Symmetric polynomials of algebras related with 2 × 2 generic traceless matrices

Let [Formula: see text] and [Formula: see text] be two generic traceless matrices of size [Formula: see text] with entries from a commutative associative polynomial algebra over a field [Formula: see text] of characteristic zero. Consider the associative unitary algebra [Formula: see text], and its Lie subalgebra [Formula: see text] generated by [Formula: see text] and [Formula: see text] over the field [Formula: see text]. It is well known that the center [Formula: see text] of [Formula: see text] is the polynomial algebra generated by the algebraically independent commuting elements [Formula: see text], [Formula: see text], [Formula: see text]. We call a polynomial [Formula: see text] symmetric, if [Formula: see text]. The set of symmetric polynomials is equal to the algebra [Formula: see text] of invariants of symmetric group [Formula: see text]. Similarly, we define the Lie algebra [Formula: see text] of symmetric polynomials in the Lie algebra [Formula: see text]. We give the description of the algebras [Formula: see text] and [Formula: see text], and we provide finite sets of free generators for [Formula: see text], and [Formula: see text] as [Formula: see text]-modules.

Varieties of unitary algebras with small growth of codimensions

In the last years, the sequence of codimensions of PI-algebras has been studied by several authors and the classification of unitary algebras, up to equivalence, with at most cubic codimension growth was given by Giambruno, La Mattina and Petrogradsky in 2007. In this paper, we establish a new approach by studying the possibilities of specific proper codimensions of a unitary algebra with growth [Formula: see text] in order to present a complete list of varieties generated by unitary algebras with polynomial growth [Formula: see text]. Also, we classify, up to PI-equivalence, the unitary algebras with growth [Formula: see text] whose leading coefficient of the polynomial describing the codimension sequence achieves the largest and the smallest possible value.

Weyl modules and Weyl functors for hyper-map algebras

We investigate the representations of the hyperalgebras associated to the map algebras [Formula: see text], where [Formula: see text] is any finite-dimensional complex simple Lie algebra and [Formula: see text] is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.

THE CENTRAL POLYNOMIALS OF THE INFINITE-DIMENSIONAL UNITARY AND NONUNITARY GRASSMANN ALGEBRAS

We describe the T-space of central polynomials for both the unitary and the nonunitary infinite-dimensional Grassmann algebra over a field of characteristic p≠2 (infinite field in the case of the unitary algebra).

THE CENTRAL POLYNOMIALS OF THE FINITE-DIMENSIONAL UNITARY AND NONUNITARY GRASSMANN ALGEBRAS

We describe the T-space of central polynomials for both the unitary and the nonunitary finite dimensional Grassmann algebra over a field of characteristic p ≠ 2 (infinite field in the case of the unitary algebra).