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}.
Friedrich Knop has given an example of a polynomial algebra A in characteristic two with an unusual SL_2 action. We exhibit a nontrivial class in H^1(SL_2,A), not detected by Frobenius kernels.
We study the existence of a Batalin-Vilkovisky differential on the complete cohomology ring of a Frobenius algebra.We construct a Batalin-Vilkovisky differential on the complete cohomology ring in the case of Frobenius algebras with diagonalizable Nakayama automorphisms.
Let G be a simple classical algebraic group over an algebraically closed field of positive characteristic. We describe the support variety of a simple G-module over the r-th Frobenius kernel of G, in terms of its calculation over the first Frobenius kernel. We then use this result to compute the block varieties of the Frobenius kernels of G.
We study cup product and cap product in Tate-Hochschild theory for a finite dimensional Frobenius algebra. We show that Tate-Hochschild cohomology ring equipped with cup product is isomorphic to singular Hochschild cohomology ring introduced by Wang. An application of cap product occurs in Tate-Hochschild duality; as in Tate (co)homology of a finite group, the cap product with the fundamental class of a finite dimensional Frobenius algebra provides certain duality result between Tate-Hochschild cohomology and homology groups. Moreover, we characterize minimal complete resolutions over a finite…
In his article "Unitary Representations and Complex Analysis", David Vogan gives a characterization of the continuous invariant Hermitian forms defined on the compactly supported sheaf cohomology groups of certain homogeneous analytic sheaves defined on open orbits in generalized flag manifolds. In the last section of the manuscript, Vogan raises a question about the possibility of a topological Frobenius reciprocity for these representations. In this article we give a specific version of the topological reciprocity in the regular, antidominant case and use it to study the existence of continu…
We verify that universal classes in the cohomology of $GL_N$ determine explicit cohomology classes of Frobenius kernels $G_{(r)}$ of various linear algebraic groups $G$ . We consider the relationship of $\varprojlim_r H^*(U_{(r)},k)$ to the rational cohomology $H^*(U,k)$ of many unipotent algebraic groups $U$. The second half of this paper investigates in detail the cohomology of Frobenius kernels $(U_3)_{(r)}$ of the Heisenberg group $U_3 \subset GL_3$.
The paper studies the dimensions of irreducible components of commuting varieties of (restricted) nilpotent $r$-tuples in a classical Lie algebra $\mathfrak{g}$ 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 $G_r$ for each $r\ge 1$, where $G$ is the simply connected, simple algebraic group such that $\text{Lie}(G)=\mathfrak{g}$. Explicit calculations for rank two groups are also presented.
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 ri…
Let $G$ be a simple simply connected group scheme defined over ${\mathbb F}_{p}$ and $k$ be an algebraically closed field of characteristic $p>0$. Moreover, let $B$ be a Borel subgroup of $G$ and $U$ be the unipotent radical of $B$. In this paper the authors compute the third cohomology group for $B$ and its Frobenius kernels, $B_{r}$, with coefficients in a one-dimensional representation. These computations hold with relatively mild restrictions on the characteristic of the field. As a consequence of our calculations, the third ordinary Lie algebra cohomology group for ${\mathfrak u}=\text{Li…
Let $(SL_2)_r$ be the $r$-th Frobenius kernels of the group scheme $SL_2$ defined over an algebraically field of characteristic $p>2$. In this paper we give for $r\ge 1$ a complete description of the cohomology groups for $(SL_2)_r$. We also prove that the reduced cohomology ring $\opH^\bullet((SL_2)_r,k)_{\red}$ is Cohen-Macaulay. Geometrically, we show for each $r\ge 1$ that the maximal ideal spectrum of the cohomology ring for $(SL_2)_r$ is homeomorphic to the fiber product $G\times_B\fraku^r$. Finally, we adapt our calculations to obtain analogous results for the cohomology of higher Frobe…
We adapt a construction due to Troesch to the category of strict polynomial superfunctors in order to construct complexes of injective objects whose cohomology is isomorphic to Frobenius twists of the (super)symmetric power functors. We apply these complexes to construct injective resolutions of the even and odd Frobenius twist functors, to investigate the structure of the Yoneda algebra of the Frobenius twist functor, and to compute other extension groups between strict polynomial superfunctors.
In this article we show that almost all blocks of all Frobenius-Lusztig kernels are of wild representation type extending results of Feldvoss andWitherspoon, who proved this result for the principal block of the zeroth Frobenius-Lusztig kernel. Furthermore we verify the conjecture that there are infinitely many Auslander-Reiten components for a finite dimensional algebra of infinite representation type for selfinjective algebras whose cohomology satisfies certain finiteness conditions.
In this paper we show that the tree class of a component of the stable Auslander-Reiten quiver of a Frobenius-Lusztig kernel is one of the three infinite Dynkin diagrams. For the special case of the small quantum group we show that the periodic components are homogeneous tubes and that the non-periodic components have shape Z[A_\infty] if the component contains a module for the infinite-dimensional quantum group.
Let $G$ be a simply-connected semisimple algebraic group scheme over an algebraically closed field of characteristic $p > 0$. Let $r \geq 1$ and set $q = p^r$. We show that if a rational $G$-module $M$ is projective over the $r$-th Frobenius kernel $G_r$ of $G$, then it is also projective when considered as a module for the finite subgroup $\Gfq$ of $\Fq$-rational points in $G$. This salvages a theorem of Lin and Nakano (\emph{Bull.\ London Math.\ Soc.} 39 (2007) 1019--1028). We also show that the corresponding statement need not hold when the group $G$ is replaced by the unipotent radical $U$…
We provide a new virtual description of the symmetric group action on the cohomology of ordered configuration space on SU_2 up to translations. We use this formula to prove the Moseley-Proudfoot-Young conjecture. As a consequence we obtain the graded Frobenius character of the Orlik-Terao algebra of type A_n.
Let $G$ be a simple, simply connected algebraic group over an algebraically closed field $k$ of characteristic $p>0$. Let $σ: G \rightarrow G$ be a surjective endomorphism of $G$ such that the fixed point set $G(σ)$ is a Suzuki or Ree group. Then, let $G_σ$ denote the scheme-theoretic kernel of $σ.$ Using methods of Jantzen and Bendel-Nakano-Pillen, we compute the $1$-cohomology for the Frobenius kernels with coefficients in the induced modules, $H^{1}(G_σ, H^{0}(λ))$, and the $1$-cohomology for the Frobenius kernels with coefficients in the simple modules, $H^{1}(G_σ, L(λ))$ for the Suzuki an…
We introduce a family $\mathbb{M}_{r;f,η}$ of infinitesimal supergroup schemes, which we call multiparameter supergroups, that generalize the infinitesimal Frobenius kernels $\mathbb{G}_{a(r)}$ of the additive group scheme $\mathbb{G}_{a}$. Then, following the approach of Suslin, Friedlander, and Bendel, we use functor cohomology to define characteristic extension classes for the general linear supergroup $GL_{m|n}$, and we calculate how these classes restrict along homomorphisms $ρ: \mathbb{M}_{r;f,η} \rightarrow GL_{m|n}.$ Finally, we apply our calculations to describe (up to a finite surjec…
For a finite dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite dimensional Hopf kernel and with image the universal enveloping algebra. In this article we define and describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases the Frobenius-Lusztig kernel is associated to a different Lie algebra than the initial …
This paper surveys results on the connections between the cohomology for algebraic groups, finite groups and Frobenius kernels that were presented at the Workshop and Summer School on Lie and Representation Theory at East China Normal University during July 2009.
Let k be an algebraically closed field of characteristic p>0 and let G be a connected reductive group over k. Let B be a Borel subgroup of G and let g and b be the Lie algebras of G and B. Denote the first Frobenius kernels of G and B by G_1 and B_1. Furthermore, denote the algebras of polynomial functions on G and g by k[G] and k[g], and similar for B and b. The group G acts on k[G] via the conjugation action and on k[g] via the adjoint action. Similarly, B acts on k[B] via the conjugation action and on k[b] via the adjoint action. We show that, under certain mild assumptions, the cohomology …
Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex $\mathcal{D}^*(A,A)$ computes the singular Hochschild cohomology of $A$. We construct a cyclic (or Calabi-Yau) $A$-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an $L$-infinity algebra structure extending the classical Gers…
