Skew-symmetric Elements Research Articles

Involutions of RA Loops

AbstractLet L be an RA loop, that is, a loop whose loop ring over any coefficient ring R is an alternative, but not associative, ring. Let ℓ ⟼ ℓθ denote an involution on L and extend it linearly to the loop ring RL. An element α ∈ RL is symmetric if αθ = α and skew-symmetric if αθ = –α. In this paper, we show that there exists an involution making the symmetric elements of RL commute if and only if the characteristic of R is 2 or θ is the canonical involution on L, and an involution making the skew-symmetric elements of RL commute if and only if the characteristic of R is 2 or 4.

Values of Noncommutative Polynomials, Lie Skew-Ideals and Tracial Nullstellensätze

A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for involutions of the first kind and four for involutions of the second kind) and this classification result is used to characterize noncommutative polynomials via their values in these algebras. As an application, we deduce that a polynomial is a sum of commutators and a polynomial identity of $d\times d$ matrices if and only if all of its values in the algebra of $d\times d$ matrices have zero trace.

The Universal Multiplicative Envelope of the Free Malcev Superalgebra on One Odd Generator

ABSTRACT A base of the universal multiplicative envelope for the free Malcev superalgebra on one odd generator is constructed. Some corollaries for central skew-symmetric elements in free Malcev and free alternative algebras are obtained. In particular, the Malcev Grassmann algebra is constructed.

A Characterization of Lie Algebras of Skew-Symmetric Elements

A characterization of Lie algebras of skew-symmetric elements of associative algebras with involution is obtained. It is proved that a Lie algebra L is isomorphic to a Lie algebra of skew-symmetric elements of an associative algebra with involution if and only if L admits an additional (Jordan) trilinear operation {x,y,z} that satisfies the identities $$\{x,y,z\}=\{z,y,x\},$$ $$[[x,y],z]=\{x,y,z\}-\{y,x,z\},$$ $$[\{x,y,z\},t]=\{[x,t],y,z\}+\{x,[y,t],z\}+\{x,y,[z,t]\},$$ $$\{\{x,y,z\},t,v\}=\{\{x,t,v\},y,z\}-\{x,\{y,v,t\},z\}+\{x,y,\{z,t,v\}\},$$ where [x,y] stands for the multiplication in L.

Quasiclassical Lie Algebras

In this paper we study finite dimensional non-semisimple Lie algebras that can be obtained as Lie algebras of skew-symmetric elements of associative algebras with involution. We call such algebras quasiclassical and characterize them in terms of existence of so-called “∗-plain” representations. We show that the theory of ∗-plain representations for quasiclassical Lie algebras is almost equivalent to representation theory of associative algebras with involution.

-identities of non-associative algebras

The main class of algebras considered in this paper is the class of algebras of Lie type. This class includes, in particular, associative algebras, Lie algebras and superalgebras, Leibniz algebras, quantum Lie algebras, and many others. We prove that if a finite group acts on such an algebra by automorphisms and anti-automorphisms and satisfies an essential -identity, then satisfies an ordinary identity of degree bounded by a function that depends on the degree of the original identity and the order of . We show in the case of ordinary Lie algebras that if is a Lie algebra, a finite group acts on by automorphisms and anti-automorphisms, and the order of is coprime to the characteristic of the field, then the existence of an identity on skew-symmetric elements implies the existence of an identity on the whole of , with the same kind of dependence between the degrees of the identities. Finally, we generalize Amitsur's theorem on polynomial identities in associative algebras with involution to the case of alternative algebras with involution.

Primitive Superalgebras with Superinvolution

Our main purpose is to provide for primitive associative superalgebras a structure theory analogous to that for algebras [5,6,10] and to classify primitive superrings with superinvolution having a minimal one-sided superideal. We were led to this problem by our work on finite dimensional central simple Jordan superalgebras over fields of characteristic not 2 [9] (see also [7]). Of course, just as symmetric elements give rise to Jordan superalgebras, skewsymmetric elements give rise to Lie superalgebras [8,4]. The results and methods are closely related to those of structure theory of associative rings and central simple associative algebras with involution [5, Chap. I;6, Chaps. II, III;1, Chap. X;10, Chap. 2]. Some of the results have been announced in [13].

Irreducible structurable bimodules for algebras generated by skew-symmetric elements

Irreducible bimodules over finite-dimensional central structurable algebras generated by skew-symmetric elements are described.

A theorem of artin for skew-alternative algebras

In the article, we prove a theorem stating that every skew-alternative algebra over a field Ф of characteristic ≠ 2, 3 is associative whenever it is generated by two skew-symmetric elements. This is an analog of the well-known theorem of Artin which states that any alternative algebra generated by two elements is associative.

Annihilators in rings with involution

In this paper we examine the annihilators of algebraic expressions in the symmetric, or skew-symmetric, elements of a ringR with involution. LetL be the set of elements ofR each of which is annihilated on the right by some power of each symmetric element. IfR contains no nonzero nil left ideal thenL=0. The subset ofL annihilated by a fixed power of each symmetric element is always a nil left ideal ofR of bounded index. WhenR is an algebra over a fieldF, letA be the left ideal of elements annihilated on the right by some polynomial in each symmetric element. If eitherF is uncountable or the degrees of the polynomials are bounded, thenA generates an algebraic ideal ofR. Similar results are obtained by using skew-symmetric elements instead of symmetric elements.

Skew-symmetric elements in the group algebra of a symmetric group

Skew-symmetric elements in the group algebra of a symmetric group

Invariant Subgroups in Rings with Involution

Let R be a ring with involution*. In this paper, we study additive subgroups A of R which are invariant under all mappings of the form ϕ x : a → xax*. That is, xAx* ⊆ A for all x ∈ R. Obvious examples of such subgroups A are ideals of R, the set of symmetric elements, and the set of skew-symmetric elements. We will prove that when R is *-prime, these examples are essentially the only ones.

The lie structure in prime rings with involution

Consider the Lie subring K, the skew-symmetric elements of an associative ring R with involution ∗. I. N. Herstein and W. E. Baxter have investigated the Lie structure of K and [ K, K]when R is a simple ring with involution ∗. We extend these results to show that, in prime rings with involution ∗ which are 2-torsion-free, any Lie ideal of K or [ K, K] contains [ J ∩ K, K]for some nonzero ∗-ideal J of R or is contained in the center of R.

Some properties of skew-symmetric elements of a ring

Let R* be a ring with unit element I. For every A ∈ R*, let the mapping A → A′ be an involutory anti-automorphism, i.e. a one-to-one mapping of R* onto itself with the following properties:Let C* denote the set of all symmetric elements of R*, i.e. C′ = C for all C ∈ C*.An element A ∈ R* is said to be skew-symmetric if A′ + A = 0 (the zero element of the ring).