Related papers: A Standard Form in (some) Free Fields: How to cons…
Let $\mathfrak g$ be a complex semisimple Lie algebra. We define what it means for a finite dimensional representation of $\mathfrak g$ to be rectangular and completely classify faithful rectangular representations. As an application, we…
The centrepiece of this paper is a normal form for primitive elements which facilitates the use of induction arguments to prove properties of primitive elements. The normal form arises from an elementary algorithm for constructing a…
We describe a simple approach to factorize non-commutative (nc) polynomials, that is, elements in free associative algebras (over a commutative field), into atoms (irreducible elements) based on (a special form of) their minimal linear…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
By means of several examples, we motivate that universal properties are the simplest way to solve a given mathematical problem, explaining in this way why they appear everywhere in mathematics. In particular, we present the co-universal…
We establish representation types (finite, tame or wild) of finite dimensional Munn algebras with semisimple bases. As an application, we establish representation types of finite 0-simple semigroups and their mutually annihilating unions.
A model of representations of a Lie algebra is a representation which a direct sum of all irreducible finite dimensional representations taken with multiplicity $1$. In the paper an explicit construction of a model of representation for all…
A method to construct free field realizations for the form factors of diagonal factorized scattering theories is described. Form factors are constructed from linear functionals over an associative `form factor algebra', which in particular…
We generalize Kudryavtseva and Mazorchuk's concept of canonical form of elements in Kiselman's semigroups to the setting of a Hecke-Kiselman monoid $\mathbf{HK}_\Gamma$ associated with a simple oriented graph $\Gamma$. We use confluence…
The representations of the observable algebra of a low dimensional quantum field theory form the objects of a braided tensor category. The search for gauge symmetry in the theory amounts to finding an algebra which has the same…
A Gelfand model for a semisimple algebra A over C is a complex linear representation that contains each irreducible representation of A with multiplicity exactly one. We give a method of constructing these models that works uniformly for a…
Most algorithms constructing bases of finite-dimensional vector spaces return basis vectors which, apart from orthogonality, do not show any special properties. While every basis is sufficient to define the vector space, not all bases are…
We construct an action of a free resolution of the Frobenius properad on the differential forms of a closed oriented manifold. As a consequence, the forms of a manifold with values in a semi-simple Lie algebra have an additional structure…
This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local…
Standard noncommutative Gr\"obner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields. This paper describes Gr\"obner basis procedures for one-sided ideals in finitely presented noncommutative…
Normal forms allow the use of a restricted class of coordinate transformations (typically homogeneous polynomials) to put the bifurcations found in nonlinear dynamical systems into a few standard forms. We investigate here the consequences…
In this note, we initiate a study of the finite-dimensional representation theory of a class of algebras that correspond to noncommutative deformations of compact surfaces of arbitrary genus. Low dimensional representations are investigated…
In this paper we are concerned with the simplest normal form computation of a family of Hopf-zero vector fields without a first integral. This family of vector fields are the classical normal forms of a larger family of vector fields with…
In this paper we study algebras of modular forms on unitary groups of signature $(n,1)$. We give a necessary and sufficient condition for an algebra of unitary modular forms to be free in terms of the modular Jacobian. As a corollary we…
A canonical minimal free resolution of an arbitrary co-artinian lattice ideal over the polynomial ring is constructed over any field whose characteristic is 0 or any but finitely many positive primes. The differential has a closed-form…