Related papers: Irregular and singular loci of commuting varieties

We study the moduli space of four dimensional ordinary Lie algebras, and their versal deformations. Their classification is well known; our focus in this paper is on the deformations, which yield a picture of how the moduli space is…

Consider a finite-dimensional algebra $A$ and any of its moduli spaces $\mathcal{M}(A,\mathbf{d})^{ss}_{\theta}$ of representations. We prove a decomposition theorem which relates any irreducible component of…

We classify irreducible representations of compact connected Lie groups whose orbit space is isometric to the orbit space of a representation of a compact Lie group of dimension~$7$, $8$ or $9$. They turn out to be closely related to…

We prove that the moduli spaces of K3 surfaces with non-symplectic involution are rational for four deformation types. With the previous results, this establishes the rationality of those moduli spaces except two classical cases.

Let $k$ be algebraically closed field of characteristic zero, let $G$ be a commutative algebraic group over $k$ such that the linear part of $G$ is isomorphic to $\mathbb{G}_a$, and let $X$ be a closed subvariety of $G$. We show that the…

We describe the variety of Jordan superalgebras of dimension $4$ whose even part is a Jordan algebra of dimension $1$ or $3$. We prove that the variety is the union of Zariski closures of the orbits of $11$ and $21$ rigid superalgebras,…

We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations…

In this paper we give a lower bound for the codimension of the Andreotti-Mayer loci in the moduli space of principally polarized complex abelian varieties. We also present a conjecture on this codimension.

In this paper, we proved that a compression space of rank $2$ is equivalent to an irreducible representation over a Lie algebra.

We give another proof that a reductive algebraic group is geometrically reductive. We show that a quotient of the semi-stable locus (by a linear action of a reductive algebraic group on a projective scheme) exists, and from this Haboush's…

We give upper bounds for the number of irreducible representations of dimension at most n for a compact semisimple Lie group. In particular, we prove that there are at most n irreducible representations of dimension at most n for a simple…

Let $\mathfrak{g}$ be the $p$-dimensional Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Let $\mathscr{N}={x\in\ggg\mid x^{[p]}=0}$ be the nilpotent variety of $\mathfrak{g}$, and…

Here we prove that a commuting variety associated with a symmetric pair (g, g_0) is irreducible for (so_{n+m}, so_n + so_m) and reducible for (gl_{n+m}, {gl}_n + gl_m) with n>m, (so_{2n}, gl_n) with odd n, (E_6, {so}_{10} + k).

Based on results for real deformation parameter q we introduce a compact non- commutative structure covariant under the quantum group SOq(3) for q being a root of unity. To match the algebra of the q-deformed operators with necesarry…

Let $X$ be an affine algebraic variety endowed with an action of complexity one of an algebraic torus $\mathbb{T}$. It is well known that homogeneous locally nilpotent derivations on the algebra of regular functions $\mathbb{K}[X]$ can be…

This paper aims to describe the restricted Kac modules of restricted Hamiltonian Lie superalgebras of odd type over an algebraically closed field of characteristic $p>3$. In particular, a sufficient and necessary condition for the…

We classify irreducible representations of connected compact Lie groups whose orbit space is isometric to the orbit space of a representation of a finite extension of (positive dimensional) toric group. They turn out to be exactly the…

We present some results about the irreducible representations appearing in the exterior algebra $\Lambda \mathfrak{g}$, where $ \mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$. For Lie algebras of type $B$, $C$ or $D$ we prove that…

Irregular cusp of an orthogonal modular variety is a cusp where the lattice for Fourier expansion is strictly smaller than the lattice of translation. Presence of such a cusp affects the study of pluricanonical forms on the modular variety…

The property of degeneration of modular graded Lie algebras, first investigated by B. Weisfeiler, is analyzed. Transitive irreducible graded Lie algebras $L=\sum_{i\in \mathbb Z}L_i,$ over an algebraically closed field of characteristic…