Skew-symmetric Elements Research Articles

Some Commutativity Criteria for Prime Rings with Involution Involving Symmetric and Skew Symmetric Elements

Some Commutativity Criteria for Prime Rings with Involution Involving Symmetric and Skew Symmetric Elements

Finite Generation of Lie Derived Powers of Skew Lie Algebras

Let [Formula: see text] be a finitely generated associative algebra over a field of characteristic different from 2. Herstein asked when the Lie algebra [Formula: see text] is finitely generated. Recently, it was shown that for a finitely generated nil algebra [Formula: see text] all derived powers of [Formula: see text] are finitely generated Lie algebras. Let [Formula: see text] be the Lie algebra of skew-symmetric elements of an associative algebra with involution. We consider all derived powers of the Lie algebra [Formula: see text] and prove that for any finitely generated associative nil algebra with an involution, all derived powers of [Formula: see text] are finitely generated Lie algebras.

Lie identities and images of Lie polynomials for the skew-symmetric elements of UTm

ABSTRACT Let UT m be the algebra of all m × m upper triangular matrices over a field whose characteristic is different from 2. Given an involution * of the first kind on UT m , we will obtain the minimal integer t such that the Lie polynomial [[z 1, z 2], …, [z 2t−1, z 2t ]] is an identity for . Afterwards, under a mild technical restriction on , we will describe all multilinear Lie polynomials whose image evaluated on K(m, *) is the space formed by all strictly upper triangular matrices of K(m, *).

Skew-symmetric elements of rational group algebras

Let RG be the group ring of a finite group G over a commutative ring R with 1. An element x in RG is said to be skew-symmetric with respect to an involution $$\sigma $$ of RG if $$\sigma (x)=-x.$$ A structure theorem for the Lie algebra of skew-symmetric elements of FG is given where F is an algebraic extension of $$\mathbb {Q}$$ which generalizes some previously known results in this direction.

On the decomposition of metabolic involutions

The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is studied for skew-symmetric elements whose squares lie in the square of the underlying field.

Simplicity of the Lie Algebra of Skew Symmetric Elements of a Leavitt Path Algebra

For a field F of characteristic not 2 and a directed row-finite graph Γ let L(Γ) be the Leavitt Path Algebra with standard involution *. We study the lie algebra of K = K(L(Γ), *) of * −skew-symmetric elements and find necessary and sufficient conditions for the Lie algebra [K, K] to be simple.

Oriented group involutions in group algebras: a survey

Let \(\circledast :{\mathbb {F}}G\rightarrow {\mathbb {F}}G\) denote the involution obtained as a linear extension of an involution of G, twisted by the homomorphism \(\sigma :G\rightarrow \{\pm 1\}\). In this survey we gather some results concerning to the Lie properties of symmetric and skew-symmetric elements and the corresponding group identities satisfied by the set of symmetric units, and when these identities determine the structure of the whole group algebra \({\mathbb {F}}G\) [resp. unit group \({\mathcal {U}}({\mathbb {F}}G)\)].

Skew-symmetric Elements in Nonlinear Involutions in Group Rings

Given an involution in a group G, it can be extended in various ways to an involution in the group ring RG, where R is a ring, not necessarily commutative. In this paper nonlinear extensions are considered and necessary and sufficient conditions are given on the group G, its involution, the ring R and the extension for the set of skew-symmetric elements to be commutative and for it to be anticommutative.

Finite generation of Lie algebras associated with associative algebras

Let F be a field of characteristic not 2. An associative F-algebra R gives rise to the commutator Lie algebra R(−)=(R,[a,b]=ab−ba). If the algebra R is equipped with an involution ⁎:R→R then the space of the skew-symmetric elements K={a∈R|a⁎=−a} is a Lie subalgebra of R(−). In this paper we find sufficient conditions for the Lie algebras [R,R] and [K,K] to be finitely generated.

On some Montgomery's results on algebras with involution in superalgebras

We study semiprime superalgebras with superinvolution whose symmetric or skewsymmetric elements are assumed to be regular or nilpotent, and superalgebras in which xx* = 0 implies x*x = 0.

Superalgebras with superinvolution whose symmetric or skewsymmetric elements are regular

We study semiprime superalgebras with superinvolution whose symmetric elements are not zero divisors, and semiprime superalgebras with superinvolution, with nonzero odd part, whose skewsymmetric elements are not zero divisors. We prove that, in both cases, such superalgebras are a domain or the subdirect sum of a domain and its opposite.

Involutions and Anticommutativity in Group Rings

AbstractLet g denote an involution on a group G. For any (commutative, associative) ring R (with 1), * extends linearly to an involution of the group ring RG. An element α ∊ RG is symmetric if α* = α and skew-symmetric if α* = -α. The skew-symmetric elements are closed under the Lie bracket, [αβ] = αβ - βα. In this paper, we investigate when this set is also closed under the ring product in RG. The symmetric elements are closed under the Jordan product, α˚α = αβ +βα. Here, we determine when this product is trivial. These two problems are analogues of problems about the skew-symmetric and symmetric elements in group rings that have received a lot of attention.

ORIENTED INVOLUTIONS AND SKEW-SYMMETRIC ELEMENTS IN GROUP RINGS

Let G be a group with involution * and σ : G → {±1} a group homomorphism. The map ♯ that sends α = ∑ αgg in a group ring RG to α♯ = ∑ σ(g)αgg* is an involution of RG called an oriented group involution. An element α ∈ RG is symmetric if α♯ = α and skew-symmetric if α♯ = -α. The sets of symmetric and skew-symmetric elements have received a lot of attention in the special cases that * is the inverse map on G or σ is identically 1, while the general case has been almost ignored. In this paper, we determine the conditions under which the set of elements that are skew-symmetric relative to a general oriented involution form a subring of RG. This is the sequel to another paper where the analogous problem for the symmetric elements was studied, with a small oversight that is corrected here.

On ideals of the group algebra of an infinite symmetric group over a field of characteristic p

We prove that any nonzero ideal of the group algebra of the infinite symmetric group over a field of nonzero characteristic contains skew-symmetric and symmetric elements of sufficiently large order. Using this result, we reduce the question of the classification of the ideals of the group algebra of the infinite symmetric group to the classification of certain subspaces of the tensor square of a finitely generated free associative algebra.

Charting the real four-qubit Pauli group via ovoids of a hyperbolic quadric of PG(7, 2)

The geometry of the real four-qubit Pauli group, being embodied in the structure of the symplectic polar space W(7, 2), is analyzed in terms of ovoids of a hyperbolic quadric of PG(7, 2), the seven-dimensional projective space of order 2. The quadric is selected in such a way that it contains all 135 symmetric elements of the group. Under such circumstances, the third element on the line defined by any two points of an ovoid is skew-symmetric, as is the nucleus of the conic defined by any three points of an ovoid. Each ovoid thus yields 36/84 elements of the former/latter type, accounting for all 120 skew-symmetric elements of the group. There are a number of notable types of ovoid-associated subgeometries of the group, of which we mention the following: a subset of 12 skew-symmetric elements lying on four mutually skew lines that span the whole ambient space; a subset of 15 symmetric elements that corresponds to two ovoids sharing three points; a subset of 19 symmetric elements generated by two ovoids on a common point; a subset of 27 symmetric elements that can be partitioned into three ovoids in two unique ways; a subset of 27 skew-symmetric elements that exhibits a 15 + 2 × 6 split similar to that exhibited by an elliptic quadric of PG(5, 2); and a subset of seven skew-symmetric elements formed by the nuclei of seven conics having two points in common, which is an analog of a Conwell heptad of PG(5, 2). The strategy we employed is completely novel and unique in its nature, mainly due to the fact that hyperbolic quadrics in binary projective spaces of higher dimensions have no ovoids. Such a detailed dissection of the geometry of the group in question may, for example, be crucial in getting further insights into the still-puzzling black-hole-qubit correspondence/analogy.

Exponents of identities of group rings

In the paper, quantitative characteristics of identities of group algebras of finite groups are studied. A series of relations connecting the PI-exponents of identities with involution and Lie identities of the group algebra itself and of its Lie subalgebra of skew-symmetric elements are obtained.

The Lie algebra of skew-symmetric elements and its application in the theory of Jordan algebras

We prove that the Lie algebra of skew-symmetric elements of the free associative algebra of rank 2 with respect to the standard involution is generated as a module by the elements [a, b] and [a, b]3, where a and b are Jordan polynomials. Using this result we prove that the Lie algebra of Jordan derivations of the free Jordan algebra of rank 2 is generated as a characteristic F-module by two derivations. We show that the Jordan commutator s-identities follow from the Glennie-Shestakov s-identity.

On the Lie algebra of skew-symmetric elements of an enveloping algebra

Let L be a restricted Lie algebra over a field of characteristic p > 2 and denote by u ( L ) its restricted enveloping algebra. We establish when the Lie algebra of skew-symmetric elements of u ( L ) under the principal involution is solvable, nilpotent, or satisfies an Engel condition.

The Yang–Mills functional and Laplace's equation on quantum Heisenberg manifolds

In this paper, we discuss the Yang–Mills functional and a certain family of its critical points on quantum Heisenberg manifolds using noncommutative geometrical methods developed by A. Connes and M. Rieffel. In our main result, we construct a certain family of connections on a projective module over a quantum Heisenberg manifold that gives rise to critical points of the Yang–Mills functional. Moreover, we show that there is a relationship between this particular family of critical points of the Yang–Mills functional and Laplace's equation on multiplication-type, skew-symmetric elements of quantum Heisenberg manifolds; recall that Laplacian is the leading term for the coupled set of equations making up the Yang–Mills equation.

On Certain Semiprime Associative Superalgebras

In this note we emphasise the relationship between the structure of an associative superalgebra with superinvolution and the structure of the Lie substructure of skewsymmetric elements. More explicitly, we show that if A is a semiprime associative superalgebra with superinvolution and K is the Lie superalgebra of skewsymmetric elements satisfying [K 2, K 2] = 0, then A is a subdirect product of orders in simple superalgebras each at most 4-dimensional over its center.