Symmetric polynomials in the variety generated by Grassmann algebras

Let [Formula: see text] denote the variety generated by infinite-dimensional Grassmann algebras, i.e. the collection of all unitary associative algebras satisfying the identity [Formula: see text], where [Formula: see text]. Consider the free algebra [Formula: see text] in [Formula: see text] generated by [Formula: see text]. We call a polynomial [Formula: see text] symmetric if it is preserved under the action of the symmetric group [Formula: see text] on generators, i.e. [Formula: see text] for each permutation [Formula: see text]. The set of symmetric polynomials forms the subalgebra [Formula: see text] of invariants of the group [Formula: see text] in [Formula: see text]. The commutator ideal [Formula: see text] of the algebra [Formula: see text] has a natural left [Formula: see text]-module structure, and [Formula: see text] is a left [Formula: see text]-module. We give a finite free generating set for the [Formula: see text]-module [Formula: see text].

The Nowicki conjecture for free metabelian Lie algebras

Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables over a field [Formula: see text] of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations [Formula: see text] (known as Weitzenböck derivations), the algebra of constants [Formula: see text] is finitely generated. When the Weitzenböck derivation [Formula: see text] acts on the polynomial algebra [Formula: see text] in [Formula: see text] variables by [Formula: see text], [Formula: see text], [Formula: see text], Nowicki conjectured that [Formula: see text] is generated by [Formula: see text] and [Formula: see text] for all [Formula: see text]. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free [Formula: see text]-generated metabelian Lie algebra [Formula: see text], with few trivial exceptions, the algebra [Formula: see text] is not finitely generated. However, the vector subspace [Formula: see text] of the commutator ideal [Formula: see text] of [Formula: see text] is finitely generated as a [Formula: see text]-module. In this paper, we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the [Formula: see text]-module [Formula: see text].

Weitzenböck derivations of free metabelian associative algebras

By the classical theorem of Weitzenböck the algebra of constants [Formula: see text] of a nonzero locally nilpotent linear derivation [Formula: see text] of the polynomial algebra [Formula: see text] in several variables over a field [Formula: see text] of characteristic 0 is finitely generated. As a noncommutative generalization one considers the algebra of constants [Formula: see text] of a locally nilpotent linear derivation [Formula: see text] of a finitely generated relatively free algebra [Formula: see text] in a variety [Formula: see text] of unitary associative algebras over [Formula: see text]. It is known that [Formula: see text] is finitely generated if and only if [Formula: see text] satisfies a polynomial identity which does not hold for the algebra [Formula: see text] of [Formula: see text] upper triangular matrices. Hence the free metabelian associative algebra [Formula: see text] is a crucial object to study. We show that the vector space of the constants [Formula: see text] in the commutator ideal [Formula: see text] is a finitely generated [Formula: see text]-module, where [Formula: see text] acts on [Formula: see text] and [Formula: see text] in the same way as on [Formula: see text]. For small [Formula: see text], we calculate the Hilbert series of [Formula: see text] and find the generators of the [Formula: see text]-module [Formula: see text]. This gives also an (infinite) set of generators of the algebra [Formula: see text].

RINGS SATISFYING GENERALIZED ENGEL CONDITIONS

Let R be an associative ring and let x, y ∈ R. Define the generalized commutators as follows: [x, 0y] = x and [x, ky] = [x, k-1y]y - y[x, k-1y](k = 1, 2, …). In this paper we study some generalized Engel rings, i.e. [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)y] = 0), [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)yn(x, y)] = 0) and [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)yn(x, y)]r(x, y) = 0). Among other results, it is proved that every Artinian [Formula: see text]-ring is strictly Lie-nilpotent. Also, we show that in each of the following cases R has nil commutator ideal: (1) if R is a [Formula: see text]-ring with unity and k, n independent of y; (2) if R is a locally bounded [Formula: see text]-ring (defined below); (3) if R is an algebraic algebra over a field in which R* is a bounded Engel group or a soluble group.

On Braided Lie Structures of Algebras in the Categories of Weak Hopf Bimodules

Let H be a weak Hopf algebra. In this paper, it is proved that the monoidal category [Formula: see text] of weak Hopf bimodules studied in Wang [19] is equivalent to the monoidal category [Formula: see text] of weak Yetter–Drinfel'd modules introduced in Böhm [2]. When H has a bijective antipode, a braiding in the category [Formula: see text] is constructed by the braiding on [Formula: see text], generalizing the main result in Schauenburg [14]. Finally, the braided Lie structures of an algebra A in the category [Formula: see text] are investigated, by showing that if A is a sum of two braided commutative subalgebras, then the braided commutator ideal of A is nilpotent.

Jordan homomorphisms revisited

Let θ be a Jordan homomorphism from an algebraAinto an algebraB. We find various conditions under which the restriction of θ to the commutator ideal ofAis the sum of a homomorphism and an antihomomorphism. Algebraic results, obtained in the first part of the paper, are applied to the second part dealing with the case whereAandBareC*-algebras.

NONCOMMUTATIVE GRÖBNER BASES FOR THE COMMUTATOR IDEAL

Let I denote the commutator ideal in the free associative algebra on m variables over an arbitrary field. In this article we prove there are exactly m! finite Gröbner bases for I, and uncountably many infinite Gröbner bases for I with respect to total division orderings. In addition, for m = 3 we give a complete description of its universal Gröbner basis.

Weakly periodic and subweakly periodic rings

Our objective is to study the structure of subweakly periodic rings with a particular emphasis on conditions which imply that such rings are commutative or have a nil commutator ideal. Related results are also established for weakly periodic (as well as periodic) rings.