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Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that…

In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone's representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof…

We investigate the representation theory of the valenced Temperley-Lieb algebras in mixed characteristic. These algebras, as described in characteristic zero by Flores and Peltola, arise naturally in statistical physics and conformal field…

In this note, we observe a relation between dialgebras (in particular, Leibniz algebras) and conformal algebras. The purpose is to show how the methods of conformal algebras help solving problems on dialgebras, and, conversely, how the…

We define an equivalence relation on propositions and a proof system where equivalent propositions have the same proofs. The system obtained this way resembles several known non-deterministic and algebraic lambda-calculi.

In this paper motivated by the celebrated fundamental theorem of algebra and its standard proof utilizing Liouville's Theorem, we prove the fundamental theorem of algebra type results for both commutative and noncommutative polynomials in…

We view difference algebra as the study of algebraic objects in the topos of difference sets. The methods of topos theory and categorical logic enable us to develop difference homological algebra, identify a solid foundation for difference…

We construct integral forms for the universal enveloping algebras of certain twisted multiloop algebras and explicit integral bases for these integral forms.

We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view. A structure theorem of ideals in the Onsager algebra is obtained with the connection to the finite-dimensional representations. We also…

Let $p$ denote a prime number. In this note, we focus on the modular Terwilliger algebras of association schemes defined in [3]. We define the primary module of a modular Terwilliger algebra of an association scheme and determine all its…

The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology,…

Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of…

We discuss generalizations of the Temperley-Lieb algebra in the Potts and XXZ models. These can be used to describe the addition of different types of integrable boundary terms. We use the Temperley-Lieb algebra and its one-boundary,…

We give a closed formula for the graded decomposition numbers of the blob algebra over a field of characteristic zero at a root of unity.

We characterize vertex algebras (in a suitable sense) as algebras over a certain graded co-operad. We also discuss some examples and categorical implications of this characterization.

We introduce two classes of algebras coming from partial triangulations of marked surfaces. The first one, called frozen algebra of a partial triangulation, is generally of infinite rank and contains frozen Jacobian algebras of…

This paper investigates the homology of the Brauer algebras, interpreted as appropriate Tor-groups, and shows that it is closely related to the homology of the symmetric group. Our main results show that when the defining parameter of the…

In this paper, we introduce the notion of derivations of Lie 2-algebras and construct the associated derivation Lie 3-algebra. We prove that isomorphism classes of non-abelian extensions of Lie 2-algebras are classified by equivalence…

We take a categorical approach to describe ternary derivations and ternary automorphisms of triangular algebras. New classes of automorphisms and derivations of triangular algebras are also introduced and studied.

We provide a clarification of the classification of two-dimensional algebras over an arbitrary base field. Using this clarification, we determine the number of non-isomorphic two-dimensional algebras over a finite field.