Related papers: Implication Zroupoids and Identities of Associativ…
This is a survey on the geometric classification of different varieties of algebras (nilpotent, nil-, associative, commutative associative, cyclic associative, Jordan, Kokoris, standard, noncommutative Jordan, commutative power-associative,…
We describe those group algebras over fields of characteristic different from 2 whose units symmetric with respect to the classical involution, satisfy some group identity.
An algebraic variety is called $\mathbb{A}^{1}$-cylindrical if it contains an $\mathbb{A}^{1}$-cylinder, i.e. a Zariski open subset of the form $Z\times\mathbb{A}^{1}$ for some algebraic variety Z. We show that the generic fiber of a family…
In this note, we define and investigate ideal covering numbers of associative rings (not assumed to be commutative or unital): three invariants defined as the minimal number of proper left, right, or two-sided ideals whose union equals the…
An associative ring $A$ gives rise to the Lie ring $A^{(-)}=(A,[a,b ]=ab-ba)$. The subject of isomorphisms of Lie rings $A^{(-)}$ and $[A,A]$ has attracted considerable attention in the literature. We prove that if the identity element of…
Let $K$ be a field of characteristic zero and let $\mathfrak{M}_5$ be the variety of associative algebras over $K$, defined by the identity $[x_1,x_2][x_3,x_4,x_5]$. It is well-known that such variety is a minimal variety and that is…
A base of a relatively free associative algebra with the identity x^3=0 over a field of arbitrary characteristic is found. As an application a minimal generating system of the 3x3 matrix invariant algebra is determined.
It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let $B$ denote an integral square matrix and $\langle B \rangle$ denote the subring of the full matrix…
This paper concerns an associative graded algebra A that is the enveloping algebra of a Lie algebra with exponential growth. The algebra A is 3-Calabi-Yau. There is a Z-form of A so for every field k there is an algebra A_k. An algebraic…
Equivalence classes of gapped Hamiltonians compatible with given symmetry constraints, such as those underlying topological insulators, can be defined in many ways. For the non-chiral classes modelled by vector bundles over Brillouin tori,…
We apply the notion of a full convex subcategory to a wide range of algebras including tilted, quasi-tilted, shod, weakly shod, left and right glued, laura, simply connected, strongly simply connected, left supported, and cluster-tilted. In…
We introduce the notion of relation type of an affine algebra and prove that it is well defined by using the Jacobi-Zariski exact sequence of Andr\'e-Quillen homology. In particular, the relation type is an invariant of an affine algebraic…
We study intertwining operator algebras introduced and constructed by Huang. In the case that the intertwining operator algebras involve intertwining operators among irreducible modules for their vertex operator subalgebras, a number of…
In this paper, we consider four types of subvarieties of the variety of associative algebras. We study these subvarieties from the point of view of operads and show their connections with well-known classes of algebras, such as dendriform…
Let $X$ be a differentiable manifold endowed with a transitive action $\alpha:A\times X\longrightarrow X$ of a Lie group $A$. Let $K$ be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms…
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 $4$,…
The "$10$-fold way" refers to the combined classification of the $3$ associative division algebras (of real, complex and quaternionic numbers) and of the $7$, ${\mathbb Z}_2$-graded, superdivision algebras (in a superdivision algebra each…
A classical result in additive combinatorics, which is a combination of Balog-Szemer\'edi-Gowers theorem and a variant of Freiman's theorem due to Ruzsa, says that if a subset $A$ of $\mathbb{F}_p^n$ contains at least $c |A|^3$ additive…
The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX-relation). In…
An isomorphism between two hermitian unitals is proved, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).