Extensions for Frobenius kernels

Bendel, Christopher P.; Nakano, Daniel K.; Pillen, Cornelius

doi:10.1016/j.jalgebra.2003.04.003

Cited by 24 publications

(27 citation statements)

References 11 publications

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“…The results above demonstrate that H 2 (G r , V ) (−r) indeed has a good filtration for p 3, V = H 0 (λ), λ a dominant weight, and arbitrary rank. In [3], it was verified that Donkin's conjecture holds for H 1 (G r , H 0 (λ)) for all primes and all ranks. An interesting question would be to see if H m (G r , H 0 (λ)) admits a good filtration in higher ranks for all m 0 and all primes.…”

Section: Lemma (A) Suppose P = 3 and φ Is Of Type G 2 Then As B-momentioning

confidence: 92%

“…Jantzen computed H 1 (G 1 , H 0 (λ)) for all λ and all primes, and his computations showed that there is a generic answer for p > 3 which is much less restrictive than the p > h condition. In recent work [3], the authors extended his results to compute H 1 (G r , H 0 (λ)) for all p, λ, and r. The main goal of this paper is to demonstrate how to compute H 2 (G r , H 0 (λ)) for all p 3, λ ∈ X(T ) + and r 1. It is interesting to note that our results do not rely on Frobenius splittings, but rather strongly on our previous calculations for H 1 (G r , H 0 (λ)) (e.g., see Theorem 6.1).…”

Section: 1mentioning

confidence: 95%

“…if Φ of type C n , n 3, and w = s α n−1 s α n , ω n−2 − ω n−1 + ω n if Φ of type C n , n 3, and w = s α n s α n−2 , ω 2 − ω 3 if Φ of type F 4 , and w = s α 2 s α 1 , 2ω 3 − ω 4 if Φ of type F 4 , and w = s α 3 s α 2 , ω 2 − ω 3 + ω 4 if Φ of type F 4 , and w = s α 2 s α 4 , 2ω 1 if Φ of type G 2 , and w = s α 1 s α 2 , ω 2 − ω 1 if Φ of type G 2 , and w = s α 2 s α 1 .…”

Section: 1mentioning

confidence: 99%

“…For the structure of the induced modules, we argue as in [3,Proposition 3.4] using [3,Lemma 3.3]. To summarize, if N is an indecomposable B-module with factors σ 1 , σ 2 and R i ind G B σ j = 0 for i 1 and each j , then ind G B N has a filtration with factors H 0 (σ 1 ), H 0 (σ 2 ) where either (or both) factor is omitted if σ j is not dominant.…”

Section: G R -Cohomologymentioning

confidence: 99%

See 3 more Smart Citations

Second cohomology groups for Frobenius kernels and related structures

Bendel

Nakano

Pillen

2007

Advances in Mathematics

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Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p. Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B-modules. We also consider the standard induced modules obtained by inducing a simple B-module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie(U ) with coefficients in k.

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Section: Lemma (A) Suppose P = 3 and φ Is Of Type G 2 Then As B-momentioning

confidence: 92%

Section: 1mentioning

confidence: 95%

Section: 1mentioning

confidence: 99%

Section: G R -Cohomologymentioning

confidence: 99%

See 2 more Smart Citations

Second cohomology groups for Frobenius kernels and related structures

Bendel

Nakano

Pillen

2007

Advances in Mathematics

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“…For higher values of r, the problem on computing the cohomology of G r turns out to be complicated. Bendel, Nakano, and Pillen have made some progress in the two papers [BNP1] [BNP2] where they explicitly calculated the first and second degrees of H i (G r , H 0 (λ)) with H 0 (λ) = ind G B (λ) the induced module of the highest weight λ. In the special case when G = SL 2 , the author computed H i (G r , H 0 (λ)) for each i, r ≥ 1 and dominant weight λ [Ngo1].…”

mentioning

confidence: 99%

On nilpotent commuting varieties and cohomology of Frobenius kernels

Ngo

2015

Journal of Algebra

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The paper studies the dimensions of irreducible components of commuting varieties of (restricted) nilpotent r-tuples in a classical Lie algebra g defined over an algebraically closed field k. As applications, we obtain some new results on the structure of the (even) cohomology ring of Frobenius kernels Gr for each r ≥ 1, where G is the simply connected, simple algebraic group such that Lie(G) = g. Explicit calculations for rank two groups are also presented.

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Bounding Cohomology for Finite Groups and Frobenius Kernels

Bendel

Nakano

Parshall

et al. 2015

Algebr Represent Theor

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Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ : G → G be a strict endomorphism (i. e., the subgroup G(σ) of σ-fixed points is finite). Also, let Gσ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the degree m cohomology H m (G(σ), L) of any irreducible kG(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. A similar result holds for the degree m cohomology of Gσ. These bounds are actually established for the degree m extension groups Ext m G(σ) (L, L ′ ) between irreducible kG(σ)-modules L, L ′ , with again a similar result holding for Gσ. In these Ext m results, of interest in their own right, the bounds depend also on L, or, more precisely, on length of the p-adic expansion of the highest weight associated to L. All bounds are, nevertheless, independent of the characteristic p. These results extend earlier work of Parshall and Scott for rational representations of algebraic groups G.We also show that one can find bounds independent of the prime for the Cartan invariants of G(σ) and Gσ, and even for the lengths of the underlying PIMs. These bounds, which depend only on the root system of G and the "height" of σ, provide in a strong way an affirmative answer to a question of Hiss, for the special case of finite groups G(σ) of Lie type in the defining characteristic.

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Extensions for Frobenius kernels

Cited by 24 publications

References 11 publications

Second cohomology groups for Frobenius kernels and related structures

Second cohomology groups for Frobenius kernels and related structures

On nilpotent commuting varieties and cohomology of Frobenius kernels

Bounding Cohomology for Finite Groups and Frobenius Kernels

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