arXiv subjects
Publications and source records attributed to A. N. Zubkov.
We describe certain almost-simple algebraic supergroups over an algebraically closed field of odd or zero characteristic. In addition to supergroups with simple Lie superalgebras from Kac's theorem, we construct new supergroups whose Lie superalgebra is either non-simple or simple but is not part of Kac's list.
We develop a fragment of the theory of Duflo-Serganova functor over a field of odd characteristic. We elaborate a method of computing the symmetry supergroup $\widetilde{\mathbb{G}_x}$ of this functor, recently introduced by A.Sherman, for a wide class of supergroups $\mathbb{G}$, and apply it to the case when $\mathbb{G}$ is $\mathrm{GL}(m|n)$ or $\mathrm{Q}(n)$, and a square zero odd element $x\in \mathrm{Lie}(\mathbb{G})$ has minimal or maximal rank. For any quasi-reductive supergroup $\mathbb{G}$, which has a pair of specific parabolic supersubgroups, we prove a criterion of injectivity of a $\mathbb{G}$-supermodule, involving vanishing of Duflo-Serganova functor on it.
Working over an algebraically closed field $\Bbbk$, we prove that all orbits of a left action of an algebraic group superscheme $G$ on a superscheme $X$ of finite type are locally closed. Moreover, such an orbit $Gx$, where $x$ is a $\Bbbk$-point of $X$, is closed if and only if $G_{ev}x$ is closed in $X_{ev}$, or equivalently, if and only if $G_{res}x$ is closed in $X_{res}$. Here $G_{ev}$ is the largest purely even group super-subscheme of $G$ and $G_{res}$ is $G_{ev}$ regarded as a group scheme. Similarly, $X_{ev}$ is the largest purely even super-subscheme of $X$ and $X_{res}$ is $X_{ev}$ regarded as a scheme. We also prove that $\mathrm{sdim}(Gx)=\mathrm{sdim}(G)-\mathrm{sdim}(G_x)$, where $G_x$ is the stabilizer of $x$.
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 algebraic group superschemes. Then we present the applications, including the Barsotti-Chevalley Theorem in the super context, and an explicit construction of the quotient superscheme $\mathbb{G}/\mathbb{H}$ of an algebraic group superscheme $\mathbb{G}$ by a group super-subscheme $\mathbb{H}$.
We introduce the notion of Krull super-dimension of supermodules over certain super-commutative Noetherian super-rings. We investigate how this notion relates to the notion of odd regular sequence introduced by T.Schmitt and how it behaves with respect to the transition to the graded and bigraded supermodules and super-rings associated with the original ones. We also apply these results to the super-dimension theory of superschemes of finite type and their morphisms.
We introduce the notion of a super-representation of a quiver. For super-representations of quivers over a field of characteristic zero, we describe the corresponding (super)algebras of polynomial semi-invariants and polynomial invariants.
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 Kähler superdifferentials.
Invariants of $G_2$ and $Spin(7)$, both acting on several copies of octonions, have been decribed in \cite{schw2} over a ground field of characteristic zero. In the current manuscript, we extend this result to an arbitrary infinite field of odd characteristic. More precisely, we prove that the corresponding algebras of invariants are generated by the same invariants of degree at most $4$ as in the case of a field of characteristic zero.
The notion of mixed representations of quivers can be derived from ordinary quiver representations by considering the dual action of groups on "vertex" vector spaces together with the usual action. A generating system for the algebra of semi-invariants of mixed representations of a quiver is determined. This is done by reducing the problem to the case of bipartite quivers of the special form and by introducing a function DP on three matrices, which is a mixture of the determinant and two pfaffians.
We consider a generalization of Donkin-Koppinen filtrations for coordinate superalgebras of general linear supergroups. More precisely, if $G=GL(m|n)$ is a general linear supergroup of (super)degree $(m|n)$, then its coordinate superalgebra $K[G]$ is a natural $G\times G$-supermodule. For every finitely generated ideal $Γ\subseteq Λ\timesΛ$, the largest subsupermodule $O_Γ(K[G])$ of $K[G]$, which has all composition factors of the form $L(λ)\otimes L(μ)$ where $(λ, μ)\inΓ$, has a decreasing filtration $O_Γ(K[G])=V_0\supseteq V_1\supseteq...$ such that $\bigcap_{t\geq 0}V_t=0$ and $V_t/V_{t+1}\simeq V_-(λ_t)^*\otimes H_-^0(λ_t)$ for each $t\geq 0$. Here $H_-^0(λ)$ is a costandard $G$-supermodule, and $V_-(λ)$ is a standard $G$-supermodule, both of highest weight $λ\inΛ$ (see \cite{z}). We deduce the existence of such a filtration from more general facts about standard and costandard filtrations in certain highest weight categories which will be proved in Section 4. Until now, analogous results were known only for highest weight categories with finite sets of weights. We believe that the reader will find the results of Section 4 interesting on its own. Finally, we apply our main result to describe invariants of (co)adjoint action of $G$.
Kantor and Trishin described the algebra of polynomial invariants of the adjoint representation of the Lie supergalgebra $gl(m|n)$ and a related algebra $A_s$ of what they called pseudosymmetric polynomials over an algebraically closed field $K$ of characteristic zero. The algebra $A_s$ was investigated earlier by Stembridge who called the elements of $A_s$ supersymmetric polynomials and determined generators of $A_s$. The case of positive characteristic $p$ has been recently investigated by La Scala and Zubkov. They formulated two conjectures describing generators of polynomial invariants of the adjoint action of the general linear supergroup $GL(m|n)$ and generators of $A_s$, respectively. In the present paper we prove both conjectures.
This paper is a survey on invariants of representations of quivers and their generalizations. We present the description of generating systems for invariants and relations between generators.
In this article we consider sheaf quotients of affine superschemes by finite supergroups that act on them freely. More precisely, if a finite supergroup $G$ acts on an affine superscheme $X$ freely, then the quotient $K$-sheaf $\tilde{X/G}$ is again an affine superscheme $Y$, where $K[Y]\simeq K[X]^G$. Besides, $K[X]$ is a finitely presented projective $K[X]^G$-module.
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 $G$ is an affine supergroup and $H$ is its normal supersubgroup, then we prove that a dur $K$-sheaf $\tilde{\tilde{G/H}}$ is again affine supergroup. Additionally, if $G$ is algebraic, then a $K$-sheaf $\tilde{G/H}$ is also algebraic supergroup and it coincides with $\tilde{\tilde{G/H}}$. In particular, any normal supersubgroup of an affine supergroup is faithfully exact.
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.
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.