By Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, Cornelius Pillen
Permit ? be a fancy th root of team spirit for a wierd integer >1 . For any complicated easy Lie algebra g , permit u ? =u ? (g) be the linked "small" quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra that are realised as a subalgebra of the Lusztig (divided strength) quantum enveloping algebra U ? and as a quotient algebra of the De Concini-Kac quantum enveloping algebra U ? . It performs an enormous function within the illustration theories of either U ? and U ? in a fashion analogous to that performed by means of the limited enveloping algebra u of a reductive team G in optimistic attribute p with recognize to its distribution and enveloping algebras. usually, little is understood in regards to the illustration thought of quantum teams (resp., algebraic teams) whilst l (resp., p ) is smaller than the Coxeter quantity h of the underlying root approach. for instance, Lusztig's conjecture about the characters of the rational irreducible G -modules stipulates that p=h . the most bring about this paper presents a shockingly uniform resolution for the cohomology algebra H (u ? ,C) of the small quantum staff
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Extra info for Cohomology for quantum groups via the geometry of the nullcone
Then ηC = (lm , s ) (lm −1 , l − 1, s + 1) if m is even (s even), if m is odd (s odd). In the ﬁrst case, when m and s are both even, let b be the number of distinct even nonzero parts. So b = 1 and by [CM, p. 92] the component group A(CηC ) ∼ = (Z/2Z)b−1 is trivial. Here the component group is deﬁned as CG (x)/CG (x)o for any x ∈ CJ , where G is the adjoint group P Sp2n . Thus, CG (x) is generated by CG (x)o and the center of G, which is contained in PJ . Thus, CG (x) ⊆ CPJ (x) here also. Now suppose that m and s are odd.
1) holds, namely, that CG (x) ⊆ PJ , where x ∈ uJ . 1. Case 1: Φ has type An . Without loss of generality we can assume that G = GLn (k). The centralizer is connected so CG (x) = CG (x)0 . 4] CG (x)0 ⊆ PJ . Case 2: Φ has type Bn . Let N = 2n + 1 and write N = lm + s where 0 ≤ s ≤ l − 1 and m > 0. 1. Set η = (lm , s ) and recall that N (Φ0 ) = OηB where ηB is the B-collapse of η. For type Bn we have ηB = (lm , s ) (lm , s − 1, 1) if s is odd or s = 0, if s is even and s = 0. 3]. For x ∈ uJ (with N (Φ0 ) = G · uJ ), let Q be the parabolic subgroup obtained from a standard triple in g involving x.
Let ⎧ ⎪ (1, −1, . . , 1, −1, 0, . . , 0) for Xn = An , Cn , or Dn ; ⎪ ⎪ ⎨ r times z times δX = ⎪ (1, −1, . . , 1, −1 , 0, . . , 0, 1) for Xn = Bn . ⎪ ⎪ ⎩ r times z times As above, one can verify that j , δX > roots in J. Also, we conclude that ⎧ r(r−1) ⎪ ⎨ 2 |X| = r 2 ⎪ ⎩ r(r − 1) and 0 for all ⎧ ⎪ ⎨rz |X| = 2rz ⎪ ⎩ 2rz + r + z Hence, maxλ λ, δX j that correspond to simple Xn = An ; X n = Bn , C n ; Xn = Dn . Xn = An ; Xn = Cn , Dn ; X n = Bn . ⎧ r(r + z − 1) ⎪ ⎪ ⎪ ⎨(r + 1 )(2r + 2z) 2 = 2|X| + |X| = ⎪r(2r + 2z) ⎪ ⎪ ⎩ r(2(r + z − 1)) Xn Xn Xn Xn = An ; = Bn ; = Cn ; = Dn .
Cohomology for quantum groups via the geometry of the nullcone by Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, Cornelius Pillen