Search

The main purpose of this paper is to provide a survey of results concerning the Z-gradings on the infinitedimensional Grassmann algebra E over a field of characteristic zero. First, we provide graded identities and central polynomials for E equipped with fine gradings on E by the semigroup (Z*, ×). We also describe briefly techniques in order to illustrate some important methods to exhibit graded identities and central polynomials of E for other abelian groups. In particular, over a field of characteristic zero, so-called 2-induced gradings of full support were considered. In order to obtain these descriptions, we strongly use elementary number theory as a tool, providing an interesting connection between this area and PI-Theory. [ABSTRACT FROM AUTHOR]

Let F be the free metabelian associative algebra generated by x and y over a field of characteristic zero. We call a polynomial f F symmetric, if f(x, y) = f(y, x). The set of all symmetric polynomials coincides with the algebra FS2 of invariants of the symmetric group S2. In this paper, we give the full description of the algebra FS2. [ABSTRACT FROM AUTHOR]

This paper deals with an exterior algebra of a vector space whose base field is of positive characteristic. In this work, a minimal set of generators forming the annihilator of even neat elements of such an exterior algebra is exhibited. The annihilator of some special type of even neat elements is determined to prove the conjecture established in [3]. Moreover, a vector space basis for the annihilators under consideration is calculated. [ABSTRACT FROM AUTHOR]

Let F be an algebraically closed field of characteristic zero. In this paper we deal with matrix superalgebras (i.e. algebras graded by Z2, the cyclic group of order 2) endowed with a pseudoinvolution. The first goal is to present the classification of the pseudoinvolutions that it is possible to define, up to equivalence, in the full matrix algebra Mn(F) of n × n matrices and on its subalgebra UTn(F) of upper-triangular matrices. Along the way we shall give the generators of the T -ideal of identities for the algebras M2(F), UT2(F) and UT3(F), endowed with all possible inequivalent pseudoinvolutions. [ABSTRACT FROM AUTHOR]

Let Pn:= H * ((RP 8)n) ~= Z2[x1, x2,..., xn] be the polynomial algebra of n generators x1, x2,..., xn with the degree of each xi being 1. We investigate the Peterson hit problem for the polynomial algebra Pn, regarded as a module over the mod-2 Steenrod algebra, A. For n > 4, this problem remains unsolvable, even with the aid of computers in the case of n = 5. In this article, we study the hit problem for the case n = 6 in degree ds = 6(2s - 1) + 3.2s, with s an arbitrary nonnegative integer. By considering Z2 as a trivial A-module, then the hit problem is equivalent to the problem of finding a basis of Z2 -vector space Z2APn. The main goal of the current article is to explicitly determine an admissible monomial basis of the Z2 vector space Z2APn for n = 6 in some degrees. One of the most important applications of the hit problem is to investigate homomorphism introduced by Singer, which is a homomorphism fn: TorA from the homology of the Steenrod algebra to the subspace of (Z2APn)d consisting of all the GL(n; Z2) -invariant classes. It is a useful tool in describing the homology groups of the Steenrod algebra, TorA n, n+d(Z2, Z2). The behavior of the sixth Singer algebraic transfer in degree ds = 6(2s - 1) + 3.2s was also discussed at the end of this paper. [ABSTRACT FROM AUTHOR]

By a result of Margarete Wolf in 1936, we know that the algebra KXdSym(d) of symmetric polynomials in noncommuting variables is not finitely generated. In 1984, Koryukin proved that if we equip the homogeneous component of degree n with the additional action of Sym(n) by permuting the positions of the variables, then the algebra of invariants KXdG of every reductive group G is finitely generated. First, we make a short comparison between classical invariant theory of finite groups and its noncommutative counterpart. Then, we expose briefly the results of Wolf. Finally, we present the main result of our paper, which is, over a field of characteristic 0 or of characteristic p > d, the algebra KXdSym(d) with the action of Koryukin is generated by the elementary symmetric polynomials. [ABSTRACT FROM AUTHOR]

The notion of a F -manifold algebras is an algebraic description of a F -manifold. In this paper, we introduce the notion of Hom-F -manifold algebras which is generalisation of F -manifold algebras and Hom-Poisson algebras. We develop the representation theory of Hom-F -manifold algebras and generalize the notion of Hom-pre-Poisson algebras by introducing the Hom-pre-F -manifold algebras which give rise to a Hom-F -manifold algebra through the subadjacent commutative Hom-associative algebra and the subadjacent Hom-Lie algebra. Using O-operators on a Hom-F -manifold algebras we construct a Hom-pre-F -manifold algebras on a module. Then, we study Hom-pre-Lie formal deformations of commutative Hom-associative algebra and we prove that Hom-F -manifold algebras are the corresponding semiclassical limits. Finally, we study Hom-Lie infinitesimal deformations and extension of Hom-pre-Lie n-deformation to Hom-pre-Lie (n + 1)-deformation of a commutative Hom-associative algebra via cohomology theory. [ABSTRACT FROM AUTHOR]

The stresses within the soft tissue are not constant for some shell surfaces. They vary with position along the mantle edge. In this paper, we show that elliptical geometry is more convenient to describe this type of surface. Thus, we introduce the elliptical kinematics along an initial curve and construct some accretive surfaces with an elliptical cross-section. In fact, these surfaces are not only curves with an elliptical cross-sectional curve, but also the material points of the surface follow an elliptical trajectory during their formation. This situation can be easily explained through elliptical motion and elliptical quaternion algebra. Then, we investigate the relationship between velocity and eccentricity of the surfaces and compare it to the case of circular motion. Furthermore, we visualize some examples to support the theoretical results through the MAPLE program. [ABSTRACT FROM AUTHOR]

“Fusion rules” are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras, in turn, are closely related to 3-transposition groups and Vertex operator algebras. In this paper we consider fusion rules for semisimple idempotents, following Albert in the power-associative case. We examine the notion of an axis in the non-commutative setting and show that the dimension d of any algebra A generated by a pair a, b of (not necessarily Jordan) axes of respective types (λ, δ) and (λ ′, δ ′ ) must be at most 5; d cannot be 4. If d ≤ 3 we list all the possibilities for A up to isomorphism. We prove a variety of additional results and mention some research questions at the end. [ABSTRACT FROM AUTHOR]

In this paper, we give categorical properties such as pullback, finite product, finite limit, coproduct, colimit and pushout in XHom -- Lie/A of the category of Hom-Lie crossed modules. [ABSTRACT FROM AUTHOR]

We describe the GLn(F) -invariant functions on Mn(G) (where G is the infinite dimensional Grassmann algebra) and show that not all of them are trace polynomials, if n = 3. [ABSTRACT FROM AUTHOR]

For an Arens-Michael algebra A we consider a class of A-...-bimodules which are invertible with respect to the projective bimodule tensor product. We call such bimodules topologically invertible over A. Given a Fréchet-Arens-Michael algebra A and a topologically invertible Fréchet A-...-bimodule M, we construct an Arens-Michael algebra LA(M) which serves as a topological version of the Laurent tensor algebra ...A(M). Also, for a fixed algebra B we provide a condition on an invertible B-bimodule N which allows us to explicitly describe the Arens-Michael envelope of LB(N) as a topological Laurent tensor algebra. In particular, we provide an explicit description of the Arens-Michael envelope of an invertible Ore extension A[x, x-1; α] for a metrizable algebra A. [ABSTRACT FROM AUTHOR]

Let Q = (a,b / R) denote the quaternion algebra over the reals which is by the Frobenius Theorem either split or the division algebra H of Hamilton's quaternions. We have presented explicitly in [4] the matrix of a typical derivation of Q. Given a derivation d ∈ Der(H), we show that the matrix D ∈ Ma (ℝ) that represents d on the linear subspace ... of pure quaternions provides a pair of idempotent matrices AdjD and - D² that correspond bijectively to the involutary matrix Ʃof a quaternion involution σ and present several equations involving these matrices. In particular, we deal with commuting derivations of H and introduce some results to guarantee commutativity. We also mention briefly eigenspace decomposition of a derivation. [ABSTRACT FROM AUTHOR]

Let A be an associative algebra endowed with an automorphism or an antiautomorphism f of order = 2. One associates to A, in a natural way, a numerical sequence cfn (A), n = 1, 2,..., called the sequence of f-codimensions of A which is the main tool for the quantitative investigation of the polynomial identities satisfied by A. In [13] it was proved that such a sequence is eventually nondecreasing in case f is an antiautomorphism. Here we prove that it still holds in case f is an automorphism and present some recent results about the asymptotics of cfn(A). [ABSTRACT FROM AUTHOR]

We deal with varieties of nonassociative algebras having polynomial growth of codimensions. We describe some results obtained in recent years in the class of left nilpotent algebras of index two. Recently the authors established a correspondence between the growth rates for left nilpotent algebras of index two and the growth rates for commutative or anticommutative metabelian algebras that allows to transfer the results concerning varieties of left nilpotent algebras of index two to varieties of commutative or anticommutative metabelian algebras. [ABSTRACT FROM AUTHOR]

Let A be an associative algebra over a field of characteristic zero. A central polynomial is a polynomial of the free associative algebra that takes central values of A. In this survey, we present some recent results about the exponential growth of the central codimension sequence and the proper central codimension sequence in the setting of algebras with involution and algebras graded by a finite group. [ABSTRACT FROM AUTHOR]

Let GL(n) = GL(n, C) denote the complex general linear group and let G GL(n) be one of the classical complex subgroups O(n), SO(n), and Sp(2k) (in the case n = 2k). We take a finite dimensional polynomial GL(n) - module W and consider the symmetric algebra S(W). Extending previous results for G = SL(n), we develop a method for determining the Hilbert series H(S(W)G, t) of the algebra of invariants S(W)G. Our method is based on simple algebraic computations and can be easily realized using popular software packages. Then we give many explicit examples for computing H(S(W)G, t). As an application, we consider the question of regularity of the algebra S(W)O(n). For n = 2 and n = 3 we give a complete list of modules W, so that if S(W)O(n) is regular then W is in this list. As a further application, we extend our method to compute also the Hilbert series of the algebras of invariants (S2V)G and (2V)G, where V = Cn denotes the standard GL(n) -module. [ABSTRACT FROM AUTHOR]

Let A be an algebra over a field F with char(F) 1= 2. If A is generated as an algebra by [[A,A], [A,A]], then for every skew-symmetric bilinear map F: A×A X, where X is an arbitrary vector space over F, the condition that F(x2, x) = 0 for all x A implies that F(xy, z) + F(zx, y) + F(yz, x) = 0 for all x, y, z A. This is applicable to the question of whether A is zero Lie product determined and is also used in proving that a Jordan homomorphism from A onto a semiprime algebra B is the sum of a homomorphism and an antihomomorphism. [ABSTRACT FROM AUTHOR]

We present recent results about Capelli polynomials with involution or graded involution and their asymptotics. In the associative case, the asymptotic equality between the codimensions of the T -ideal generated by the Capelli polynomial of rank k2 + 1 and the codimensions of the matrix algebra Mk(F) was proved. This result was extended to superalgebras. Recently, similar results have been determined by the authors in the case of algebras with involution and superalgebras with graded involution. [ABSTRACT FROM AUTHOR]