It is safe to say that the theory of modular representations of finite groups is not a part of the average mathematician's toolkit. Matrix representations of finite groups over the complex field, and the resulting characters (traces of matrices), occur rather widely in both pure and applied mathematics. But replacing complex numbers by elements of a finite or other field of prime characteristic...

Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers (i.e. directed graphs) with relations the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the quivers, leading naturally to a combinatorial setting. In this paper we study a refined version, so-called q-Cartan matrices, where each nonzero path is weighted ...

Abstract For a given inverse semigroup action on topological space, one can associate an étale groupoid. We prove that there exists correspondence between the certain subsemigroups and open wide subgroupoids in case is strongly tight. Combining with recent result of Brown et al., we obtain Cartan intermediate subalgebras groupoid C*-algebra.

This article is motivated by the theory of distinguished curves in parabolic geometries, as developed in [2]. A parabolic geometry is, by definition, modelled on a homogeneous space of the form G/P where G is a real semisimple Lie group and P is a parabolic subgroup. (There is also a complex theory which corresponds to the choices of complex G’s and P ’s with specific curvature restrictions for...

The dualism between classical matter and geometry is extended to include elementary hadronic matter described in quantized form in terms of a gauge theory based on the group SO(4; 1) containing a built-in fundamental length parameter R of geometric origin of the order of a Fermi. Quantized matter at subnuclear distances is described in terms of generalized quantum mechanical wave functions ((rs...

We present a variant of the Theory of Lorentzian (i. e. with a hyperbolic generalized Cartan matrix) Kac–Moody algebras recently developed by V. A. Gritsenko and the author. It is closely related with and strongly uses results of R. Borcherds. This theory should generalize well-known Theories of finite Kac–Moody algebras (i. e. classical semi-simple Lie algebras corresponding to positive genera...

There is a common description of different intrinsic geometric flows in two dimensions using Toda field equations associated to continual Lie algebras that incorporate the deformation variable t into their system. The Ricci flow admits zero curvature formulation in terms of an infinite dimensional algebra with Cartan operator ∂/∂t. Likewise, the Calabi flow arises as Toda field equation associa...

We discuss the formulation of cosmological topologically massive (simple) supergravity theory in three-dimensional Riemann-Cartan space-times. use language exterior differential forms and properties Majorana spinors on 3-dimensional space-times to explicitly demonstrate local supersymmetry action density involved. Coupled field equations that are complete both their bosonic fermionic sectors de...

We study the existence of natural and projectively equivariant quantizations for differential operators acting between order 1 vector bundles over a smooth manifold M . To that aim, we make use of the Thomas-Whitehead approach of projective structures and construct a Casimir operator depending on a projective Cartan connection. We attach a scalar parameter to every space of differential operato...