WILLIAM CHIN AND IAN M. MUSSON. ABSTRACT. We determine the ...... N. JACOBSON, Lie algebras (Dover, New York, 1979). 11. A. JOSEPH and G.
THE CORADICAL FILTRATION FOR QUANTIZED ENVELOPING ALGEBRAS WILLIAM CHIN AND IAN M. MUSSON
ABSTRACT We determine the coradical filtration on a quantized enveloping algebra t/9(g) = U defined over Q(q). As applications, we determine the biideals of U and describe the group of Hopf algebra automorphisms of U.
1. Introduction 1.1. Let g be a finite dimensional complex simple Lie algebra of rank n with Cartan matrix (aw). We can find integers dte{\,2,3} (1 ^ / ^ n) such that (d ( a y ) is symmetric. Given integers M,N,d^ 0, we set
[M+Nl
_ [M+N]d
0Am/Am_v Let U+ (respectively U~) be the subalgebra of U generated by El,...,En (respectively Fl,...,Fn). Then as an algebra, grU is the smash product of U~ ® U+ by the commutative group algebra Ao. As a coalgebra, grU is isomorphic to U. In [11], the filtration {AJ on U is used to show that U is a domain. 1.6. As a first application of Theorem A, we show, in Subsection 4.1 that any biideal of U contains the ideal generated by all the E{ and F{, and so has finite codimension (Theorem C). Since for any Lie algebra I) the Hopf ideals of U(\)) have the form rt£/(!)) for n an ideal of I), this result is the 'quantum analogue' of the fact that g is simple. 1.7. In addition, in Subsection 4.3 we determine the Hopf algebra maps between quantized enveloping algebras (Theorem D). We denote the group of Hopf algebra automorphisms of a Hopf algebra / / b y Aut Hopf //. We show that AutHopf UQ(Q) is the semidirect product of a group of 'diagonal' automorphisms by the group of automorphisms of the Dynkin diagram. Since the Hopf algebra automorphisms of U(Q) preserve the set g of primitive elements of £/(g), we have AutHopf U(Q) = Autg. By [4, Chapter VIII, 5.3, Corollaire 1] or [10, Chapter IX, Exercise 9], Autg is the semidirect product of the group of elementary automorphisms of g by the group of automorphisms of the Dynkin diagram. Thus the group of Hopf algebra automorphisms is smaller in the quantum case because the connected component of the identity is only a torus. 1.8. We introduce some notation which will be used throughout this paper. Let R be the root system associated to g, and a l s ...,a f l a base for the simple roots. Let P = ZR be the root lattice and ( , ) the bilinear form on P determined by (a,, OLj) = dxair For 1 ^ / < n, let a" = 2a i /(a f , a() (so that (a",a ; ) = ai} as in [13, 1.2]), and lets,, be the simple reflection defined bys,.(/?) = fi— (a",/?)af. Write f-Ffor the Weyl group generated by s1,...,sn. For 1 ^ i:^ n, let Tt be the /^-algebra automorphism of U defined in [13, Theorem 3.1]. If we Wand w = 5( st ...s{ is a reduced expression for w, then the automorphism Tw= T; Tt...T( depends only on w and not on the reduced expression for it [13, Theorem 3.2]. Let w0 = st ... st be a reduced expression for the longest element of W. Following [13, Appendix], we order the positive roots R+ by setting
Fix 7, and set 0 = ft, w{0) = stx... stj_. We define Efi = T^EJ and /J = Tw{P)(Ft). If 0 = where /? = /?'+/?" and 0 ^ /?' ^ /?, all terms of the form (fi+yt, — yt) must cancel mod(#— 1)/, as do those of the form ()> 0. The induction follows easily from cases 1-3. COROLLARY.
If Tlp(w) = 0, then ft is a simple root.
Proof. We may assume that the Cartan matrix is not of type G2. We need the following observation about root systems which follows easily from the rank 2 case. Suppose that n,r\ are roots, with n ^ ±rj and {fx,rf) # 0. Then if /J. and n have equal lengths, we have (fi,rjv) =(rj,nv) = ± 1, while if rj is short and /* is long, we have (M, 1V) = ± 2 and O7,//) = ± 1. For / = 1, ...,t, we have m,. = -(fii+1, avm). Since all the fit are conjugate to a under W, they all have equal lengths. It follows from these remarks that there exists a} e {1,2} such that mt e {0, + a}} for all / with x(J) = j . Here we are also using our assumption that /?(+1 is not simple, so fii+l # aT(I). For j = \,...,n, set hj = # {/1 T(/) = j , mi = a}} - # {/1 T(/) = 7, mt = - at). Then
In addition, h^w) = Yjl-i^r ^ foU°ws we obtain the result from the lemma.
tnat
^ ^ ) > 0 if ^ is not simple, and then
3.7 Proof of Theorems A andB. As in the proof of the lemma in 1.4, C = UQ(b~) and D = Ug(b+) are pointed Hopf algebras with coradical A = kG. Now set
t=l
t-1
m
m
C(w) = C(l) ,
£(m) = D(\)
for m ^ 1,
and C(0) = D(0) = A. Since the relations defining Ug(b~) and (/9(b+) are homogeneous in the Ft and Et, this makes Cand D graded bialgebras. Thus U = C®ADisa. graded coalgebra with U(m) = y[jC(i)®AD(m-i). Since A1 = C(l) + £>(l) + i4, we have A™ = Y,Zo ^(0- Thus to prove Theorem B (and Theorem A), it suffices to show that U is coradically graded, using the lemma in 2.2. By the lemma in 2.3, it is enough to show that C and D are coradically graded. To show that D is coradically graded, we need to show that any skew primitive element of D is contained in Z)(0) + D(l). We make a further reduction. Write D = UQ® U+, and for a e P, let Da denote the homogeneous term of degree a in the P-grading (see 1.8). Suppose that xeDa is (g, 1) primitive. Set ® D*-1a' + a " = a > a " ^ °)' a n d f ° r a n v group-like /? e Uo, Da, =
{yeDa\A(y)=y®hmodIa}.
WILLIAM CHIN AND IAN M. MUSSON
59
Note that Da can be written uniquely in the form 0 = cr0a, where U' is a nontrivial map of Hopf algebras, (j) is injective by Theorem C. For each /en, 0 ^ ) is a nontrivial ( 0 ( ^ ) , 1) primitive in U'. Hence by Theorem A, there exists r = a(i) e m such that {K^ = K'r, and we have
for suitable at, bt, ct e k. It is clear that a: n -* m is injective. Now if a(i) = r, a(j) = s, then applying to the equation A ^ ^ X r 1 — qaith root of unity and of semisimple groups in characteristic/?: independence of/?', Aste'risque 220 (1994). 3. I. BOCA, 'The coradical nitration of f/,(sl (2)) at roots of unity', Comm. Algebra 22 (1994) 5769-5776. 4. N. BOURBAKI, Groupes et algebres de Lie (Hermann, Paris, 1975). 5. W. CHIN, R. G. LARSON, I. M. MUSSON and J. TOWBER, 'The first two terms of the coradical filtration
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Department of Mathematics DePaul University Chicago, IL 60614 USA
Department of Mathematical Sciences University of Wisconsin at Milwaukee, Milwaukee, WI 53201 USA
