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Vladimir L. Popov

Publications and source records attributed to Vladimir L. Popov.

49 records · Page 3Linked to original sources
arXiv2008

Irregular and singular loci of commuting varieties

We prove that the singular locus of the commuting variety of a noncommutative reductive Lie algebra is contained in the irregular locus and we compute the codimension of the latter. We prove that one of the irreducible components of the irregular locus has codimension 4. This yields the lower bound of the codimension of the singular locus, in particular, implies that it is at least 2. We also prove that the commuting variety is rational.

math.AG↗
arXiv2006

Tensor product decompositions and open orbits in multiple flag varieties

For a connected semisimple algebraic group $G$, we consider some special infinite series of tensor products of simple $G$-modules whose $G$-fixed point spaces are at most one-dimensional. We prove that their existence is closely related to the existence of open $G$-orbits in multiple flag varieties and address the problem of classifying such series.

math.RT↗
arXiv2006

On the Cayley degree of an algebraic group

A connected linear algebraic group G is called a Cayley group if the Lie algebra of G endowed with the adjoint G-action and the group variety of G endowed with the conjugation G-action are birationally G-isomorphic. In particular, the classical Cayley map, X \mapsto (I_n-X)/(I_n+X), between the special orthogonal group SO_n and its Lie algebra so_n, shows that SO_n is a Cayley group. In an earlier paper (see math.AG/0409004) we classified the simple Cayley groups defined over an algebraically closed field of characteristic zero. Here we consider a new numerical invariant of G, the Cayley degree, which "measures" how far G is from being Cayley, and prove upper bounds on Cayley degrees of some groups.

math.AG↗
arXiv2006

Cayley groups

The classical Cayley map, X --> (I_n-X)/(I_n+X), is a birational isomorphism between the special orthogonal group SO_n and its Lie algebra so_n, which is SO_n-equivariant with respect to the conjugating and adjoint actions respectively. We ask whether or not maps with these properties can be constructed for other algebraic groups. We show that the answer is usually "no", with a few exceptions. In particular, we show that a Cayley map for the group SL_n exists if and only if n <= 3. This answers an old question of Luna.

math.AG↗
arXiv2005

Finite linear groups, lattices, and products of elliptic curves

Let $V$ be a finite dimensional complex linear space and let $G$ be an irreducible finite subgroup of $\GL(V)$. For a $G$-invariant lattice $Λ$ in $V$ of maximal rank, we give a description of structure of the complex torus $V/Λ$. In particular, we prove that for a wide class of groups, $V/Λ$ is isogenous to a self-product of an elliptic curve, and that in many cases $V/Λ$ is isomorphic to a product of mutually isogenous elliptic curves with complex multiplication. We show that there are $G$ and $Λ$ such that the complex torus $V/Λ$ is not an abelian variety but one can always replace $Λ$ by another $G$-invariant lattice $Δ$ such that $V/Δ$ is a product if elliptic curves with complex multiplication. We amplify these results with a criterion, in terms of the character and the Schur $\mathbf Q$-index of $G$-module $V$, of the existence of a nonzero $G$-invariant lattice in $V$.

math.AG↗
arXiv2005

On polarizations in invariant theory

Given a reductive algebraic group $G$ and a finite dimensional algebraic $G$-module $V$, we study how close is the algebra of $G$-invariant polynomials on $V^{\oplus n}$ to the subalgebra generated by polarizations of $G$-invariant polynomials on $V$. We address this problem in a more general setting of $G$-actions on arbitrary affine varieties.

math.AG↗
arXiv2005

Problems for problem session

Below are the problems that I formulated at Open Problems Session of {\it Workshop on Group Actions on Rational Varieties}, McGill University and University of Montreal, Canada, March 2002. To appear in: "Affine Algebraic Geometry" conference Proceedings volume in Contemporary Mathematics series of the Amer. Math. Soc. Ed. by Jaime Gutierrez, Vladimir Shpilrain, and Jie-Tai Yu.

math.AG↗
arXiv2005

Generically multiple transitive algebraic group actions

With every nontrivial connected algebraic group $G$ we associate a positive integer ${\rm gtd}(G)$ called the generic transitivity degree of $G$ and equal to the maximal $n$ such that there is a nontrivial action of $G$ on an irreducible algebraic variety $X$ for which the diagonal action of $G$ on $X^n$ admits an open orbit. We show that ${\rm gtd}(G)\leqslant 2$ (respectively, ${\rm gtd}(G)=\nobreak 1$) for all solvable (respectively, nilpotent) $G$, and we calculate ${\rm gtd}(G)$ for all reductive $G$. We prove that if $G$ is nonabelian reductive, then the above maximal $n$ is attained for $X=G/P$ where $P$ is a proper maximal parabolic subgroup of $G$ (but not only for such homogeneous spaces of $G$). For every reductive $G$ and its proper maximal parabolic subgroup $P$, we find the maximal $r$ such that the diagonal action of $G$ (respectively, a Levi subgroup $L$ of ~$P$) on $(G/P)^r$ admits an open $G$-orbit (respectively, $L$-orbit). As an application, we obtain upper bounds for the multiplicities of trivial components in some tensor product decompositions. As another application, we classify all the pairs $(G, P)$ such that the action of $G$ on $(G/P)^3$ admits an open orbit, answering a question of {\sc M. Burger}.

math.AG↗
arXiv2004

Self-dual projective algebraic varieties associated with symmetric spaces

We discover a class of projective self-dual algebraic varieties. Namely, we consider actions of isotropy groups of complex symmetric spaces on the projectivized nilpotent varieties of isotropy modules. For them, we classify all orbit closures $X$ such that $X=\check X$ where $\check X$ is the projective dual of $X$. We give algebraic criteria of projective self-duality for the considered orbit closures.

math.AP↗
arXiv2004

Automorphism groups of finite dimensional simple algebras

We show that if a field k contains sufficiently many elements(for instance, if k is infinite), and K is an algebraically closed field containing k, then every linear algebraic k-group over K is k-isomorphic to Aut(A\otimes_kK), where A is a finite dimensional simple algebra over k.

math.RA↗
arXiv2002

Invariant tensor fields and orbit varieties for finite algebraic transformation groups

Let $X$ be a smooth algebraic variety endowed with an action of a finite group $G$ such that there exists the geometric quotient $π_X:X\to X/G$. We characterize rational tensor fields $τ$ on $X/G$ such that the {\it pull back} of $τ$ is regular on $X$: these are precisely all $τ$ such that $\operatorname{div}_{R_{X/G}}(τ)\ge 0$ where $R_{X/G}$ is the {\it reflection divisor} of $X/G$ and $\operatorname{div}_{R_{X/G}}(τ)$ is the {\it $R_{X/G}$-divisor} of $τ$. We give some applications, in particular to the generalization of Solomon's theorem. In the last section we show that if $V$ is a finite dimensional vector space and $G$ a finite subgroup of $\operatorname{GL}(V)$, then each automorphism $ψ$ of $V/G$ admits a biregular lift $ϕ: V\to V$ provided that $ψ$ maps the regular stratum to itself and $ψ_*(R_{X/G})=R_{X/G}$.

math.AG↗