arXiv subjects

Christopher M. Drupieski

Publications and source records attributed to Christopher M. Drupieski.

25 records · Page 2Linked to original sources

arXiv2011

Cohomology rings for quantized enveloping algebras

We compute the structure of the cohomology ring for the quantized enveloping algebra (quantum group) $U_q$ associated to a finite-dimensional simple complex Lie algebra $\mathfrak{g}$. We show that the cohomology ring is generated as an exterior algebra by homogeneous elements in the same odd degrees as generate the cohomology ring for the Lie algebra $\mathfrak{g}$. Partial results are also obtained for the cohomology rings of the non-restricted quantum groups obtained from $U_q$ by specializing the parameter $q$ to a non-zero value $ε\in \mathbb{C}$.

math.QA↗

arXiv2011

Second cohomology for finite groups of Lie type

Let $G$ be a simple, simply-connected algebraic group defined over $\mathbb{F}_p$. Given a power $q = p^r$ of $p$, let $G(\mathbb{F}_q) \subset G$ be the subgroup of $\mathbb{F}_q$-rational points. Let $L(λ)$ be the simple rational $G$-module of highest weight $λ$. In this paper we establish sufficient criteria for the restriction map in second cohomology $H^2(G,L(λ)) \rightarrow H^2(G(\mathbb{F}_q),L(λ))$ to be an isomorphism. In particular, the restriction map is an isomorphism under very mild conditions on $p$ and $q$ provided $λ$ is less than or equal to a fundamental dominant weight. Even when the restriction map is not an isomorphism, we are often able to describe $H^2(G(\mathbb{F}_q),L(λ))$ in terms of rational cohomology for $G$. We apply our techniques to compute $H^2(G(\mathbb{F}_q),L(λ))$ in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type.

math.RT↗

arXiv2011

Cohomology for infinitesimal unipotent algebraic and quantum groups

In this paper we study the structure of cohomology spaces for the Frobenius kernels of unipotent and parabolic algebraic group schemes and of their quantum analogs. Given a simple algebraic group $G$, a parabolic subgroup $P_J$, and its unipotent radical $U_J$, we determine the ring structure of the cohomology ring $H^\bullet((U_J)_1,k)$. We also obtain new results on computing $H^\bullet((P_J)_1,L(λ))$ as an $L_J$-module where $L(λ)$ is a simple $G$-module with high weight $λ$ in the closure of the bottom $p$-alcove. Finally, we provide generalizations of all our results to the quantum situation.

math.RT↗

arXiv2011

First cohomology for finite groups of Lie type: simple modules with small dominant weights

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple, simply connected algebraic group defined over $\mathbb{F}_p$. Given $r \geq 1$, set $q=p^r$, and let $G(\mathbb{F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $H^1(G(\mathbb{F}_q),L(λ))$ where $L(λ)$ is the simple $G$-module of highest weight $λ$. Under certain very mild conditions on $p$ and $q$, we are able to completely describe the first cohomology group when $λ$ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $λ$ is a minimal nonzero dominant weight.

math.GR↗

arXiv2011

Representations and cohomology for Frobenius-Lusztig kernels

Let $U_ζ$ be the quantum group (Lusztig form) associated to the simple Lie algebra $\mathfrak{g}$, with parameter $ζ$ specialized to an $\ell$-th root of unity in a field of characteristic $p>0$. In this paper we study certain finite-dimensional normal Hopf subalgebras $U_ζ(G_r)$ of $U_ζ$, called Frobenius-Lusztig kernels, which generalize the Frobenius kernels $G_r$ of an algebraic group $G$. When $r=0$, the algebras studied here reduce to the small quantum group introduced by Lusztig. We classify the irreducible $U_ζ(G_r)$-modules and discuss their characters. We then study the cohomology rings for the Frobenius-Lusztig kernels and for certain nilpotent and Borel subalgebras corresponding to unipotent and Borel subgroups of $G$. We prove that the cohomology ring for the first Frobenius-Lusztig kernel is finitely-generated when $\g$ has type $A$ or $D$, and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general.

math.RT↗

arXiv2010

Differentiating the Weyl generic dimension formula and support varieties for quantum groups

The authors compute the support varieties of all irreducible modules for the small quantum group $u_ζ(\mathfrak{g})$, where $\mathfrak{g}$ is a simple complex Lie algebra, and $ζ$ is a primitive $\ell$-th root of unity with $\ell$ larger than the Coxeter number of $\mathfrak{g}$. The calculation employs the prior calculations and techniques of Ostrik and of Nakano--Parshall--Vella, as well as deep results involving the validity of the Lusztig character formula for quantum groups and the positivity of parabolic Kazhdan-Lusztig polynomials for the affine Weyl group. Analogous support variety calculations are provided for the first Frobenius kernel $G_1$ of a reductive algebraic group scheme $G$ defined over the prime field $\mathbb{F}_p$.

math.GR↗

arXiv2009

On injective modules and support varieties for the small quantum group

Let $u_ζ(g)$ denote the small quantum group associated to the simple complex Lie algebra $g$, with parameter $q$ specialized to a primitive $\ell$-th root of unity $ζ$ in the field $k$. Generalizing a result of Cline, Parshall and Scott, we show that if $M$ is a finite-dimensional $u_ζ(g)$-module admitting a compatible torus action, then the injectivity of $M$ as a module for $u_ζ(g)$ can be detected by the restriction of $M$ to certain root subalgebras of $u_ζ(g)$. If the characteristic of $k$ is positive, then this injectivity criterion also holds for the higher Frobenius--Lusztig kernels $U_ζ(G_r)$ of the quantized enveloping algebra $U_ζ(g)$. Now suppose that $M$ lifts to a $U_ζ(g)$-module. Using a new rank variety type result for the support varieties of $u_ζ(g)$, we prove that the injectivity of $M$ for $u_ζ(g)$ can be detected by the restriction of $M$ to a single root subalgebra.

math.RT↗