相关论文: Incidence Hilbert schemes and infinite dimensional…

Let $S$ be the affine plane $\C^2$ together with an appropriate $\mathbb T = \C^*$ action. Let $\hil{m,m+1}$ be the incidence Hilbert scheme. Parallel to \cite{LQ}, we construct an infinite dimensional Lie algebra that acts on the direct…

We construct geometrically the generating fields of a W algebra which acts irreducibly on the direct sum of the cohomology rings of the Hilbert schemes of n points on a projective surface for all n. We compute explicitly the commutators…

For different cohomology theories (including the Hochschild homology, Hodge cohomology, Chow groups, and Grothendieck groups of coherent sheaves), we identify the cohomology of moduli space of rank 1 perverse coherent sheaves on the blow-up…

We construct a new family of infinite-dimensional quasi-graded Lie algebras on hyperelliptic curves. We show that constructed algebras possess infinite number of invariant functions and admit a decomposition into the direct sum of two…

We prove a number of results on integrability and extendability of Lie algebras of unbounded skew-symmetric operators with common dense domain in Hilbert space. By integrability for a Lie algebra $\mathfrak{g}$, we mean that there is an…

This review paper contains a concise introduction to highest weight representations of infinite dimensional Lie algebras, vertex operator algebras and Hilbert schemes of points, together with their physical applications to elliptic genera…

We compute the cohomology with trivial coefficients of Lie algebras $\mathfrak{m}_0$ and $\mathfrak{m}_2$ of maximal class over the field $\mathbb{Z}_2$. In the infinite-dimensional case, we show that the cohomology rings…

We integrate the Lifting cocycles $\Psi_{2n+1},\Psi_{2n+3},\Psi_{2n+5},...$ ([Sh1], [Sh2]) on the Lie algebra $\Dif_n$ of holomorphic differential operators on an $n$-dimensional complex vector space to the cocycles on the Lie algebra of…

It is well known that $n$-dimensional projective group gives rise to a non-homogenous representation of the Lie algebra $sl(n+1)$ on the polynomial functions of the projective space. Using Shen's mixed product for Witt algebras (also known…

A theorem of G\"ottsche establishes a connection between cohomological invariants of a complex projective surface $S$ and corresponding invariants of the Hilbert scheme of $n$ points on $S.$ This relationship is encoded in certain infinite…

We study the polyhedral geometry of the hyperplanes orthogonal to the weights of the first and the second fundamental representations of $sl_n$ inside the dual fundamental Weyl chamber. We obtain generating functions that enumerate the…

Suppose the ground field to be algebraically closed and of characteristic different from $2$ and $3$. All Heisenberg Lie superalgebras consist of two super versions of the Heisenberg Lie algebras, $\frak{h}_{2m,n}$ and $\frak{ba}_n$ with…

In this paper we review various strikingly parallel algebraic structures behind Hilbert schemes of points on surfaces and certain finite groups called the wreath products. We explain connections among Hilbert schemes, wreath products,…

We study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of $\mathfrak{gl}_n$ restricted to the dual fundamental Weyl chamber. We obtain…

We provide a unified approach, via deformations of incidence algebras, to several important types of representations with finiteness conditions, as well as the combinatorial algebras which produce them. We show that over finite dimensional…

We describe infinite-dimensional Leibniz algebras whose associated Lie algebra is the Witt algebra and we prove the triviality of low-dimensional Leibniz cohomology groups of the Witt algebra with the coefficients in itself.

A real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant) is called a Hilbert-Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in…

E(2) is studied as the automorphism group of the Heisenberg algebra H. The basis in the Hilbert space K of functions on H on which the unitary irreducible representations of the group are realized is explicitely constructed. The addition…

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in…

Finite and Infinite-dimensional representations of symmetry algebras play a significant role in determining the spectral properties of physical Hamiltonians. In this paper, we introduce and apply a practical method to construct infinite…