Related papers: Notes on normed algebras, 4

We give various formulas to compute the number of all involutions, i.e. elements of order 2, in an incidence algebra $I(X,\mathbb{K})$, where $X$ is a finite poset (star, Y and Rhombuses) and $\mathbb{K}$ is a finite field of characteristic…

This work is a collection of old and new aplications of Galois cohomology to the clasification of algebraic and arithmetical objects.

We consider the functions in two variables on an arbitrary poset, for which the convolution operation is defined. We obtain the generalization of incidence algebra and describe its properties: invertibility, the Jackobson radical,…

These notes deal with some topics related to limits of norms, functions on the unit circle, and so on.

This note is designed to show some classes of differential-difference equations admitting Lax representation which generalize evolutionary equations known in the literature.

We continue the classification of isomorphism classes of k-involutions of exceptional algebraic groups. In this paper we classify k-involutions for split groups of type F4 over certain fields, and their fixed point groups.The classification…

In recent years, many results have been established regarding classifications of varieties whose colength sequences are bounded by a fixed constant. In this work, we explore this theme in the setting of algebras endowed with a graded…

We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a…

In loop quantum gravity, states of quantum geometry are represented by classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is worthwhile to enumerate and distinguish these classes. This paper looks at the case of 4-regular…

We introduced positive cones in an earlier paper as a notion of ordering on central simple algebras with involution that corresponds to signatures of hermitian forms. In the current paper we describe signatures of hermitian forms directly…

To an orthogonal or unitary involution on a central simple algebra of degree 4, or to a symplectic involution on a central simple algebra of degree 8, we associate a Pfister form that characterises the decomposability of the algebra with…

The space of derivations of finite dimensional evolution algebras associated to graphs over a field with characteristic zero has been completely characterized in the literature. In this work we generalize that characterization by describing…

A brief overview of some computer algebra methods for computations with nested integrals is given. The focus is on nested integrals over integrands involving square roots. Rewrite rules for conversion to and from associated nested sums are…

This article studies the set of R-equivalence classes of the group of proper projective similitudes of an algebra with involution of the first kind. The main results concern base fields of characteristic different from 2 over which every…

Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…

These are the notes from my courses on the arithmetic of quadratic forms.

The hermitian u-invariants of a central simple algebra with involution are studied. In this context, a new technique is obtained to give bounds for the behavior of these invariants under a quadratic field extension. This is applied to…

We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad.

These notes, associated with a topics course, deal with some special features of summability and supremum norms which are often useful.

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…