Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall,'s Cohomology for quantum groups via the geometry of the PDF

By Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, Cornelius Pillen

ISBN-10: 0821891758

ISBN-13: 9780821891759

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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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[2]| = r 2 ⎪ ⎩ r(r − 1) and 0 for all ⎧ ⎪ ⎨rz |X[1]| = 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[2]| + |X[1]| = ⎪r(2r + 2z) ⎪ ⎪ ⎩ r(2(r + z − 1)) Xn Xn Xn Xn = An ; = Bn ; = Cn ; = Dn .

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Cohomology for quantum groups via the geometry of the nullcone by Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, Cornelius Pillen


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