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

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

Allow ? be a posh th root of solidarity for a strange integer >1 . For any advanced easy Lie algebra g , permit u ? =u ? (g) be the linked "small" quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which are realised as a subalgebra of the Lusztig (divided energy) quantum enveloping algebra U ? and as a quotient algebra of the De Concini-Kac quantum enveloping algebra U ? . It performs an incredible position within the illustration theories of either U ? and U ? in a fashion analogous to that performed via the limited enveloping algebra u of a reductive staff G in optimistic attribute p with appreciate to its distribution and enveloping algebras. usually, little is understood concerning the illustration conception of quantum teams (resp., algebraic teams) whilst l (resp., p ) is smaller than the Coxeter quantity h of the underlying root process. for instance, Lusztig's conjecture in regards to the characters of the rational irreducible G -modules stipulates that p=h . the most lead to this paper offers an incredibly uniform solution for the cohomology algebra H (u ? ,C) of the small quantum team

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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.

1 − 2, 2 + 4} 3. 1. Richardson orbits. Let G be a complex, simple and simply connected algebraic group with root system Φ. For J ⊂ Π, the (standard) parabolic subgroup PJ = LJ UJ ⊇ B of G has a dense (open) orbit CJ in the Lie algebra uJ of UJ under the adjoint action of PJ . In particular, if J = ∅, then LJ = T , and PJ = B, the Borel subgroup corresponding to Φ− . The corresponding Richardson orbit CJ is the G-orbit G · x for any x ∈ CJ . Therefore, when J = ∅, CJ is the regular or principal nilpotent orbit.

J. PARSHALL, AND C. 10). The module M is therefore isomorphic to a “Steinberg” type module on Uζ (lJ ) that remains irreducible if viewed as a uζ (lJ )-module. The highest weight of M is −w0,J (w · 0) and the lowest weight of M is −w · 0. Note that the module M does depend on the choice of w. 2. 7, the Ad-action induces an action of Uζ (pJ ) (and hence also of uζ (pJ )) on Uζ (uJ ). This defines an action of Uζ (pJ ) on the cohomology H• (Uζ (uJ ), C). 8. 1 below, we determine Homuζ (lJ ) ((indUζζ (b)J w · 0)∗ , H• (Uζ (uJ ), C)).

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