LECTURE 1: MOTIVATION

Transcription

1 LECTURE 1: MOTIVATION STEVEN SAM AND PETER TINGLEY 1. Towards quantum groups Let us begn by dscussng what quantum groups are, and why we mght want to study them. We wll start wth the related classcal objects. So, let G be a complex smple Le group and let g be ts Le algebra (tangent space at the dentty e G), wth Le bracet [, ]. A very frutful branch of mathematcs has been concerned wth studyng the representaton theory of G. Ths s often accomplshed by studyng the representaton theory of g. Recall that a representaton of g s a map φ: g End(V ) such that φ([x, y]) = φ(x)φ(y) φ(y)φ(x), whch s actually the same thng as a representaton of an assocatve algebra U(g), called the unversal envelopng algebra. Formally, U(g) s smply the assocatve algebra generated by g subject to the relatons [x, y] = xy yx. People sometmes say The quantum group U (g) s a deformaton of U(g). What mght ths mean? One should have U (g) = U(g)[[]] as a vector space, and the multplcaton n U (g) should be the same as the multplcaton n U(g) modulo. Ths s a bt nave though, snce: Theorem 1.1. Every deformaton of U(g) s trval,.e., always somorphc to U(g)[[]] as an algebra. Proof. Ths follows from the fact that the Hochschld cohomology group H 2 (U(g), U(g)) s 0. A better dea s to deform U(g) as a Hopf algebra Hopf algebras. What s the Hopf algebra structure on U(g)? It s the structure we need to defne tensor product and dualty of representatons. Let s frst loo at tensor product. We now that the acton of G on V W s gven by g(v w) = gv gw. Ths nduces an acton of g on V W, gven by X(v w) = Xv w + v Xw for x g. We thn of ths as a map : U(g) U(g) U(g) X X X (X g) and extend multplcatvely. Call ths comultplcaton. We must also consder the trval representaton V 0 where X g acts by 0 and scalars act as usual. Ths gves a map ε: U(g) V 0 that pcs out the constant term. Ths s the count. Fnally, we consder the fact that g has dual representatons: X g acts on V by (Xf)(v) = f( Xv), so ths gves a map S : U(g) U(g) X X, extended to all of U(g) as an algebra antautomorphsm. Ths gves the antpode. Recall that (V W ) = W V. One can show that ths forces S to be a coalgebra-antautomorphsm. Date: February 11,

2 2 STEVEN SAM AND PETER TINGLEY Defnton 1.2. A Hopf algebra over a feld F conssts of the data (H, m, ι,, ε, S), where H s a vector space, m: H H H ι: F H : H H H ε: H F S : H H such that (1) (H, m, ) s an algebra, (2) (H,, ɛ) s a coalgebra, (3) and ɛ are maps of algebras (usng the tensor product algebra structure on H H), (4) m, ι are maps of coalgebras, (5) S s an algebra-antautomorphsm and a coalgebra-antautomorphsm, and (6) the dagram commutes. H H S 1 H H H or 1 S ι ε m U(g) Remar 1.3. The commutatve dagram n the defnton of a Hopf algebra can be explaned n varous ntrnsc ways. For nstance, Davd Jordan ponted out the followng: Consder the convoluton product on End(H), where the product s defned by φ ψ(x) := m (φ ψ) (X). Then ι ɛ s the dentty element n ths convoluton algebra. The above dagrams mply that the antpode S s a two sded nverse for the dentty map 1 End(H) (note: ths s the dentty map of vector spaces, not the dentty n the convoluton algebra). In partcular, ths mples that any balgebra can have at most one antpode. Hence the antpode should not really be consdered extra data n a Hopf algebra, but rather exstence of an antpode s a condton on a balgebra Unqueness of deformatons. The followng unqueness statement s a strong motvaton for studyng quantum groups: f somethng you are nterested n can be deformed n a unque way, you wll almost certanly learn somethng nterestng by studyng that deformaton. Here rgd means there s a good noton of (both left and rght) duals, and monodal means there s a tensor product. Theorem 1.4 (see [2, Example 2.24]). There s a unque non-trval deformaton of the category of fnte-dmensonal modules of U(g) as a rgd monodal category. Proof. One shows that possble deformatons of a tensor category C are parameterzed by H 3 (C), and obstructons by H 4 (C), where H s Davydov Yetter cohomology. Here a tensor category s a rgd monodal category, wth the extra assumptons that 1 s smple and every object has fnte length, whch both hold of U(g)-rep. In the case of U(g)-rep, Davydov Yetter cohomology agrees wth Le algebra cohomology, so H 3 (U(g)-rep) = C and H 4 (U(g)-rep) = 0. See [2, 4]. In ths semnar, we wll more often dscuss deformatons of algebras than deformatons of categores. Here the stuaton s somewhat less clear, although we do have the followng result (the orgnal deformaton s due to Drnfel d and Jmbo, and the precse statement here can be found n [4]. Theorem 1.5. There exsts a non-trval deformaton of U(g) (as a Hopf algebra) whose category of representatons realzes the unque deformaton of Rep U(g).

3 LECTURE 1: MOTIVATION 3 Unfortunately, ths deformaton of U(g) s not unque, n the sense that there are non-somorphc Hopf algebras U (g) over C[[]] whch are deformatons of U(g). However, there are some unqueness statements one can mae: Theorem 1.6 (see [4, Theorem 5]). The deformaton of U(g) s unque up equvalences ncludng somorphsm, twstngs (as quas-hopf algebras that preserves the fact that algebras n queston are Hopf algebras ths one s qute nontrval), and change of parameter. The nterestng type of equvalence s twstng. The dea of ths operaton s smple: For any nvertble element J U(g) U(g), one can try to mae a new coproduct J := J J 1. The result s no longer a Hopf algebra, but f one s wllng to wor wth quas-hopf algebras (.e., Hopf algebras whch are not strctly assocatve, but rather have a non-trval assocator map), one can modfy the assocator such that the result s a new quas-hopf algebra. In some cases the result s n fact stll a strct Hopf algebra. We wll not dscuss the noton of twstng n detal; for our purposes, t s enough to now that there s a unqueness statement, but that t nvolves a qute non-trval noton of equvalence. There s also another way to get a unqueness clam, whch was n fact the frst unqueness result of Drnfel d. One ntroduces some extra structure, namely the Cartan nvoluton E F of U(g). The deformaton s unque f t also deforms ths new structure: Theorem 1.7 ([3], see also [4, Theorem 3]). U (g) s the unque (up to change of deformaton parameter ) Hopf-algebra deformaton of U(g) subject to the addtonal condtons (1) U h (g) contans a commutatve sub-hopf algebra C such that C/C U(h), where U(h) = H I s the unversal envelopng algebra of the Cartan subalgebra of g. (2) C s nvarant under an algebra nvoluton θ whch s also a coalgebra ant-automorphsm, and such that θ nduces the Cartan nvoluton E F on U(g) = U (g)/u (g) How deformatons can loo dfferent. Havng made some unqueness statements, let us consder how deformatons of U(g) can loo dfferent. The dea s that of deformaton quantzaton: gong bac to the orgnal group G, a deformaton of G should be a deformaton of the algebra of functons on G. The deformed product wll defne a Posson bracet by {f, g} := (f g g f)/ (mod ). If two deformatons lead to non-equvalent Posson bracets, one would thn they are non-equvalent. Transferrng the noton of a Posson bracet to U(g), one ends up wth the noton of a co-posson Hopf algebra. Ths s a Hopf algebra along wth an addtonal map δ : U(g) U(g) U(g), satsfyng some compatblty (see [1]). Any deformaton U h (g) of U(g) gves rse to a co-posson Hopf structure by δ(x) := (X) op (X). It turns out that one can fnd deformatons that gve rse to non-somorphc co-posson Hopf structures on U(g). So, from ths deformaton quantzaton pont of vew, one would conclude that the deformaton s non-unque. However, all such deformatons wll have equvalent categores of representatons How s the deformed category of U(g)-rep dfferent? As a category, t s not dfferent snce algebra deformatons of U(g) are trval. But as a -category (really, as a rgd monodal category), t has changed. To see ths, loo at V W and W V. These are always somorphc. Over U(g), the somorphsm s just the flp map. So, we get an acton of the symmetrc group S n on tensor product V 1 V n of representatons of (g) by swtchng orders. Over U (g), there s an somorphsm V W σ br (W V ), but t s not flp on the underlyng vector spaces. However, one can choose a natural famly of such somorphsm such that the acton of the brad group Br n on tensor products V 1 V n of representatons of U)q(g). In fact, one just needs to chec the brad relaton. See Fgure 1.

4 4 STEVEN SAM AND PETER TINGLEY U V W U V W W V U = W V U Fgure 1. The brad relaton. The crossng should be nterpreted as an somorphsm from a tensor product of two representatons to the tensor product n the other order. The appearance of the brad group suggests connectons to not theory. In fact, ths leads to the celebrated quantum group not nvarants, although there s stll some wor to do. 2. Towards crystals 2.1. Drnfeld Jmbo quantum groups. Let A = (a,j ) be the Cartan matrx for g. For U(g), have Chevalley generators E, F, and H wth relatons [H, E j ] = a,j E j [H, F j ] = a,j F j The comultplcaton s gven by [E, F j ] = δ,j H ad(e ),j E j = 0 ad(f ),j F j = 0. H H H E E E F F F. Let us not consder the antpode for now, as t s n fact determned by the other data. Let D be the dagonal matrx such that DA s symmetrc, and d = D,. Now U (g) has the same generators wth deformed relatons: q [H, E j ] = a,j E j [H, F j ] = a,j F j exp(d H ) exp( d H ) [E, F j ] = δ,j exp(d ) exp( d ),j [ ] 1 a,j E E j E,j = 0 =0 exp(d ),j [ ] 1 a,j F F j F,j = 0. =0 exp(d ) [ n Here s the q-bnomal coeffcent, whch s just a polynomal n q. ]

5 The comultplcaton s gven by LECTURE 1: MOTIVATION 5 H H H E E exp(d H ) + 1 E F F 1 + exp( d H ) F. Remar 2.1. Ths loos non-trval as a deformaton of an algebra structure, but as dscussed above t must be somorphc to the trval deformaton. But, n most cases, no explct somorphsm s nown! Stated another way, no one nows how to wrte down the coproduct structure f we don t change the multplcaton rules (although ths can be done for sl 2 ). We want to be able to specalze to numbers, but we can t because we have to worry about convergence. So we renormalze. Let q = exp(). For now, assume d = 1 for smplcty (smplylaced case). Also let K = exp(h ), whch we can thn of as q H. We get a new algebra wth generators E, F, K ±1, and relatons K K 1 = 1 K K j = K j K [H, E j ] = a,j E j [H, F j ] = a,j F j K K 1 [E, F j ] = δ,j q q 1,j =0,j =0 [ ] 1 a,j [ ] 1 a,j q q E E j E,j = 0 F F j F,j = 0. and the comultplcaton becomes H H H E E K + 1 E F F 1 + K 1 F. Now everythng s defned over Z[q, q 1, (q q 1 ) 1 ]. However, essentally because we were forced to ntroduce the generators K, ths s no longer a deformaton of U(g). However, t has many of the propertes of a deformaton. The goal of the theory of crystals s to draw a hghest weght representaton of U q (g) (n the lmt at q ) as a colored drected graph.

6 6 STEVEN SAM AND PETER TINGLEY Example 2.2. g = sl 3, λ = ω 1 + ω 2, so V (λ) s the adjont representaton. Weght space decomposton: F 2 F 1 F 2 F 1 F 2 F 1 F 2 F 1 The vertces of the graph should correspond to a bass of V (λ), so for nstance the should somehow be separated nto two vertces (snce ths weght space s two dmensonal) There are a few problems Problem 1: In the mages of F 1 and F 2 and the ernels of F 1 and F 2 are dstnct. So there are four dstngushed 1-dmensonal subspaces n, and t seems mpossble to separate t nto two vertces. Passng to the q lmt wll fx ths, but Problem 2: We currently cannot plug n q =, as we are worng over C[q, q 1, (q q 1 ) 1 ]. Problem 3: We would really le the reverse arrows to correspond to the operators E. However, one can have E F = [n] := (q n q 1 )/(q q 1 ), whch s not 1. All of these problems wll be fxed by ntroducng the Kashwara operators Ẽ and F, and the noton of a crystal lattce. Ths wll be done next wee. References [1] V. Char and A. Pressley, A gude to quantum groups, Cambrdge unversty press, Cambrdge (1994). [2] Damen Calaque and Pavel Etngof, Lectures on tensor categores, arxv: v4. [3] V. G. Drnfel d, Quas-Hopf algebras, Algebra Analz 1 (1989). Translaton n Lenngrad Math. J. 1 (1990), no. 6, [4] Steven Shnder, Deformaton Cohomology for balgebras and Quas-balgebras, Contemporary Math. 134 (1992),