Related papers: On functor points of affine supergroups

Recently, Venkatesh extended the category equivalence between affine algebraic groups and Harish-Chandra pairs, which was proved by the author in the supersymmetric context, to the situation of the Verlinde category in positive…

This paper dwells upon two aspects of affine supergroup theory, investigating the links among them. First, I discuss the "splitting" properties of affine supergroups, i.e. special kinds of factorizations they may admit - either globally, or…

We show basic results on super-manifolds and super Lie groups over a complete field of characteristic $\ne 2$, extensively using Hopf-algebraic techniques. The main results are two theorems. The first main theorem shows a category…

We prove a category equivalence between algebraic supergroups and Harish-Chandra pairs over a commutative ring which is $2$-torsion free. The result is applied to re-construct the Chevalley $\mathbb{Z}$-supergroups constructed by Fioresi…

We develop a general theory of algebraic group superschemes, which are not necessarily affine. Our key result is a category equivalence between those group superschemes and Harish-Chandra pairs, which generalizes the result known for affine…

Over an arbitrary field of characteristic $\ne 2$, we define the notion of Harish-Chandra pairs, and prove that the category of those pairs is anti-equivalent to the category of algebraic affine supergroup schemes. The result is applied to…

Let G=U(p,q) and K=U(p)xU(q). In arXiv:0801.1530, the authors construct a functor from the category of Harish-Chandra modules for the pair (G,K) to the category of representations of the degenerate affine Hecke algebra of type B_n. In this…

We define exact functors from categories of Harish-Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of…

It is known that there exists a natural functor $\Phi$ from Lie supergroups to super Harish-Chandra pairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair, yielding an equivalence of…

We give the definitions of affine algebraic supervariety and affine algebraic group through the functor of points and we relate them to the other definitions present in the literature. We study in detail the algebraic supergroup $SL(m|n)$…

Torsors under affine groups are generalized in the super context by super-torsors under affine super-groups. We investigate those super-torsors by using Hopf-algebra language and techniques. It is explicitly shown, under suitable…

We construct a new class of algebras resembling enveloping algebras and generalizing orthogonal Gelfand-Zeitlin algebras and rational Galois algebras studied by [EMV,FuZ,RZ,Har]. The algebras are defined via a geometric realization in terms…

We generalize to the super context, the known fact that if an affine algebraic group $G$ over a commutative ring $k$ acts freely (in an appropriate sense) on an affine scheme $X$ over $k$, then the dur sheaf $X\tilde{\tilde{/}}G$ of…

We define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan [KK] on the structure of Verma modules in the…

Let G=GL(N), K=GL(p)xGL(q), where p+q=N, and n be a positive integer. We construct a functor from the category of Harish-Chandra modules for the pair (G,K) to the category of representations of the degenerate affine Hecke algebra of type…

For a given finite dimensional Hopf algebra $H$ we describe the set of all equivalence classes of cocycle deformations of $H$ as an affine variety, using methods of geometric invariant theory. We show how our results specialize to the…

It is well-known that the Harish-Chandra transform, $f\mapsto\mathcal{H}f,$ is a topological isomorphism of the spherical (Schwartz) convolution algebra $\mathcal{C}^{p}(G//K)$ (where $K$ is a maximal compact subgroup of any arbitrarily…

We prove a variety results on tensor product factorizations of finite dimensional Hopf algebras (more generally Hopf algebras satisfying chain conditions in suitable braided categories). The results are analogs of well-known results on…

We impose a rather unknown algebraic structure called a `hyperstructure' to the underlying space of an affine algebraic group scheme. This algebraic structure generalizes the classical group structure and is canonically defined by the…

The Harish-Chandra Fourier transform, $f\mapsto\mathcal{H}f,$ is a linear topological algebra isomorphism of the spherical (Schwartz) convolution algebra $\mathcal{C}^{p}(G//K)$ (where $K$ is a maximal compact subgroup of any arbitrarily…