Jose Brox

University of Valladolid | UVA · Department of Algebra, Geometry and Topology

We study the differential identities of the algebra M_k(F) of k × k matrices over a field F of characteristic zero when its full Lie algebra of derivations, L = Der(M_k(F)), acts on it. We determine a set of 2 generators of the ideal of differential identities of M_k(F) for k ≥ 2. Moreover, we obtain the exact values of the corresponding differenti...

We study the categorical-algebraic condition that internal actions are weakly representable (WRA) in the context of varieties of (non-associative) algebras over a field. Our first aim is to give a complete characterization of action accessible, operadic quadratic varieties of non-associative algebras which satisfy an identity of degree two and to...

We study polynomial identities satisfied by the mutation product xpy − yqx on the underlying vector space of an associative algebra A, where p, q are fixed elements of A. We simplify known results for identities in degree 4, proving that only two identities are necessary and sufficient to generate them all; in degree 5, we show that adding one new...

This proceedings volume presents a selection of peer-reviewed contributions from the Second Non-Associative Algebras and Related Topics (NAART II) conference, which was held at the University of Coimbra, Portugal, from July 18–22, 2022. The conference was held in honor of mathematician Alberto Elduque, who has made significant contributions to the...

A square matrix is $k$-Toeplitz if its diagonals are periodic sequences of period $k$. We find universal formulas for the determinant, the characteristic polynomial, some eigenvectors, and the entries of the inverse of any tridiagonal $k$-Toeplitz matrix (in particular, of any tridiagonal matrix) over any commutative unital ring, expressed in terms...

In this paper, we study ad-nilpotent elements of semiprime rings R with involution \(*\) whose indices of ad-nilpotence differ on \({{\,\mathrm{Skew}\,}}(R,*)\) and R. The existence of such an ad-nilpotent element a implies the existence of a GPI of R, and determines a big part of its structure. When moving to the symmetric Martindale ring of quoti...

In this paper, we study ad-nilpotent elements in Lie algebras arising from semiprime associative rings R free of 2-torsion. With the idea of keeping under control the torsion of R, we introduce a more restrictive notion of ad-nilpotent element, pure ad-nilpotent element, which is a only technical condition since every ad-nilpotent element can be ex...

As originally defined by Ashraf and Mozumder, multiplicative (generalized)-skew derivations must satisfy two identities. In this short note we show that, as a consequence of the simultaneous satisfaction of both identities, a multiplicative (generalized)-skew derivation of a prime ring is either a multiplicative (generalized) derivation (i.e., not...

We find closed-form algebraic formulas for the elements of the inverses of tridiagonal 2- and 3-Toeplitz matrices which are symmetric and have constant upper and lower diagonals. These matrices appear, respectively, as the impedance matrices of resonator arrays in which a receiver is placed over every 2 or 3 resonators. Consequently, our formulas a...

Let $L$ be a Lie algebra over a field $F$ of characteristic zero or $p > 3$. An element $c \in L$ is called Clifford if $\ad_c^3=0$ and its associated Jordan algebra $L_c$ is the Jordan algebra $F \oplus X$ defined by a symmetric bilinear form on a vector space $X$ over $F$. In this paper we prove the following result: Let $R$ be a centrally closed...

We determine the Jordan algebras associated to ad-nilpotent elements of index at most 3 in the Lie algebras $R^−$ and $Skew(R, ∗)$ for semiprime rings $R$ without or with involution $∗$. To do so we first characterize these ad-nilpotent elements.

In this note we extend the Lie inner ideal structure of simple Artinian rings with involution, initiated by Benkart and completed by Benkart and Fernández Lopez, to centrally closed prime rings with involution of characteristic not 2, 3 or 5. New Lie inner ideals (which we call special) occur when making this extension. We also give a purely algebr...

In this paper, we prove that the ideals generated by two elements x, y in a nondegenerate Lie algebra L over a ring of scalars Φ with ${\frac 1 2, \frac 1 3}$ are orthogonal if and only if [x, [y, L]] = 0.

We characterize, in terms of its idempotents, the Leavitt path algebras of an arbitrary graph that satisfies Condition (L) or Condition (NE). In the latter case, we also provide the structure of such algebras. Dual graph techniques are considered and demon-strated to be useful in the approach of the study of Leavitt path algebras of arbitrary graph...