arXiv subjects

Caroline B. Wright

Publications and source records attributed to Caroline B. Wright.

3 recordsLinked to original sources

arXiv2013

Bounding the dimensions of rational cohomology groups

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple simply-connected algebraic group over $k$ that is defined and split over the prime field $\mathbb{F}_p$. In this paper we investigate situations where the dimension of a rational cohomology group for $G$ can be bounded by a constant times the dimension of the coefficient module. We then demonstrate how our results can be applied to obtain effective bounds on the first cohomology of the symmetric group. We also show how, for finite Chevalley groups, our methods permit significant improvements over previous estimates for the dimensions of second cohomology groups.

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arXiv2010

Second cohomology groups for algebraic groups and their Frobenius kernels

Let $G$ be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic $p > 0$. Let $T$ be a maximal split torus in $G$, $B \supset T$ be a Borel subgroup of $G$ and $U$ its unipotent radical. Let $F: G \rightarrow G$ be the Frobenius morphism. For $r \geq 1$ define the Frobenius kernel, $G_r$, to be the kernel of $F$ iterated with itself $r$ times. Define $U_r$ (respectively $B_r$) to be the kernel of the Frobenius map restricted to $U$ (respectively $B$). Let $X(T)$ be the integral weight lattice and $X(T)_+$ be the dominant integral weights. The computations of particular importance are $\h^2(U_1,k)$, $\h^2(B_r,\la)$ for $\la \in X(T)$, $\h^2(G_r,H^0(\la))$ for $\la \in X(T)_+$, and $\h^2(B,\la)$ for $\la \in X(T)$. The above cohomology groups for the case when the field has characteristic 2 one computed in this paper. These computations complete the picture started by Bendel, Nakano, and Pillen for $p \geq 3$ \cite{BNP2}.

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arXiv2008

Cohomology of quantum groups: An analog of Kostant's Theorem

We prove the analog of Kostant's Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostant's cohomology formula holds for quantum groups at a generic parameter $q$, recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra $\mathfrak{g} = \mathfrak{sl}(n)$. We also show that Kostant's formula holds when $q$ is specialized to an $\ell$-th root of unity for odd $\ell \ge h-1$ (where $h$ is the Coxeter number of $\mathfrak{g}$) when the highest weight of the coefficient module lies in the lowest alcove. This can be regarded as an extension of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.

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