The topological center of the spectrum of the Weyl algebra$W$, i.e. the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i\in\mathbb {N}\}$ , is characterized in a recent paper by$Jabbari$and$Namioka$(Ellis group and the topological center of the flow generated by the map $n\mapsto \lambda^{n^{k}}$ , to appear in$Milan J. Math.$). By the techniques essentially used in the cited paper, the topological center of the spectrum of the subalgebra$W$_{$k$}, the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i=0,1,2,\ldots,k\}$ , will be characterized, for all$k$∈ℕ. Also an example of a non-minimal dynamical system, with the enveloping semigroup Σ, for which the set of all continuous elements of Σ is not equal to the topological center of Σ, is given.

Let$p$be a prime integer, 1≤$s$≤$r$be integers and$F$be a field of characteristic different from$p$. We find upper and lower bounds for the essential$p$-dimension ed_{$p$}( $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$ ) of the class $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$ of central simple algebras of degree$p$^{$r$}and exponent dividing$p$^{$s$}. In particular, we show that ed($Alg$_{8,2})=ed_{2}($Alg$_{8,2})=8 and ed_{$p$}( $$ Al{{g}_{{{{p}^2},p}}} $$ )=$p$^{2}+$p$for$p$odd.

Let $A$ be a finite dimensional k-algebra standardly stratified for a partial order $\leqslant$ and $\Delta$ be the direct sum of all standard modules. In this paper we study the extension algebra $E = Ext^{\ast}_A (Δ,Δ)$ of standard modules, characterize the stratification property of $E$ for $\geqslant$ and $\geqslant ^{op}$, and obtain a sufficient condition for $E$ to be a generalized Koszul algebra (in a sense which we define).

We introduce the notions of a D-standard abelian category and a K-standard additive category. We prove that for a finite dimensional algebra A, its module category is D-standard if and only if any derived autoequivalence on Ais standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. We prove that if the subcategory of projective A-modules is K-standard, then the module category is D-standard. We provide new examples of D-standard module categories.

We study endomorphism algebras of 2-term silting complexes in derived categories of hereditary finite dimensional algebras, or more generally of Ext-finite hereditary abelian categories. Module categories of such endomorphism algebras are known to occur as hearts of certain bounded t-structures in such derived categories. We show that the algebras occurring are exactly the algebras of small homological dimension, which are algebras characterized by the property that each indecomposable module either has injective dimension at most one, or it has projective dimension at most one.