646 research outputs found
-supermodules with good and Weyl filtrations
The purpose of this paper is to prove necessary and sufficient criteria for a
-supermodule to have a good or Weryl filtration. We also introduce the
notion of a Steinberg supermodule analogous to the classical notion of
Steinberg module. We prove that the Steinberg supermodule inherits some
properties of the Steinberg module. Some new series of finite-dimensional
tilting supermodules are found
Some homological properties of in arbitrary characteristic
We show that Penkov's approach to a superanalog of Borel-Bott-Weil theorem
for over a field of zero characteristic can be extended for a
perfect field of arbitrary odd characteristic. We also prove some partial
version of Kempf's vanishing theorem and characteristic free formula for Euler
characteristic , where is a Borel subgroup of
Affine quotients of supergroups
In this article we consider sheaf quotients of affine superschemes by affine
supergroups that act on them freely. The necessary and sufficient conditions
for such quotients to be affine are given. If is an affine supergroup and
is its normal supersubgroup, then we prove that a dur -sheaf
is again affine supergroup. Additionally, if is
algebraic, then a -sheaf is also algebraic supergroup and it
coincides with . In particular, any normal supersubgroup
of an affine supergroup is faithfully exact.Comment: 31 page
Invariants of mixed representations of quivers I
We introduce a new concept of mixed representations of quivers that is a
generalization of ordinary representations of quivers and orthogonal
(symplectic) representations of symmetric quivers introduced recently by
Derksen and Weyman. We describe the generating invariants of mixed
representations of quivers (First Fundamental Theorem) and prove additional
results that allow us to describe the defining relations between them in the
second article.Comment: 42 page
Invariants of mixed representations of quivers II : defining relations and applications
In the previous article we introduced the new concept of mixed
representations of quivers and described the generators of their algebras of
invariants. In this article we describe the defining relations of these
algebras. Some applications for the invariants of orthogonal or symplectic
groups acting on several matrices are given.Comment: 27 page
Solvable, reductive and quasireductive supergroups
This work was inspired by two natural questions. The first question is when
Lie(G')=Lie(G)', where G is a connected algebraic supergroup defined over a
field of characteristic zero. The second question is whether the unipotent
radical of any normal supersubgroup H of G coincides with the intersection of H
and G_u, where G_u is the unipotent radical of G. Both questions have
affirmative answers in the category of algebraic groups (in the second case one
has to assume additionally that G and H are reduced whenever char K >0).
Surprisingly, using the technique of Harish-Chandra superpairs and a complete
description of an action of an algebraic supergroup on an abelian supergroups
by supergroup automorphisms we found out rather simple counterexamples to both
questions. Besides, the second counterexample shows that the reductivity of G
does not imply that G_{ev} has even finite unipotent radical. On the other
hand, if G_{ev} is reductive, then it is easy to see that G_u is finite (odd)
supergroup. In other words, the reductivity of an algebraic supergroup does not
correspond to the reductivity of its largest even subgroup in contrast to such
properties as unipotency or solvability. In the last section of our article we
describe reductive algebraic supergroups in terms of sandwich pairs and give
necessary and sufficient conditions for an algebraic supergroup to be
quasireductive. The last result complements the recent Serganova's description
of quasireductive supergroups in terms of structural properties of their Lie
superalgebras. Our approach is focused on the normal subgroup structure
Central elements in the distribution algebra of a general linear supergroup and supersymmetric elements
In this paper we investigate the image of the center of the distribution
algebra of the general linear supergroup over a ground field of
positive characteristic under the Harish-Chandra morphism
obtained by the restriction of the natural map . We
define supersymmetric elements in and show that each image for
is supersymmetric. The central part of the paper is devoted to a
description of a minimal set of generators of the algebra of supersymmetric
elements over Frobenius kernels
Quotient sheaves of algebraic supergroups are superschemes
To generalize some fundamental results on group schemes to the super context,
we study the quotient sheaf of an algebraic supergroup by
its closed supersubgroup , in arbitrary characteristic 2. Our main
theorem states that is a Noetherian superscheme. This together
with derived results give positive answers to interesting questions posed by J.
Brundan.Comment: Revised the manuscript, improving the exposition and correcting
typos, 40 pages; the final version accepted for publicatio
Pseudocompact algebras and highest weight categories
We develop a new approach to highest weight categories with good
(and cogood) posets of weights via pseudocompact algebras by introducing
ascending (and descending) quasi-hereditary pseudocompact algebras. For
admitting a Chevalley duality, we define and investigate tilting
modules and Ringel duals of the corresponding pseudocompact algebras. Finally,
we illustrate all these concepts on an explicit example of the general linear
supergroup .Comment: 43 page
On the notion of Krull super-dimension
We introduce the notion of Krull super-dimension of a super-commutative
super-ring. This notion is used to describe regular super-rings and calculate
Krull super-dimensions of completions of super-rings. Moreover, we use this
notion to introduce the notion of super-dimension of any irreducible
superscheme of finite type. Finally, we describe nonsingular superschemes in
terms of sheaves of K\"{a}hler superdifferentials.Comment: 30 page
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