Related papers: Generalized Punctual Hilbert Schemes and $\mathfra…

In a recent paper by the authors, Lie bialgebra structures on generalized Heisenberg- Virasoro algebra L are considered. In this paper, the explicit formula of the quantization on generalized Heisenberg-Virasoro algebra is presented.

We find certain relations between flag Hilbert schemes of points on plane curves and moduli spaces of one-dimensional plane sheaves. We show that some of these moduli spaces are unirational.

We propose a new definition of so called Hamiltonian forms in n-plectic geometry and show that they have a non-trivial Lie infinity-algebra structure.

In this paper, we introduce the concept of a (lattice) skew Hilbert algebra as a natural generalization of Hilbert algebras. This notion allows a unified treatment of several structures of prominent importance for mathematical logic, e.g.…

The $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component of a closed oriented surface is a preferred component of the character variety consisting of homomorphisms from the fundamental group of the surface to the projective linear group…

In this paper we extend the Lie theory of integration in two different ways. First we consider a finite dimensional Lie algebra of vector fields and discuss the most general conditions under which the integral curves of one of the fields…

The space of generalized projective structures on a Riemann surface $\Sigma$ of genus g with n marked points is the affine space over the cotangent bundle to the space of SL(N)-opers. It is a phase space of $W_N$-gravity on…

Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, which is simply connected in each simplicial level. We use the 1-jet of the classifying space of G to construct, starting…

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the…

A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra…

We propose a Hamiltonian Lie algebroid and a momentum section over a Dirac structure as a generalization of a Hamiltonian Lie algebroid over a pre-symplectic manifold and one over a Poisson manifold. A Hamiltonian Lie algebroid and a…

This paper provides a geometric description for Lie--Hamilton systems on $\mathbb{R}^2$ with locally transitive Vessiot--Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of…

We present the method for finding of the nonlinear Poisson-Lie groups structures on the vector spaces and for their quantization. For arbitrary central extension of Lie algebra explicit formulas of quantization are proposed.

The main purpose of the paper is to study hyperkahler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkahler structures. Motivated by the work of…

In this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and discuss regularity properties (in the sense of Milnor)…

We describe various structures of algebraic nature on the space of continuous valuations on convex sets, their properties (like versions of Poincar\'e duality and hard Lefschetz theorem), and their relations and applications to integral…

Partial Isometries are important constructs that help give nontrivial solutions once a simple solution is known. We generalize this notion to Extended Partial Isometries and include operators which have right inverses but no left inverses…

In this paper, we give three bases for the cohomology groups of the Hilbert scheme of two points on projective space. Then, we use these bases to compute all effective and nef cones of higher codimensional cycles on the Hilbert scheme.…

I connect an old result of mine on a Lie algebra generalization of the Amitsur-Levitski theorem with equations for sheets in a reductive Lie algebra and with recent results of Kostant-Wallach on the variety of singular elements in a…