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