Standardly stratified lower triangular K-algebras with enough idempotents

In this paper we study the lower triangular matrix $\mathbb{K}$-algebra $\mathrm{\Lambda }:=\left[\begin{array}{cc}\hfill T\hfill & \hfill 0\hfill \\ \hfill M\hfill & \hfill U\hfill \end{array}\right]$, where U and T are basic $\mathbb{K}$-algebras with enough idempotents and M is an U-T-bimodule where $\mathbb{K}$ acts centrally. Moreover, we characterise in terms of U, T and M when, on one hand, the lower triangular matrix $\mathbb{K}$-algebra Λ is standardly stratified in the sense of [15]; and on the other hand, when Λ is locally bounded in the sense of Gabriel [10]. Finally, we also study several properties relating the projective dimensions in the categories of finitely generated modules $\mathrm{mod}(U)$, $\mathrm{mod}(T)$ and $\mathrm{mod}(\mathrm{\Lambda })$.     

Bounds on multiplicities of spherical spaces over finite fields

Let G be a reductive group scheme of type A acting on a spherical scheme X. We prove that there exists a number C such that the multiplicity $\dim \mathrm{Hom}(\rho ,\mathbb{C}[X(F)])$ is bounded by C, for any finite field F and any irreducible representation ρ of $G(F)$. We give an explicit bound for C. We conjecture that this result is true for any reductive group scheme and when F ranges (in addition) over all local fields of characteristic 0.Different aspects of this conjecture were studied in [3], [11], [6], [7].     

On the antipode of Hopf algebras with the dual Chevalley property

In this paper, we study the antipode of a finite-dimensional Hopf algebra H with the dual Chevalley property and obtain an annihilation polynomial for the antipode. This generalizes an old result given by Taft and Wilson in 1974. As consequences, we show that 1) the quasi-exponent of H is the same as the exponent of its coradical, that is, $\mathrm{qexp}(H)=\exp ?(H_0)$; 2) $\mathrm{qexp}(H?\mathbb{k}〈S^2〉)=\mathrm{qexp}(H)$.     

Endomorphisms and cores of quadratic forms graphs in odd characteristic

A graph G is called a pseudo-core if every endomorphism of G is either an automorphism or a colouring. A graph G is a core if every endomorphism of G is an automorphism. Let ${\mathbb{F}}_q$ be the finite field with q elements where q is a power of an odd prime number. The quadratic forms graph, denoted by $\mathrm{Quad}(n,q)$ where $n\ge 2$, has all quadratic forms on ${\mathbb{F}}_q^n$ as vertices and two vertices f and g are adjacent whenever $\mathrm{rk}(f-g)=1$ or 2. We prove that every $\mathrm{Quad}(n,q)$ is a pseudo-core. Further, when n is even, $\mathrm{Quad}(n,q)$ is a core. When n is odd, $\mathrm{Quad}(n,q)$ is not a core. On the other hand, we completely determine the independence number of $\mathrm{Quad}(n,q)$.     

On the universal completion of pointfree function spaces

This paper approaches the construction of the universal completion of the Riesz space $\mathrm{C}(L)$ of continuous real functions on a completely regular frame L in two different ways. Firstly as the space of continuous real functions on the Booleanization of L. Secondly as the space of nearly finite Hausdorff continuous functions on L. The former has no counterpart in the classical theory, as the Booleanization of a spatial frame is not spatial in general, and it offers a lucid way of representing the universal completion as a space of continuous real functions. As a corollary we obtain that $\mathrm{C}(L)$ and $\mathrm{C}(M)$ have isomorphic universal completions if and only if the Booleanization of L and M are isomorphic and we characterize frames L such that $\mathrm{C}(L)$ is universally complete as almost Boolean frames. The application of this last result to the classical case $\mathrm{C}(X)$ of the space of continuous real functions on a topological space X characterizes those spaces X for which $\mathrm{C}(X)$ is universally complete. Finally, we present a pointfree version of the Maeda-Ogasawara-Vulikh representation theorem and use it to represent the universal completion of an Archimedean Riesz space with weak unit as a space of continuous real functions on a Boolean frame.     

GE2-rings and a graph of unimodular rows

For a commutative ring A we consider a related graph, $\mathrm{\Gamma }(A)$, whose vertices are the unimodular rows of length 2 up to multiplication by units. We prove that $\mathrm{\Gamma }(A)$ is path-connected if and only if A is a ${\mathrm{GE}}_2$-ring, in the terminology of P. M. Cohn. Furthermore, if $Y(A)$ denotes the clique complex of $\mathrm{\Gamma }(A)$, we prove that $Y(A)$ is simply connected if and only if A is universal for ${\mathrm{GE}}_2$. More precisely, our main theorem is that for any commutative ring A the fundamental group of $Y(A)$ is isomorphic to the group $K_2(2,A)$ modulo the subgroup generated by symbols.     

Free complex Banach lattices

The construction of the free Banach lattice generated by a real Banach space is extended to the complex setting. It is shown that for every complex Banach space E there is a complex Banach lattice ${\text{FBL}}_{\mathbb{C}}[E]$ containing a linear isometric copy of E and satisfying the following universal property: for every complex Banach lattice $X_{\mathbb{C}}$, every operator $T:E\to X_{\mathbb{C}}$ admits a unique lattice homomorphic extension $\stackrel{?}{T}:{\text{FBL}}_{\mathbb{C}}[E]\to X_{\mathbb{C}}$ with $6\stackrel{?}{T}6=6T6$. The free complex Banach lattice ${\text{FBL}}_{\mathbb{C}}[E]$ is shown to have analogous properties to those of its real counterpart. However, examples of non-isomorphic complex Banach spaces E and F can be given so that ${\text{FBL}}_{\mathbb{C}}[E]$ and ${\text{FBL}}_{\mathbb{C}}[F]$ are lattice isometric. The spectral theory of induced lattice homomorphisms on ${\text{FBL}}_{\mathbb{C}}[E]$ is also explored.