In the frst case yc = axc = a(σ(c)x + D(c)) = a(σ(c))a 1 y + ad(c), and we see that the new automorphsm σ s the composton Ad(a)σ, and D = ad, where Ad(a)(x) = axa 1. In the case a = 1, we have yc = (x + b)c = σ(c)x + D(c) + bc = σ(c)y + D(c) + bc σ(c)b, so that σ = σ and D = D + ad σ b. Summarzng we have Proposton Changng σ, D to Ad(a)σ, ad (resp. to σ, D + ad σ b) does not change the twsted dervaton algebra up to somorphsm. Remark It s clear that f A has no zero dvsors, then the algebra A σ,d [x] and A σ [x, x 1 ] also have no zero dvsors. Gven a twsted polynomal algebra A σ,d [x], we can construct a natural fltraton gven by deg(p(x)) = n where p(x) = a n x n a 0, a n 0. The assocated graded algebra s clearly A σ [x]. Defnton We shall say that the algebra A σ [x] s a smple degeneraton of A σ,d [x]. Example Let A be an algebra over a feld k of characterstc 0, let x 1,...,x n be a set of generators of A and let Z 0 be a central subalgebra of A. For each = 1,...,n, denote by A the subalgebra of A generated by x 1,...,x, and let Z 0 = Z 0 A. We assume the followng three condtons hold for each = 1,...,n: 1. x x j = b j x j x + P j f > j, where b j k, P j A 1, 2. σ (x j ) = b j x j for j < defne an automorphsm of A D (x j ) = P j for j <. We obtan A = A 1 σ,d [x ] as twsted polynomal algebra. For every, we may consder the twsted polynomal algebra A defned by the relaton x x j = b j x j x for j <. We call ths the assocated quas polynomal algebra. Theorem Under the above assumptons, the quas polynomal algebra A = A n s obtaned from A by a sequence of smple degeneratons. Proof. See [DCP93] theorem 5.3, page 56.

33 3. Twsted polynomal algebras Representaton theory of twsted dervaton algebras We want to analyze some nterestng cases of the prevous constructons for whch the resultng algebras are fnte modules over ther centers and thus we can develop for them the noton of degree and a good representaton theory. Let us frst make a reducton, consder a fnte dmensonal semsmple algebra A over an algebrac closed feld k, let ke be the fxed ponts of the center of A under σ where the e are the central dempotents. We have D(e ) = D(e 2 ) = 2D(e )e, hence D(e ) = 0. It follows that, decomposng A = Ae, each component Ae s stable under σ and D and thus we have A σ,d [x] = (Ae ) σ,d [x]. Ths allows us to restrct our analyss to the case n whch 1 s the only central dempotent. The second reducton s descrbed by the followng: Lemma Consder the algebra A = k n wth σ the cyclc permutaton of the summands and let D be a twsted dervaton of ths algebra relatve to σ. Then D s an nner twsted dervaton. Proposton Let σ be the cyclc permutaton of the summand of the algebra k n. Then () k n [ σ x, x 1 ] s an Azumaya algebra of degree k over ts center k [x n, x n ]. () k n [ σ x, x 1 ] k[x n,x n ] k [ x, x 1] s the algebra of n n matrces over k [ x, x 1]. Proof. [DCP93] proposton 6.1, page 56. Assume now that A s semsmple and that σ nduces a cyclc permutaton of the central dempotents Lemma () A = M d (k) n. () Let D be a twsted dervaton of A relatve to σ. Then the par (σ, D) s equvalent to the par (σ, 0) where Proof. [DCP93] lemma 6.2, page 57. Summarzng we have σ (a 1, a 2,...,a n ) = (a n, a 1,...,a n 1 )

34 3. Twsted polynomal algebras 28 Proposton Let A be a fnte dmensonal semsmple algebra over an algebrac closed feld k. Let σ be an automorphsm of A whch nduces a cyclc permutaton pf order n of the central dempotents of A. Let D be a twsted dervaton of A relatve to σ. Then: A σ,d [x] = Md (k) k n σ [x], A σ,d [x, x 1 ] = Md (k) k n σ [x, x 1 ]. Ths last algebra s Azumaya of degree dk. We can now globalze the prevous constructon. Let A be a prme algebra (.e aab = 0, a, b A, mples that a = 0 or b = 0) over the feld k and let Z be the center of A. Then Z s a doman and A s torson free over Z. Assume that A s a fnte module over Z. Then A embeds n a fnte dmensonal central smple algebra Q(A) = A Z Q(Z), where Q(Z) s the rng of fracton of Z. If Q(Z) denotes the algebrac closure of Q(Z) we have that A Z Q(Z) s the full rng M d (Q(Z)). Then d s called the degree of A. Let σ be an automorphsm of the algebra A and let D be a twsted dervaton of A relatve to σ. Assume that (a) There s a subalgebra Z 0 of Z such that Z s fnte over Z 0. (b) D vanshes on Z 0 and σ restrcted to Z 0 s the dentty. These assumptons mply that σ restrcted to Z s an automorphsm of fnte order. Let d be the degree of A and let k be the order of σ on the center Z. Defnton If A s an order n a fnte dmensonal central smple algebra and (σ, D) satsfy the prevous condtons we shall say that the trple (A, σ, D) s fnte. Assume that the feld k has characterstc 0. Then Theorem Under the above assumptons the twsted polynomal algebra A σ,d [x] s an order n a central smple algebra of degree kd. Proof. [DCP93] theorem 6.3, page 58. Corollary Under the above assumptons, A σ,d [x] and A σ [x] have the same degree. Let A a prme algebra over a feld k of characterstc 0, let x 1,...,x n be a set of generators of A and let Z 0 a central subalgebra of A. For each = 1,...,k, denote by A the subalgebra of A generated by x 1,...,x k, and let Z 0 = Z 0 A. We assume, as n example 3.2.8, that the followng three condtons hold for each = 1,...,k:

35 3. Twsted polynomal algebras 29 (a) x x j = b j x j x + P j f > j, where b j k, P j A 1. (b) A s a fnte module over Z 0. (c) Formulas σ (x j ) = b j x j for j < defne an automorphsm of A 1 whch s the dentty on Z 1 0. Note that lettng D (x j ) = P j for j <, we obtan A = A 1 σ,d [x ], so that A s an terated twsted polynomal algebra. We may consder the twsted polynomal algebra A wth zero dervatons, so that the relatons are x x j = b j x j x for j <. we call ths the assocated twsted polynomal algebra. We can state the man theorem of ths secton. Theorem Under the above assumptons, the degree of A s equal to the degree of the assocated quas polynomal algebra A. Proof. By theorem A s obtaned from A wth a sequence of smple degeneratons, hence by corollary 3.3.7, t follows that they have the same degree. 3.4 Representaton theory of twsted polynomal algebras Let k a feld and 0 q k a gven element. Gven n n skew symmetrc matrx H = (h j ) over Z, we construct the twsted polynomal algebra k H [x 1,...,x n ]. Ths s the algebra on generators x 1,...,x n and the followng defnng relatons: x x j = q h j x j x (3.2) for, j = 1,...,n. It can be vewed as an terated twsted polynomal algebra wth respect to any order of the x s. Smlarly we can defne the twsted Laurent polynomal algebra k H [x 1, x 1 1,...,x n, x 1 n ]. Note that both algebras have no zero dvsors. To study ts spectrum we start wth a smple general lemma. Lemma If M s an rreducble A σ [x] module, then there are two possbltes: () x = 0, hence M s actually an A module. () x s nvertble, hence M s actually an A σ [x, x 1 ] module. Proof. It s clear that m x and ker x are submodules of M. Corollary In any rreducble k H [x 1,...,x n ] module, each element x s ether 0 or nvertble.