Related papers: A Formalization of Elementary Linear Algebra: Part…
In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in…
We investigate linearity of amalgams of subgroups of algebraic groups along intersections with algebraic subgroups. In the process, we establish linearity of certain "doubles" of linear groups, and obtain new examples of finitely generated…
Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…
Finite-dimensional subalgebras of a Lie algebra of smooth vector fields on a circle, as well as piecewise-smooth global transformations of a circle on itself, are considered. A canonical forms of realizations of two- and three-dimensional…
The aim of this paper is twofold. First, we introduce a new class of linearizations, based on the generalization of a construction used in polynomial algebra to find the zeros of a system of (scalar) polynomial equations. We show that one…
Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear…
In linear algebra applications, elementary matrices hold a significant role. This paper presents a diagrammatic representation of all $2^m\times 2^n$-sized elementary matrices in algebraic ZX-calculus, showcasing their properties on…
The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…
For arbitrary F-algebra, in which the operation of addition is defined, I explore biring of matrices of mappings. The sum of matrices is determined by the sum in F-algebra, and the product of matrices is determined by the product of…
The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the…
We introduce two-sorted theories in the style of [CN10] for the complexity classes \oplusL and DET, whose complete problems include determinants over Z2 and Z, respectively. We then describe interpretations of Soltys' linear algebra theory…
Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and…
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
We introduce a new class of algebras called endo-commutative algebras in which the square mapping preserves multiplication, and provide a complete classification of endo-commutative algebras of dimension 2 over the field F_2 of two…
We assess the situation of our elementary Linear Algebra classes in the US holistically and through personal history recollections. Possible remedies for our elementary Linear Algebra's teaching problems are discussed and a change from…
For the second fundamental representation of the general linear group over a commutative ring $R$ we construct straightforward and uniform polynomial expressions of elementary generators as products of elementary conjugates of an arbitrary…
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
We study automorphism groups of formal matrix algebras. We also consider automorphisms of ordinary matrix algebras (in particular, triangular matrix algebras).
A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…
Lie brackets or Lie affgebra structures on several classes of affine spaces of matrices are studied. These include general normalised affine matrices, special normalised affine matrices, anti-symmetric and anti-hermitian normalised affine…