# 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

Enable ? be a posh th root of solidarity for a wierd integer >1 . For any advanced basic 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 a huge 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 staff G in optimistic attribute p with appreciate to its distribution and enveloping algebras. mostly, little is understood in regards to the illustration thought of quantum teams (resp., algebraic teams) while 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 staff

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**Example text**

If η ∈ P (m) for some positive integer m, call η an -partition. Given a pair (η, σ) in P (n), suppose that ηi = σi , 1 ≤ i ≤ r, and ηj = σj , 1 ≤ j ≤ s. In other words, the ﬁrst r rows and ﬁrst s columns in the Young diagrams associated to η and σ are the same. Suppose that (η1 , · · · , ηr ) is an -partition, and consider the pair (η, σ) in P(−1)s (m) obtained by lettting η = (ηr+1 , · · · ), σ = (σs+1 , · · · ), and m = ηr+1 + · · · ηn . , σ = σ and η = η. Then a pair (η, σ) is called irreducible if (η, σ) → (η, σ) implies that (η, σ) = (η, σ).

0, −1) r1 times r2 times in the orthonormal basis describing Φ in n+1-dimensional Euclidean space E [Bo, p. 250]. Note that the ﬁrst r1 -groupings of (1, 0, . . , 0, −1) have m + 1-components, while the last r2 -groupings of (1, 0, . . , 0, −1) have m-components. 2)). 1), let λ be a weight of Λiζ,J . Then λ is a sum of distinct positive roots not in Φ+ J . If β is a positive root then β, δA = 0, ±1, ±2 (i. e. β = i − j with i < j). Set A[t] = {β ∈ Φ+ \Φ+ J | β, δA = t}. 36 C. P. BENDEL, D.

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 . On the other hand, by using the expression of ρ for the classical Lie algebras in terms of the -basis (see [GW, p.