Related papers: Matrix algebras with degenerate traces and trace i…

Let $f_1,...,f_s \in \mathbb{K}[x_1,...,x_m]$ be a system of polynomials generating a zero-dimensional ideal $\I$, where $\mathbb{K}$ is an arbitrary algebraically closed field. We study the computation of "matrices of traces" for the…

We construct six unitary trace invariants for 2 by 2 quaternionic matrices which separate the unitary similarity classes of such matrices, and show that this set is minimal. We prove two quaternionic versions of a well known…

Let $k$ be an arbitrary field and $d$ a positive integer. For each degenerate symmetric or antisymmetric bilinear form $M$ on $k^{d}$ we determine the structure of the Lie algebra of matrices that preserve $M$, and of the Lie algebra of…

Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$, ${\mathfrak q}\colon \mathcal{G}\times\mathcal{G}\longrightarrow \mathcal{G}$ be an $\mathcal{R}$-bilinear mapping and ${\mathfrak T}_{\mathfrak…

Let $G$ be a group of order $k$. We consider the algebra $M_k(\mathbb{C})$ of $k$ by $k$ matrices over the complex numbers and view it as a crossed product with respect to $G$ by embedding $G$ in the symmetric group $S_k$ via the regular…

One studies certain degenerations of the generic square matrix over a field $k$ along with its main related structures, such as the determinant of the matrix, the ideal generated by its partial derivatives, the polar map defined by these…

Let $UT_4(F)$ be $4\times 4$ upper triangular matrix algebra over a field $F$ of characteristic zero and let $\mathcal{A}$ be the subalgebra of $UT_4(F)$ linearly generated by $\{\mathbf{e}_{ij}:1 \leq i\leq j \leq 4 \} \setminus…

Let $F$ be an infinite field and $UT(d_1,\dots, d_n)$ be the algebra of upper block-triangular matrices over $F$. In this paper we describe a basis for the $G$-graded polynomial identities of $UT(d_1,\dots, d_n)$, with an elementary grading…

Assume that $\{a_{n};\,n\geq0\}$ is a sequence of positive numbers and $\sum a_{n}^{\,-1}<\infty$. Let $\alpha_{n}=ka_{n}$, $\beta_{n}=a_{n}+k^{2}a_{n-1}$ where $k\in(0,1)$ is a parameter, and let $\{P_{n}(x)\}$ be an orthonormal polynomial…

Let $\mathcal{T}$ be a triangular algebra over a commutative ring $\mathcal{R}$ and $\mathcal{Z(T)}$ be the center of $\mathcal{T}$. Suppose that ${\mathfrak q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}$ is an…

We prove that the Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ of traceless matrices over a finite field of characteristic $p$ can be generated by $2$ elements with exceptions when $(n, p)$ is $(3, 3)$ or $(4,2)$. In the latter cases, we…

The trace algebra C_{nd} is generated by all traces of products of d generic n x n matrices. Minimal generating sets of C_{nd} and their defining relations are known for n < 3 and n = 3, d=2. This paper states a minimal generating set and…

We study the problem when every matrix over a division ring is representable as either the product of traceless matrices or the product of semi-traceless matrices, and also give some applications of such decompositions. Specifically, we…

In this paper we consider the algebra of upper triangular matrices UT$_n(F)$, endowed with a $\mathbb{Z}_2$-grading (superalgebra) and equipped with a superinvolution. These structures naturally arise in the context of Lie and Jordan…

We study the computational complexity of singularity for multilinear maps. While the determinant characterizes singularity for matrices, its multilinear analogue -- the hyperdeterminant -- is defined only in boundary format and quickly…

The deep interconnection between linear algebra and graph theory allows one to interpret classical matrix invariants through combinatorial structures. To each square matrix A over a commutative ring K, one can associate a weighted directed…

Can the cross product be generalized? Why are the trace and determinant so important in matrix theory? What do all the coefficients of the characteristic polynomial represent? This paper describes a technique for `doodling' equations from…

Motivated by the fundamental lower bounds questions in proof complexity, we initiate the study of matrix identities as hard instances for strong proof systems. A matrix identity of $d \times d$ matrices over a field $\mathbb{F}$, is a…

We consider the algebra M_k(C) of k-by-k matrices over the complex numbers and view it as a crossed product with a group G of order k by embedding G in the symmetric group S_k via the regular representation and embedding S_k in M_k(C) in…

We propound the thesis that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally reducible (to…