Crystallizing the q -Analogue of Universal Enveloping Algebras Masaki Kashiwara Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa Sakyo-ku, Kyoto 606, Japan §0. Introduction The notion of the ^-analogue of universal enveloping algebras is introduced independently by Drinfeld and Jimbo in their study of exactly solvable models in statistical mechanics. This algebra Uq(§) contains a parameter q and it becomes the universal enveloping algebra when q = 1. This parameter is the one of temparature in the context of statistical mechanics and q = 0 corresponds to the absolute temparature zero. Therefore, we can expect that the theory of L^(cj) will be simplified at q = 0. We call the study of Uq(o) &t q = 0 crystallization. Of course, we cannot deform Uq(^) at q = 0. However, we can construct the bases of representations of Uq($) at "q = 0", and the l/g(g)-module structure is described by combinatrics among them. This gives a purely combinatorial description of the tensor category of Uq(^)-modules (and hence £/(g)-modules). § 1. Crystal Bases 1.1 Definition of Uq(ç^) Let us consider the following data: (1.1) (1.2) (1.3) (1.4) (1.5) a finite-dimensional Q-vector space t, an index set J, a linearly independent subset {a f ;/ e 1} oft* and a subset {h\\i e 1} oft, an inner product (, ) on t* and a lattice P of t*. We assume that they satisfy the following conditions : (1.6) (1.7) {(hj,ocj)} is a generalized Cartan matrix (i.e. (h/,a/) = 2, (fy,a/) G Z<# for i ì j and (hf, Uj) = 0 <=> (hJ9 a,-) = 0), (a/,aOGZ>o, (1-8) M = (1.9) » af e P and 1y eP* = {h€t; (h,P) a Z}. Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990 (ë) The Mathematical Society of Japan. 1991 792 Masaki Kashiwara The Q(g)-algebra Uq(o) is then the algebra generated by the symbols e^fi (i G /) and qh (hG P*) with the following fundamental relations: (1.10) (1.11) (1.12) (1.13) qh = 1 for h = 0 and qh+h' = qhqh\ qheiq-h = qMet and qhfiq~h = q~{KcCi)U [eu fj] = Stjitt - if 1 )/(ft - ft"1) w h e r e * = 1 { a u a ù a n d U = iMht> E n ( - l ) ^ ! n ) ^ ^ " n ) = 0 and Zn(-l)nfln)fjftn) = 0 for Ì + j and ft = l-(hi,ocj). Here we used the notations [n]t = (q" — q[n)/(qi — qi'1), [n]j! = FCUiM* ef = enJ[ri\i\, fln) = f?/[n]i\. We understand ef = /f° = 0 forn < 0. an d 1.2 Operators 2/ and /,For a t/g(g)-module M and X G P, we set Mx = {u G M; tzw = # ^ w } and call it the weight space of weight X. We say that M is integrable if M = ©xepMx and if M is a union of finitely dimensional sub-Uq(qt)-modules for any i. Here Uq(Qt) is the subalgebra of Uq(o) generated by eÌ9 ft and tt. By the representation theory of Uq(sl2)9 any element u of Mx is uniquely written in the form (1.14) u = Ysfi^Un (resp. = ^eì^vn) where un G kerej n MA+nai and un = 0 except when n + (/ij,/l) > 0 and n > 0 (resp. u„ G ker/j n M^-,,«, and vn = 0 except when n > (/if, A) and n > 0). We define the endomorphisms 2; and /; on M by and Then ^ and /,- satisfy the relations symmetric to this: e(U = X e i ü « an d 1.3 Crystal Base Let A be the subring of Q(q) consisting of the rational functions regular at q = 0. Let M be an integrable Uq(s)-module. Definition 1.1. A crystal base of M is a pair (L,B) satisfying the following conditions. (1.15) (1.16) (1.17) (1.18) (1.19) (1.20) L is a free sub-y4-module of M such that M = Q(q) ®A L. B is a base of the Q-vector space L/qL. L = ®xepLx and ß = UXepBx where L* = L n M / i , f t = 5 n (Lx/qLj). êjL <= L and /jL c: L. Hence ëf and fi operate also on L/qL. ëiB c 5 u {0} and ftB aBU {0}. For ft, ft' eB,b' = fib if and only if ft = efft'. Crystallizing the #-Analogue of Universal Enveloping Algebras 793 For a crystal base (L,B), the crystal graph is the oriented colored (by i G J) graph with B as the set of vertices and ft —> ft' if ft' = fib, The crystal graph describes completely the action of e\ and fi on B U {0}. For b G B9 we set ß/(ft)=max{fc>0;gfft^O} and ^•(ft)=max{/c>0;/fft^0} ( For ft G Bx, we have (hi,X) = Cpi(b)^8i(b). Example 1.2. When g ;= s/2, Uq(sl2) is the algebra generated by e,f,t9t x with the commutation relation tet~l = q2e,tft~1 = g - 2 / and [e,/] = (t — t~l)/(q — g - 1 ). Then any irreducible (/ + 1)-dimensional representation is isomorphic to Vj = ©i=oQ(#)w/c with fuk = [k+ 1]M/C+I, euk = [1 + 1 - /c]w/c_i and tUk = qJ~2hUh Then L = ©À/fc and ß = {Mft;0<fc<!}c L/qL. Then (L,J3) is a crystal base of V\. Its crystal graph is Mo —* wi —*• * ' ' —• w/_i —• U\. 1.4 Stability by Tensor Product Let us define the comultiplication of t/g(g) by A(qh) = qh®q\ A(ej) = ei®tf1 + 1 ® eÌ9 A(fi)=fi®l + ti®fi. Then £/g(g) has a Hopf algebra structure with A as a comultiplication. By 4, the tensor product of two Uq(§)-modules has a structure of l/9(g)-module. Theorem 1 (Stability by ®). Let M\ and M2 be two integrable Uq(Q)-modu1es and let (Lj9Bj) be a crystal base ofMj (j = 1,2). Set L = L\®AL2 and B ^= {b\®b2e. L/qLibjeBj}. (i) Then (L,B) is a crystal base of M\ ®Q(q) M2. (ii) For bj G Bj (j = 1,2), we have LöI®/,'&2 if (pm) <,Si(h 794 Masaki Kashiwara 1.5 Existence and Uniqueness We set P+ = {XeP; (huX) > 0}. For X G P+, let V(X) be the irreducible integrable Uq (Q) -module generated by a vector ux of weight X satisfying e^ux = 0. Then V(Q = Uq(Q)/( ^(C/ g (g) ei + Uq(Q)fl+M) + £ / \ i Uq(Q)(q" ~ <Z<M>)) • hep* / Let L(X) be the smallest sub-^-module of M such that L(X) contains ux and L(X) is stable by the fi. Let 5(2) be the subset of L(X)/qL(X) consisting of the non-zero vectors of the form/^ '-fuux. Theorem 2 (Existence). (L(X),B(X)) is a crystal base ofV(X). Let G[nt be the category of integrable Uq(Q)-modules such that there exists a finite subset F of P such that M = ®xeF+Q_Mx. Here, ß - = X Z<oO/- Then $int is a semi-simple abelian category and any irreducible object is isomorphic to V(X) for some X e P+ ([L], [R]). Theorem 3 (Uniqueness). Let (L,B) be a crystal base of an object M in G{nt. Then there exists an isomorphism M = @jV(Xj) by which (L,B) is isomorphic to ®j(L(Xj),B(Xj)). Combining the Theorems 1, 2 and 3, we can describe completely the tensor category Gint. First note that the crystal graph of V (X) is connected. Hence the irreducible decomposition of an object in G\nt is equivalent to the connected component decomposition of the crystal graph. Then Theorem 1 tells us the crystal graph of tensor products Example. Take the case g = 5/3 (see §4 for the notation). Let {Ai} be the dual base of {hi}. Then the decompositions V(Ai) ® V(A{) = V(A2) 0 V(2Ai) and V(Ai) ® V(A2) = V(A1 + A2) © V(0) are described as follows. 1 — > • 2 >• 1 • 1 •1 . ' i i U ' i L'i' i L . 2 > • — 2 Is '1 .''Ul- Crystallizing the g-Analogue of Universal Enveloping Algebras §2. 795 Crystal Base of [/"(g) 2.1 Operators 2/ and // on t/^~(g) Let £/~(g) be the subalgebra of Uq(o) generated by the //. Then U~(Q) has the unique endomorphisms e\ and e" such that [e,,P] = (î,ef (P) - tr'e{(P))/(«, - ft"1) for any P e t/-(g). Then e| and /,• satisfy the commutation relations : (2.1) ^,fj = q;{f"A,)fM5i}. Here we consider /;- as the left multiplication operator. Then any element w of £/~(g) can be uniquely written as w= Z^" )w » with 4 W "=°- We define the endomorphisms e,- and // of t/~(g) by Kz/.-")«»)=2:/.-""1)"« Then e,-/,- — 1 holds. Let L(oo) be the smallest sub-y4-module of Uq (g) that contains 1 and that is stable by /,-. Let J5(oo) be the subset of L(oo)/qL(ao) consisting of the vectors of the form f^ • • •//, • 1. Then (2?(oo),L(oo)) has a similar property to crystal bases. Theorem 4. (i) e,-L(oo) c L(oo) and //L(oo) c L(oo). (ii) ë,jB(oo) c5(oo)U {0} and fiB(oo) c fl(oo). (in) B(oo) is a base of L(oo)/qL(co), (iv) Ifbe B(co) satisfies ejb ^ 0, then ft = f{e\b. The relation of (L(oo), 5(oo)) and (L(X),B(X)) is given by the following theorem. Theorem 5. For X G P + , let %x : L^~(g) -> V(X) be the U~(§)-linear homomorphism sending 1 to ux. (i) 7CAL(OO) = L(X). Hence %x induces the homomorphism %x '• L(oo)/'gL(oo) —> L(X)/qL(X). (ii) {ft G B(có);nx(b) ^ 0} is isomorphic to B(X) by %x. (Hi) fi o%x = nxo fu (iv) For ft G B(oo) such that lïx(b) ^ 0, eî%x(b) = nxfâb). Masaki Kashiwara 796 § 3. Global Crystal Base Let U~(Q)Z be the sub-Zfogr1]-algebra of Uq($) generated by the, f\n\ For X G P+, we set Vx(X) = U~(Q)Z ' ux. Let — be the ring automorphism of U~(Q) such that q = q~l and ft = ft. This induces the automorphism — of V(X) by Pux = Pux for P G C/-(g). Theorem 6. (i) (Q ® C/~(g)z) nL(oo) nL(oo)~ —> L(oo)/qL(oo) is an isomorphism. (ii) For any X G P+, (Q®FzW)rïL(/l)nL(yl)~ -> L(X)/qL(X) is an isomorphism. Let G denote the inverse of these isomorphisms. Then we have G(b) = G(b) for ft G L(X)/qL(X) with A G P + U {oo}. Moreover, we have G(b)ux = G\%xb) for any ft G L(oo)/qL(oo). Theorem 7. Por anj; n > 0 and i, iïu;(z)nu;(Q)z= 0 zfe^Mft), ffV(X)nvz(X)= 0 z^Mft). bG/;'B(A)\{0} W e call G(b) global crystal base. It is proven by Lusztig ([L3]) that the canonical bases introduced by himself in [L2] in the case An, Dn and En coincide with the global canonical bases introduced here. §4. Example This example is a joint work with T. Nakashima. Let us take g = sln. Hence I = {1, • • •, n - 1}, (ah aj) = 1, -1/2,0 according to i = j , \i - j \ = 1, \i - j \ > 0. Let Ai G t* be the dual base of hi and take ®ZAi as P. Then the crystal graph of B(A\) is n-1 m-4a n— >H For A = Xf=i A (1 <• h < • • • ^ ìN) we embed B(X) into B^O®'1 ®ß(^i)®''2 ® • • • by u\ !->• f [T]® • • • ®[jT] ) ® ( [T]® ' " ' ®\h\ ) ® " ' "• Then B(X) is parametrized by ( mu ® Wl2 mlh )•( ™21 m2ï-2 ) in 5(yii)®^ ® • • •. W e associate t o this b a s e t h e Y o u n g d i a g r a m Y(X) with a positive integer in each box as follows Crystallizing the ^-Analogue of Universal Enveloping Algebras mm m2\ 797 win • m2h mu mN}iN Here Y(X) is the Young diagram with the columns with length i\, • • -,/#• Theorem. By this correspondence, B(X) is equal to the set of semi-standard tableaux with shape Y(X) (i.e. {my} satisfies my < myj ifi > i and my < my if j < f). §5. Remarks The notion of crystal base is introduced in [Ki] under the form dual to the one given here. Theorem 2,3 in the case of AmBn,Cn and Dn and Theorem 1 are proven there. In [M], the crystal graph of basic representation of Uq(sln) is given, The results here have been announced in [K2]. Independently, Lusztig introduced the notion of canonical bases in the case >4„,D„,E„ ([L2]) and he showed that they coincide with global canonical bases ([L3]). References [D] Drinfeld, V. G.: Hopf algebra and the Yang-Baxter equation. Sov. Math. Dokl. 32 (1985) 254-258 [J] Jimbo, M.: A ^-difference analogue of U(§) and the Yang-Baxter equation. Lett. Math. Phys. 10 (1985) 63-69 [K] Kashiwara, M.: 1. Crystallizing the ^-analogue of universal enveloping algebras. Commun. Math. Phys. 133 (1990) 249-260 2. Bases crystallines. C. R. Acad. Sci. Paris 311 (1990) 277-280 [L] Lusztig, G.: 1. On quantum groups. J. Algebra (1990) 2. Canonical bases arising from quantized enveloping algebra. J. AMS 3. Canonical bases arising from quantized enveloping algebra, II. Preprint [M] Misra, K. C, Miwa, T.: Crystal bases for basic representation of Uq(sl(n)). Commun. Math. Phys. 134 (1990) 79-88 [R] Rosso, M. : Analogue de la forme de Killing et du théorème d'Harish-Chandra pour les groupes quantiques. Ann. Sci. Ec. Norm. Sup. 23 (1990) 445-467