Finite-dimensional Simple Algebra Research Articles

On Simple Finite-Dimensional Algebras with Infinite Basis of Identities

On Simple Finite-Dimensional Algebras with Infinite Basis of Identities

On Simple Finite-Dimensional Algebras with Infinite Bases of Identities

В 1993 г. И. П. Шестаков поставил вопрос о существовании центральной простой конечномерной алгебры над полем нулевой характеристики, тождества которой не задаются конечным набором тождеств. В 2012 г. И. М. Исаевым и автором построен пример семимерной центральной простой алгебры над любым полем, не имеющей конечного базиса тождеств. В данной работе строится пример шестимерной центральной простой алгебры над полем нулевой характеристики, не имеющей конечного базиса тождеств. Библиография: 22 названия.

A CHARACTERIZATION OF DERIVATIONS AND AUTOMORPHISMS ON SOME SIMPLE ALGEBRAS

In the present paper, we study simple algebras, which do not belong to the well-known classes of algebras (associative algebras, alternative algebras, Lie algebras, Jordan algebras, etc.). The simple finite-dimensional algebras over a field of characteristic 0 without finite basis of identities, constructed by Kislitsin, are such algebras. In the present paper, we consider two such algebras: the simple seven-dimensional anticommutative algebra \(\mathcal{D}\) and the seven-dimensional central simple commutative algebra \(\mathcal{C}\). We prove that every local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is a derivation, and every 2-local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also a derivation. We also prove that every local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is an automorphism, and every 2-local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also an automorphism.

The images of multilinear and semihomogeneous polynomials on the algebra of octonions

The generalized L'vov–Kaplansky conjecture states that for any finite-dimensional simple algebra A the image of a multilinear polynomial on A is a vector space. In this paper, we prove it for the algebra of octonions O over a field F satisfying certain specified conditions (in particular, we prove it for quadratically closed fields, and for the field R ). In fact, letting V be the space of pure octonions in O , we prove that the image set must be either { 0 } , F, V or O . We discuss possible evaluations of semihomogeneous polynomials on O and of arbitrary polynomials on the corresponding Malcev algebra.

Automorphisms and derivations of finite-dimensional algebras

Let A be a finite-dimensional algebra over a field F with char(F)≠2. We show that a linear map D:A→A satisfying xD(x)x∈[A,A] for every x∈A is the sum of an inner derivation and a linear map whose image lies in the radical of A. Assuming additionally that A is semisimple and char(F)≠3, we show that a linear map T:A→A satisfies T(x)3−x3∈[A,A] for every x∈A if and only if there exist a Jordan automorphism J of A lying in the multiplication algebra of A and a central element α satisfying α3=1 such that T(x)=αJ(x) for all x∈A. These two results are applied to the study of local derivations and local (Jordan) automorphisms. In particular, the second result is used to prove that every local Jordan automorphism of a finite-dimensional simple algebra A (over a field F with char(F)≠2,3) is a Jordan automorphism.

Two-sided zero product determined algebras

An algebra A is said to be two-sided zero product determined if every bilinear functional φ:A×A→F satisfying φ(x,y)=0 whenever xy=yx=0 is of the form φ(x,y)=τ1(xy)+τ2(yx) for some linear functionals τ1,τ2 on A. We present some basic properties and equivalent definitions, examine connections with some properties of derivations, and as the main result prove that a finite-dimensional simple algebra that is not a division algebra is two-sided zero product determined if and only if it is separable.

Local and 2-local derivations of simple n-ary algebras

In the present paper, we prove that every local and $2$-local derivation of the complex finite-dimensional simple Filippov algebra is a derivation. As a corollary we have the description of all local and $2$-local derivations of complex finite-dimensional semisimple Filippov algebras. All local derivations of the ternary Malcev algebra $M_8$ are described. It is the first example of a finite-dimensional simple algebra that admits pure local derivations, i.e. algebra admits a local derivation which is not a derivation.

Positive Cones and Gauges on Algebras With Involution

AbstractWe extend the classical links between valuations and orderings on fields to Tignol–Wadsworth gauges and positive cones on finite-dimensional simple algebras with involution. We also study the compatibility of gauges and positive cones and prove lifting results in the style of the Baer–Krull theorem for fields.

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set ${\rm sl}_n(K)$ of matrices of trace 0, or all of $M_n(K)$. This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for $n=2$ in Section 2, some decisive results for $n=3$ in Section 3, and partial information for $n\geq 3$ in Section 4, also for non-multilinear polynomials. In addition we consider the case of $K$ not algebraically closed, and polynomials evaluated on other finite dimensional simple algebras (in particular the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other researches, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of art, and in the other hand providing counterexamples showing the boundaries of generalizations.

Isotopes of simple algebras of arbitrary dimension

It is proved that every finite-dimensional algebra is embeddable in a simple finite-dimensional algebra (a suitable isotope of a matrix algebra). An isotope of the 2nd order matrix algebra over an infinite extension of the ground field may contain a trivial ideal. Every one-sided isotope of a simple unital alternative or Jordan algebra is a simple algebra. Besides, any isotope of a central simple non-Lie Maltsev algebra of characteristic other than 2 and 3 is a simple algebra. But an isotope of a simple Jordan algebra of the symmetric bilinear form on the infinite dimensional space may contain a trivial ideal.

Graded polynomial identities as identities of universal algebras

Let A and B be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose A and B are graded by a semigroup S so that the graded identical relations of A are the same as those of B. Then A is isomorphic to B as an S-graded algebra.

Simple finite-dimensional algebras without finite basis of identities

In 1993, Shestakov posed a problem of existence of a central simple finite-dimensional algebra over a field of characteristic 0 whose identities cannot be defined by a finite set (Dniester Notebook, Problem 3.103). In 2012, Isaev and the author constructed an example that gave a positive answer to this problem. In 2015, the author constructed an example of a central simple seven-dimensional commutative algebra without finite basis of identities. In this article we continue the study of Shestakov’s problem in the case of anticommutative algebras. We construct an example of a simple seven-dimensional anticommutative algebra over a field of characteristic 0 without finite basis of identities.

Classification of division gradings on finite-dimensional simple real algebras

We classify, up to isomorphism and up to equivalence, division gradings (by abelian groups) on finite-dimensional simple real algebras. Gradings on finite-dimensional simple algebras are determined by division gradings, so our results give the classification, up to isomorphism, of (not necessarily division) gradings on such algebras.Linear algebra over the field of two elements plays an interesting role in the proofs.

An Example of a Central Simple Commutative Finite-Dimensional Algebra with an Infinite Basis of Identities

In 1993 I. P. Shestakov came up with the question whether there exists a central simple finite-dimensional algebra over a field of characteristic 0, whose identities are not given by a finite set (Dniester Notebook, Question 3.103). In 2012 I. M. Isaev and the author constructed a desired example, giving an affirmative answer to the question posed. Here research into Shestakov’s question is continued for the case of commutative algebras. A seven-dimensional central simple commutative algebra over a field of characteristics 0 having no finite basis of identities is exemplified.

PI-exponents of some Simple Algebras with Unit

Over a field of characteristic zero we have constructed a discrete series of simple finite-dimensional algebras with unit. Its sequence of dimensions is not bounded, but PI-exponents are different fractional numbers strictly less than 4.

Example of Simple Finite Dimensional Algebra with No Finite Basis of Its Identities

In 1993, I. P. Shestakov formulated the following question: Are there finite dimensional central simple algebras over a field of characteristic 0 that do not have a finite basis of identities (Dniester notebook, question 3.103)? In this paper, the authors give the positive answer to Shestakov's question.

An example of a simple finite-dimensional algebra with no finite basis of identities

An example of a simple finite-dimensional algebra with no finite basis of identities

Real forms of extended Kac–Moody symmetries and higher spin gauge theories

We consider the relation between higher spin gauge fields and real Kac–Moody Lie algebras. These algebras are obtained by double and triple extensions of real forms $${\mathfrak{g}_0}$$ of the finite-dimensional simple algebras $${\mathfrak{g}}$$ arising in dimensional reductions of gravity and supergravity theories. Besides providing an exhaustive list of all such algebras, together with their associated involutions and restricted root diagrams, we are able to prove general properties of their spectrum of generators with respect to a decomposition of the triple extension of $${\mathfrak{g}_0}$$ under its gravity subalgebra $${\mathfrak{gl}(D,\mathbb {R})}$$ . These results are then combined with known consistent models of higher spin gauge theory to prove that all but finitely many generators correspond to non-propagating fields and there are no higher spin fields contained in the Kac–Moody algebra.

On codimension growth of finite-dimensional Lie superalgebras

Let A be a finite-dimensional simple algebra over a field of characteristic zero and cn(A), n=1, 2, …, its sequence of codimensions. Here we prove that e x p ( A ) = l i m n → ∞ c n ( A ) n , the PI-exponent of A, exists and is bounded from above by dim A. It is well known that, for associative or Lie or Jordan algebras, the equality exp (A)=dim A holds, provided that the main field is algebraically closed. Since simple Lie superalgebras are simple in a non-graded sense, their PI-exponent exists and here we prove that for the infinite family of Lie superalgebras of type b(t), t⩾3, the PI-exponent is strictly less than the dimension. Finally, we exhibit a seven-dimensional Lie superalgebra whose PI-exponent is strictly between 6 and 7.

Polynomial Identities of Finite Dimensional Simple Algebras

Let F be an algebraically closed field and let A and B be arbitrary finite dimensional simple algebras over F. We prove that A and B are isomorphic if and only if they satisfy the same identities.