On analogs of some classical group-theoretic results in Poisson algebras

We investigate the Poisson algebras, in which the n-th hypercenter (center) has a finite codimension. It was established that, in this case, the Poisson algebra P includes a finite-dimensional ideal K such that P/K is nilpotent (Abelian). Moreover, if the n-th hypercenter of a Poisson algebra P over some field has a finite codimension, and if P does not contain zero divisors, then P is Abelian.

On extension of classical Baer results to Poisson algebras

In this paper we prove that if P is a Poisson algebra and the n-th hypercenter (center) of P has a finite codimension, then P includes a finite-dimensional ideal K such that P/K is nilpotent (abelian). As a corollary, we show that if the nth hypercenter of a Poisson algebra P (over some specific field) has a finite codimension and P does not contain zero divisors, then P is an abelian algebra.

Residually finite dimensional algebras and polynomial almost identities

Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon,

A Hitchin Connection for a Large Class of Families of Kähler Structures

This chapter presents a Hitchin connection constructed in a setting which significantly generalizes the setting covered by the first author, which, in turn, was a generalization of the moduli space covered in the original work on the Hitchin connection. In fact, this construction provides a Hitchin connection which is a partial connection on the space of all compatible complex structures on an arbitrary but fixed prequantizable symplectic manifold which satisfies a certain Fano-type condition. The subspace of the tangent space to the space of compatible complex structures on which the constructed Hitchin connection is defined is of finite codimension if the symplectic manifold is compact. It also proves uniqueness of the Hitchin connection under a further assumption. A number of examples show that this Hitchin connection is defined in a neighbourhood of the natural families of complex structures compatible with the given symplectic form which these spaces admit.

EQUIVALENT NORMS WITH AN EXTREMELY NONLINEABLE SET OF NORM ATTAINING FUNCTIONALS

We present a construction that enables one to find Banach spaces$X$whose sets$\operatorname{NA}(X)$of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently,$X$does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2,Israel J. Math.(to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces,J. Funct. Anal. 272(2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space$X$where the set$\operatorname{NA}(X)$for the original norm is not “too large”. The construction can be applied to every Banach space containing$c_{0}$and having a countable system of norming functionals, in particular, to separable Banach spaces containing$c_{0}$. We also provide some geometric properties of the norms we have constructed.

Note on algebras which are sums of two PI subalgebras

We study an associative algebra A over an arbitrary field that is a sum of two subalgebras A1 and A2 (i.e. A = A1 + A2). Additionally we assume that Ai has an ideal of finite codimension in Ai which satisfies a polynomial identity fi = 0 for i = 1, 2. Suppose that all rings R = R1 + R2, which are sums of subrings R1 and R2, are PI rings when Ri satisfies the polynomial identity fi = 0 for i = 1, 2. We prove that A is a PI algebra.

Real nullstellensatz and *-ideals in *-algebras

For a fixed tuple of square matrices X ={X_1,...,X_g} the set I(X) of all noncommutative polynomials p in X and X∗ such that p(X) = 0 is an ideal in the ∗-algebra of all polynomials. This article concerns such zeroes and their corresponding ideals. An algebraic characterization of ideals of the form I(X) is a real nullstellensatz. A main result of this article is a strong nullstellensatz for a ∗-ideal of finite codimension in a ∗-algebra. Without the finite codimension assumption, there are examples of such ideals which do not satisfy, very liberally interpreted, any Nullstellensatz. A polynomial p in noncommuting variables (x_1,...,x_g,x∗_1,...,x_∗g) is called analytic if it is a polynomial in the variables x_j only. As shown in this article, ∗-ideals generated by analytic polyno-mials do satisfy a natural Nullstellensatz and those generated by homogeneous analytic polynomials have a particularly simple description. Another natural notion of zero of a noncommutative polynomial p is a pair (X, v) such that p(X)v = 0; here X is an n by n matrix tuple and v ∈ R^n. For fixed (X,v), the set of all such polynomials is a left ideal. The relationship between such zeroes and their left ideals is considerably more developed than is our beginning effort here. This article provides a guide to that literature.