Construction of Miniversal Deformations of Lie Algebras

We consider deformations of finite or infinite dimensional Lie algebras over a field of characteristic 0. There is substantial confusion in the literature if one tries to describe all the non-equivalent deformations of a given Lie algebra. It is known that there is in general no "universal" deformation of the Lie algebra L with a commutative algebra base A with the property that for any other deformation of L with base B there exists a unique homomorphism f: A -> B that induces an equivalent deformation. Thus one is led to seek a "miniversal" deformation. For a miniversal deformation such a homomorphism exists, but is unique only at the first level. If we consider deformations with base spec A, where A is a local algebra, then under some minor restrictions there exists a miniversal element. In this paper we give a construction of a miniversal deformation.Comment: 29 pages, (plain) Te

Laplacian spectrum for the nilpotent Kac-Moody Lie algebras

We prove that the maximal nilpotent subalgebra of a Kac-Moody Lie algebra has an (essentially unique) Euclidean metric with respect to which the Laplace operator in the chain complex is scalar on each component of a given degree. Moreover, both the Lie algebra structure and the metric are uniquely determined by this property.Comment: 11 page

Self-dual polygons and self-dual curves

We study projectively self-dual polygons and curves in the projective plane. Our results provide a partial answer to problem No 1994-17 in the book of Arnold's problems

Massey products and deformations

The classical deformation theory of Lie algebras involves different kinds of Massey products of cohomology classes. Even the condition of extendibility of an infinitesimal deformation to a formal one-parameter deformation of a Lie algebra involves Massey powers of two dimensional cohomology classes which are not powers in the usual definition of Massey products in the cohomology of a differential graded Lie algebra. In the case of deformations with other local bases, one deals with other, more specific Massey products. In the present work a construction of generalized Massey products is given, depending on an arbitrary graded commutative, associative algebra. In terms of these products, the above condition of extendibility is generalized to deformations with arbitrary local bases. Dually, a construction of generalized Massey products on the cohomology of a differential graded commutative associative algebra depends on a nilpotent graded Lie algebra. For example, the classical Massey products correspond to the Lie algebra of strictly upper triangular matrices, while the matric Massey products correspond to the Lie algebra of block strictly upper triangular matrices.Comment: 13 pages, tex documen

On the restricted Lie algebra structure for the Witt Lie algebra in finite characteristic

We show that the p-operator in the Witt algebra (the restricted Lie algebra of derivations of the quotient of the polynomial algebra over a field of characteristic p by the ideal generated by the p-th power of the indeterminant) is given by multiplication by a scalar.Comment: 6 Page

Projections of Singular Vectors of Verma Modules over Rank 2 Kac-Moody Lie Algebras

We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac-Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of $sl_{2}$ (Theorem 3). The formula is derived from a more general but less explicit formula due to Feigin, Fuchs and Malikov [Funct. Anal. Appl. 20 (1986), no. 2, 103-113]. In the simpler case of $A_{1}^{1}$ the formula was obtained in [Fuchs D., Funct. Anal. Appl. 23 (1989), no. 2, 154-156].Comment: This is a contribution to the Special Issue on Kac-Moody Algebras and Applications, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA