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We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions,…
With this paper we hope to contribute to the theory of quantales and quantale-like structures. It considers the notion of $Q$-sup-algebra and shows a representation theorem for such structures generalizing the well-known representation…
We define several versions of the cohomology ring of an associative algebra. These ring structures unify some well known operations from homological algebra and differential geometry. They have some formal resemblance with the quantum…
Starting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems.…
We give an accessible introduction into the theory of lower central series of associative algebras, exhibiting the interplay between algebra, geometry and representation theory that is characteristic for this subject, and to discuss some…
We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the…
An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is…
In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…
In math.RT/0302174 we developed a framework to study representations of groups of the form $G((t))$, where $G$ is an algebraic group over a local field $K$. The main feature of this theory is that natural representations of groups of this…
The theory of $F$-manifolds, and more generally, manifolds endowed with commutative and associative multiplication of their tangent fields, was discovered and formalised in various models of quantum field theory involving algebraic and…
A method to construct in explicit form the generators of the simple roots of an arbitrary finite-dimensional representation of a quantum or standard semisimple algebra is found. The method is based on general results from the global theory…
It is shown that, given any finite dimensional, split basic algebra $\Lambda = K\Gamma/I$ (where $\Gamma$ is a quiver and $I$ an admissible ideal in the path algebra $K \Gamma$), there is a finite list of affine algebraic varieties, the…
A central question in cognitive science is whether conceptual representations converge onto a shared manifold to support generalization, or diverge into orthogonal subspaces to minimize task interference. While prior work has discovered…
We describe the main algebraic and geometric properties of the class of algebras introduced in [arXiv:0705.1629]. We discuss their origins in symplectic geometry and associative algebra, and the notions of cohomology and representations. We…
Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion of a twisted toroidal Lie algebra. In this paper, we construct representations of twisted toroidal Lie algebras from twisted modules…
We investigate regions formed by cylinders of circles of fixed radii. We investigate graphs obtained by collapsing each level set of the functions represented by the natural projections of them to the $1$-dimensional line. Some specific…
The f-invariant is a higher version of the e-invariant that takes values in the divided congruences between modular forms; it can be formulated as an elliptic genus of manifolds with corners of codimension two. In this thesis, we develop a…
We survey some important results concerning the finite--dimensional representations of the loop algebra of a simple complex Lie algebra, and their twisted loop subalgebras. In particular, we review the parametrization and description of the…
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
The general structure of the representation theory of a $Z_2$-graded coalgebra is discussed. The result contains the structure of Fourier analysis on compact supergroups and quantisations thereof as a special case. The general linear…