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A unified treatment of both superconformal and quasisuperconformal algebras with quadratic non-linearity is given. General formulas describing their structure are found by solving the Jacobi identities. A complete classification of…
In this thesis we studied the structure coefficients and especially their dependence on $n$ in the case of a sequence of double-class algebras. The first chapter is dedicated to the study of the structure coefficients in the general cases…
By an exotic algebraic structure on the affine space ${\bf C}^n$ we mean a smooth affine algebraic variety which is diffeomorphic to ${\bf R}^{2n}$ but not isomorphic to ${\bf C}^n$. This is a survey of the recent developement on the…
A second order family of special Lagrangian submanifolds of complex m-space is a family characterized by the satisfaction of a set of pointwise conditions on the second fundamental form. For example, the set of ruled special Lagrangian…
The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair…
In \cite{BigAlg-3gen}, an explicit description of bi-quadratic algebras on three generators with PBW basis was obtained. There are four classes: I-IV. The aim of the paper is to study algebras that belong to one of the classes: class II.1.…
In [19], Zheng studied the bounded derived categories of constructible $\bar{\mathbb{Q}}_l$-sheaves on some algebraic stacks consisting of the representations of a enlarged quiver and categorified the integrable highest weight modules of…
A classical result in quantum topology is that oriented 2-dimensional topological quantum field theories (2-TQFTs) are fully classified by commutative Frobenius algebras. In 2006, Turaev and Turner introduced additional structure on…
The construction of Freudenthal's Magic Square, which contains the exceptional simple Lie algebras, in terms of symmetric composition algebras is further developed here. The para-Hurwitz algebras, which form a subclass of the symmetric…
Motivated by the physical concept of special geometry two mathematical constructions are studied, which relate real hypersurfaces to tube domains and complex Lagrangean cones respectively. Me\-thods are developed for the classification of…
In this work we have considered the complexity of the different structures as topological group on Z. We collect some new results, as well as some known results on the group of the integers in order to present: -A family of $2^\cont$…
We show how the exceptional isogenies of classical groups to orthogonal groups of quadratic spaces of dimensions up to 8 over fields of characteristic different from 2 may be obtained by explicit algebraic constructions using the…
In several classes of countable structures it is known that every hyperarithmetic structure has a computable presentation up to bi-embeddability. In this article we investigate the complexity of embeddings between bi-embeddable structures…
We equip Ellis and Brundan's version of the odd categorified quantum group for sl(2) with a differential giving it the structure of a graded dg-2-supercategory. The presence of the super grading gives rise to two possible…
By characterizing all orthogonal polynomials sequences $(P_n)_{n\geq 0}$ for which $$ (ax+b)(\triangle +2\,\mathrm{I})P_n(x(s-1/2))=(a_n x+b_n)P_n(x)+c_n P_{n-1}(x),\quad n=0,1,2,\dots, $$ where $\,\mathrm{I}$ is the identity operator, $x$…
This paper investigates the effective categoricity of ultrahomogeneous structures. It is shown that any computable ultrahomogeneous structure is $\Delta^0_2$ categorical. A structure A is said to be weakly ultrahomogeneous if there is a…
Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. They have been intensively studied and used for many applications…
The conditions under which a given manifold $M$ may be given a tangent bundle or a cotangent bundle structure are analyzed. This is an important property arising in different contexts. For instance, in the study of integrability of a given…
We introduce and study higher spherical algebras, an exotic family of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher tetrahedral algebra studied in [7],…
We review the higher gauge symmetries in double and exceptional field theory from the viewpoint of an embedding tensor construction. This is based on a (typically infinite-dimensional) Lie algebra $\frak{g}$ and a choice of representation…