Abstracts for Seminar on Algebra, Geometry and Physics

A Frobenius Lie algebra g, by definition, admits a linear form f: g -> k
such that the bilinear form f([x,y]) is non-degenerate. The talk will, in particular,
survey many interesting interactions of this notion with various aspects of
representation theory and differential geometry.

The talk is based on a joint work with M.S.Khalgui and P. Torasso.

 

Most of the results discussed in this talk are conjectural.
Let L be ample line bundle on an a projective algebraic surface S. Let g be
the genus of a smooth curve in the linear system |L|. If L is suffciently ample with
respect to d, the number of n_{L,d}of d-nodal curves in a general d-dimensional sublinear
system of |L| will be  finite. Kool-Shende-Thomas use relative Hilbert
schemes of points of the universal curve over |L| to define the numbers
n_{L,d} as BPS invariants and prove a conjecture of mine about their
generating function.

We study quadratic algebras over a field $\textbf{k}$. We prove
that the class of  $n$-generated PBW algebras with finite global
dimension and polynomial growth contains a unique (up to
isomorphism) monomial algebra, $A= \textbf{k} \langle x_1,\cdots,x_n
\rangle /(x_jx_i \mid 1 \leq i < j \leq n)$.

The main result shows that for an $n$-generated quantum binomial algebra
$A$ the following conditions are equivalent:
(i) $A$ is a PBW algebra with  finite global dimension.
(ii) $A$ is a PBW algebra with polynomial growth.

We consider the commutative algebra of densities on a manifold $M$. This algebra is endowed with the canonical scalar product and we shall consider self-adjoint operators on this algebra. We naturally come to canonical pencils of operators (acting on densities of different weights) and passing through a given operator (acting on densities of a given weight). There are singular values of weights for operators of a given order. These singular values lead to an interesting geometrical picture: 1) If $M$ is the real line we arrive at classical constructions in projective geometry.