HOPF ALGEBRAS WITH TRACE AND CLEBSCH-GORDAN COEFFICIENTS. 1. Recollections and the problem

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1 HOPF ALGEBRAS WITH TRACE AND CLEBSCH-GORDAN COEFFICIENTS CORRADO DE CONCINI Abstract. In ths lecture I shall report on some jont work wth Proces, Reshetkhn and Rosso [1]. 1. Recollectons and the problem Let G be a smply connected semsmple algebrac group over C. g =Le G. T G a maxmal torus, t =Le T, R the root system, R + postve roots, = {α 1,..., α n } the smple roots, A = (a,j ), the Cartan matrx, D =dag(d 1,..., d n ) the dagonal matrx wth DA symmetrc. B T B the correspondng Borel subgroup and ts opposte. Fx l odd (and prme wth 3 f there are G 2 components). Let ε be a prmtve l root of 1. The quantzed envelopng algebra s the C-algebra U ε (g) generated by elements {E 1,... E n }, {F 1,... F n }, K λ, λ X (T ) the character group of T, wth relatons: K λ K µ = K λ+µ K λ E K 1 λ K λ F K 1 λ = ε ˇα,λ E = ε ˇα,λ F E F j = F j E j, [E, F ] = K α Kα 1 [ 1 a j k [ 1 a j k ] ] ε ε ε ε 1 ε = ε d E 1 k aj E j E k = 0, j F 1 k aj F j F k = 0, j [ ] m [m] = ε! h [m h], [h] ε![h] ε! ε! = [h] ε... [2] ε [1] ε and [h] ε = εh ε h ε ε. acts on generators by 1 ε K µ = K µ K µ, E = E 1 + K α E, F = F Kα F The man pecularty of beng at roots of unty s that U ε (g) has a very bg center Z. Indeed, U ε (g) s a fnte Z-module. Z s the coordnate rng of an algebrac varety X of dmenson equal to dm g. So: 1) Every rreducble U ε (g)-module s fnte dmensonal. 1

2 2 CORRADO DE CONCINI 2) If Ûε(g) denotes the set of rreducble U ε (g)-modules, takng central characters we get a surjectve map γ : Ûε(g) X. Problem 1. Gven two rreducble U ε (g)-modules, V, W, descrbe the composton factors of V W and ther multplctes. Consder the sub Hopf algebra U ε (b) U ε (g) generated by the K λ and E 1,... E n. Problem 2. Gven an rreducble U ε (g)-module, V decompose ts restrcton to U ε (b). Notce that as Problem 2, also Problem 1 s a problem of branchng. In fact t conssts of decomposng the restrcton of the rreducble U ε (g) U ε (g)-module V W to the subalgebra (U ε (g)). To explan our results recall that Z contans a sub Hopf algebra Z 0 such that U ε (g) s a free Z 0 module of rank l dmg. Z 0 s the coordnate rng of an algebrac group H (the Posson dual of G) whch s the kernel of the homomorphsm p : B B T defned by p((b, b )) = p + (b)p (b ) 1 where p ± are the quotents modulo the unpotent radcals. The ncluson Z 0 Z gves a map π : X H. Theorem 1. There s a non empty Zarsk open set V H H such that f V and W are two rreducble representatons of U ε (g), such that f (h 1, h 2 ) = (πγ(v ), πγ(w )) V, as a U ε (g) module, wth m = l R+ rkg. V W = U (πγ) 1 (h 1h 2)U m As for problem 2, Z 0 + = Z 0 U ε (b) s a sub Hopf algebra of U ε (b). Z 0 + s the coordnate rng of B and the ncluson Z 0 + Z 0 nduces the homomorphsm µ : H B gven by the projecton on the second factor. Agan, takng central character we get a map γ : Ûε(b) B. Theorem 2. There s a non empty Zarsk open set V H such that f V s an rreducble representatons of U ε (g), such that f h = πγ(v ) V, as a U ε (b) module, V = U (γ ) 1 (µ(h))u m wth m = l ( R+ s)/2 where s s the number of orbts of w 0 on the set of smple roots. The man content of these theorems s that n both cases, at least genercally, the multplctes are unformly dstrbuted among the varous rreducble components. 2. Cayley-Hamlton algebras. The results we have just stated follow from a careful study of the centers of the algebras nvolved, by applyng the theory of Cayley-Hamlton algebras whch we now brefly recall. A trace on a algebra R over a feld k (here =C) s a 1-ary operaton t : R R such that (1) t s k lnear. (Ths mples that t(r) s an k subspace) (2) t(a)b = b t(a), a, b R. (Ths mples that t(r) s n the center of R),

3 HOPF ALGEBRAS WITH TRACE AND CLEBSCH-GORDAN COEFFICIENTS 3 (3) t(ab) = t(ba), a, b R. (Ths mples that t s 0 on the space of commutators [R, R],) (4) t(t(a)b) = t(a)t(b), a, b R. (Ths mples that t(r) s an subalgebra, called the trace algebra, and that t s also t(r) lnear). The most famous example s that of the algebra M n of n n matrces wth the usual trace tr. Algebras wth trace form a category. Gven (R 1, t 1 ), (R 2, t 2 ) a morphsm φ from (R 1, t 1 ) to (R 2, t 2 ) s an algebra homomorphsm such that φt 1 (a) = t 2 φ(a) for each x R 1. A morphsm φ : (R, t) (M n, tr) s called a representaton. (M n, tr) have the followng extra propertes: (1) tr(1) = n (2) Gven A M n, set p (n) a (x) = det (xi A), the characterstc polynomal, then we get the Cayley-Hamlton dentty p (n) a (a) = 0 Remark that p (n) a (x) has coeffcents whch are unversal polynomals n tr(a),..., tr(a n ). For example for n = 2, p (2) a (x) = x 2 tr(a)x (tr(a)2 tr(a 2 )). Thus gven (R, t) we can consder p (n) a (x) for every a R as a polynomal wth coeffcents n the trace algebra and defne Defnton 3. An algebra wth trace (A, t) s a n-cayley Hamlton algebra (or brefly an n-ch algebra) f 1) t(1) = n 2) p (n) a (a) = 0 for all a R. Remark that f ρ : (R, t) (M r, tr) s a representaton of a n C-H algebra, then r = n snce r = tr(1) = tr(ρ(1)) = ρ(t(1)) = ρ(n) = n. If (R, t) n-ch algebra and h s a postve nteger, (R, ht) s a hn-ch algebra and f V s a (n dmensonal) (R, t)-module, V h s a (R, ht)-module. The category of C-H algebras clearly contans free objects. The free n-ch algebra F n x 1,..., x m on m generators s the algebra generated freely by the x s and by traces of monomals modulo the deal generated by evaluatng the n th Cayley Hamlton dentty. 3. Representatons of Cayley-Hamlton algebras. The followng results descrbes the free n-ch algebra F n x 1,..., x m. Consder the algebra C n (ξ 1,..., ξ m ) of GL(n)-equvarant polynomal maps of matrces M n (k) m M n (k), ξ (A 1,..., A m ) = A, wth trace t(ψ) = tr ψ. We have a unversal map F n x 1,..., x m C n (ξ 1,..., ξ m ) Theorem 4. (Proces) (1) The unversal map F n x 1,..., x m C n (ξ 1,..., ξ m )

4 4 CORRADO DE CONCINI s an somorphsm. (2) The trace algebra T n (ξ 1,..., ξ m ) of the algebra C n (ξ 1,..., ξ m ) s the algebra of nvarants of m tuples of matrces and, f m > 1, t s the center of C n (ξ 1,..., ξ m ). (3) C n (ξ 1,..., ξ m ) s a fnte T n (ξ 1,..., ξ m ) module. The above theorem tells us n partcular that f (R, t) s a n-ch algebra whch s fntely generated also ts trace algebra A s fntely generated and R s a fnte A-module. It follows that every rreducble representaton of R (as an algebra) s fnte dmensonal. A s a rng of functons on an affne algebrac varety X, so f R denotes the set of rreducble representatons of R, we get, by takng central characters, a surjectve map γ : R X. The ponts of X tself are a modul space for a class of representatons. Indeed Theorem 5. Gven a pont x X, there s a unque semsmple, trace compatble, representaton W x of the form W x = V γ 1 (x)v m V, and each semsmple, trace compatble, representaton of R s of ths form. In fact there s a Zarsk open set (possbly empty), called the unramfed locus, n X such that f m x A s the maxmal deal n A, R/m x R s semsmple and so W x s the unque representaton lyng over x whch s compatble wth the trace. If A s the center of R and R satsfes sutable condtons, for example t s a doman, there s a non empty Zarsk open set for whch W x s rreducble. In general however, the determnaton of γ 1 (x) and of the multplctes m V s a very hard problem. If we consder (R, ht), wth h a postve nteger, then the semsmple, trace compatble, representaton correspondng to x X wth respect to the new trace s just W h x. 4. Reduced trace In our example of quantzed envelopng algebra we have no trace, so f we want to use the results about the CH algebras we need to fnd ways to canoncally ntroduce a trace. Here s a way of dong ths under some assumptons. A prme algebra R s an algebra n whch the product of two non-zero deals s non-zero. Let R be a prme algebra, assume that A s a subalgebra of the center and R s an A module of fnte type. Then: 1) A s an ntegral doman. 2) R s a torson free module. So f F s the feld of fractons of A, then R R A F and S := R A F s a fnte dmensonal smple algebra somorphc to M k (D) wth D a fnte dmensonal dvson rng. Denote by Z the center of S and of D. Then dm Z D = h 2.

5 HOPF ALGEBRAS WITH TRACE AND CLEBSCH-GORDAN COEFFICIENTS 5 If Z s an algebrac closure of Z, M k (D) Z Z = M hk (Z). Settng p := [Z : F ] = dm F Z we also have S F Z = M k (D) F Z = M hk (Z) p The number n := hkp s called degree of S over F. Consder the F lnear operator a L : S S, a L (b) := ab. We defne the reduced trace t S/F (a) = 1 hk tr(al ). Ths theorem tells us that, provded A s ntegrally closed, we have constructed a nce trace: Theorem 6. If S = R A F as before and A s ntegrally closed we have that the reduced trace t S/F maps R nto A, so we wll denote by t R/A the nduced trace. The algebras R and S wth ther reduced trace are n Cayley Hamlton algebras. Even more mportant for us s the fact that, n the stuaton above, the nce trace s essentally unque. Indeed, Theorem 7. Under the same crcumstances of Theorem 6, f τ : R A s any trace for whch R s an m Cayley Hamlton algebra then there s a postve nteger r for whch: m = rn, τ = r t R/A Notce that n partcular, ths mples that f x s a pont n X and W x s the unque semsmple representaton above x, compatble wth t R/A then Wx r s the unque semsmple representaton above x compatble wth τ. 5. Compatblty Snce our fnal goal s to analyze the restrcton of certan representaton of an algebra R to a subalgebra, what I am gong to explan now s how the reduced trace of R relatve to a subrng A restrcts to a subalgebra. Assume that we have two prme algebras R 1 R 2 wth two central subrngs A 1 R 1, A 2 R 2, A 1 A 2 such that R 1 s a fnte A 1 module, R 2 s a fnte A 2 module and A 1, A 2 are ntegrally closed. We have the reduced traces t R1/A 1 and t R2/A 2 for whch both algebras are Cayley-Hamlton. We need a crteron ensurng that the restrcton of t R2/A 2 to R 1 s a multple of t R1/A 1 (n ths case we say that they are compatble). The crteron s the followng, Theorem 8. Denote by Z 1 the center of R 1. If the algebra Z 1 A1 A 2 s a doman there s a postve nteger r such that r t R1/A 1 = t R2/A 2 on R Back to quantum algebras In ths fnal secton I am gong to explan how the theory of Cayley Hamlton algebras can be appled to the quantum group stuaton descrbed n Secton 1. As a frst step we need to show that quantzed envelopng algebras have reduced traces. Ths s not hard. One knows that both U ε (g) and U ε (b) are domans and

6 6 CORRADO DE CONCINI hence prme and that ther centers Z, Z are ntegrally closed. Z 0 and Z + 0 are clearly ntegrally closed, so we have reduced traces t Uε(g)/Z 0 t Uε(g)/Z t U ε(g) 2 /Z 2 t U ε(b)/z +. 0 Next we have to compare them. We start wth Problem 1 and the ncluson (U ε (g)) U ε (g) U ε (g). Unfortunately we have that (Z) s not contaned n Z Z. On the other hand the fact that Z 0 s a sub Hopf algebra means that (Z 0 ) Z 0 Z 0 Z Z. So, we may try to compare t Uε(g) 2 /Z 2 wth t U ε(g)/z 0. In order to use Theorem 8 we need to show that the varety X defned as the fber product φ X X q π π π X X H H m H. wth m : H H H the group multplcaton, s rreducble. Let us recall that X s obtaned as the fber product X π H p T/W l ρ G σ G//G = T/W. Where for (h, k) H, ρ((h, k)) = hk 1, σ s the quotent modulo the adjont acton, W s the Weyl group and the map l s the map nduced by the homomorphsm t t l of the torus T on W -nvarants. Puttng the two dagrams together we obtan X as the fber product From ths one proves: X q X X pφ σ ρ m π π T/W l T/W. Step 1. X s a complete ntersecton over X X whch s Cohen-Macaulay so t s Cohen-Macaulay. Step 2. normal. X s non sngular n codmenson 1 so n partcular t s reduced and Step 3. X s connected. Ths s the hardest step and one needs to use a varaton of Stenberg secton for the quotent G G//G (G//G denoted the categorcal quotent defned usng nvarant polynomal functons ). Usng the rreducblty of X we deduce that the restrcton to (Uε (g)) of t Uε(g) 2 /Z 2 s mt U ε(g)/z 0 wth m = l R+ rkg and Theorem 1 follows. The open set V H H conssts of the set of pars ((h 1, k 1 ), (h 2, k 2 )) wth h 1 k1 1, h 2k2 1, h 1h 2 k2 1 k 1 1 all regular semsmple. The soluton to Problem 2 s qute smlar wth the extra dffculty that one has to compute the center of U ε (b) frst snce ths has not been determned before. We have seen that Z 0 + s the coordnate rng of the group B. For a domnant weght λ, consder the rreducble G-module V λ of hghest weght λ. Take a hghest weght vector v V λ, a hghest weght vector φ and a lowest vector ψ n Vλ. Defne the

7 HOPF ALGEBRAS WITH TRACE AND CLEBSCH-GORDAN COEFFICIENTS 7 functon f λ (b) = φ(bv)ψ(bv) for b B. These functons generate a subrng Z n Z + 0, somorphc to the polynomal rng C[f ω 1,..., f ωr ], ω the fundamental weghts. Embed Z nto the polynomal rng Z = C[f 1/l ω 1,..., f 1/l ω r ]. Z s the rng of nvarants n Z under an obvous acton of the group Z/lZ whch we can consder as the group T l of l-torson ponts n T. Set Γ = {t T l t w0 = t 1 }. and Z + 1 = (Z ) Γ. Theorem 9. The center of the algebra U ε (b) s somorphc to Z + = Z + 0 Z Z + 1. Once we know the center of U ε (b), n order to solve Problem 2 we need to compare t Uε(g)/Z and t Uε(b)/Z +. In order to use Theorem 8 we need to show that 0 the fber product Ỹ Ỹ Y q X π H l p B wth Y =Spec Z +, s a rreducble varety. Ths s done as n the prevous case. After these facts have been establshed the proof of Theorem 2 follows rather easly. 7. Fnal remarks and problems The technque explaned can be used n other cases. One example s the so called quantzed functon algebra F ε [G]. In ths case the role played by H s played by G and we have a projecton π : F ε [G] G. Theorem 10. There s a non empty Zarsk open set V G G such that f V and W are two rreducble representatons of F ε [G], such that f (g 1, g 2 ) = (π(v ), π(w )) V, as a F ε [G] module, wth m = l R+ rkg. V W = U (π) 1 (g 1g 2)U m References [1] De Concn, C.; Proces, C.; Reshetkhn, N.; Rosso, M. Hopf algebras wth trace and representatons. Invent. Math. 161 (2005), no. 1, 1 44.