A note on the modular representation on the $\mathbb Z/2$-homology groups of the fourth power of real projective space and its application

We write $BV_h$ for the classifying space of the elementary Abelian 2-group $V_h$ of rank $h,$ which is homotopy equivalent to the cartesian product of $h$ copies of $\mathbb RP^{\infty}.$ Its cohomology with $\mathbb Z/2$-coefficients can be identified with the graded unstable algebra $P^{\otimes h} = \mathbb Z/2[t_1, \ldots, t_h]= \bigoplus_{n\geq 0}P^{\otimes h}_n$ over the Steenrod ring $\mathcal A$, where grading is by the degree of the homogeneous terms $P^{\otimes h}_n$ of degree $n$ in $h$ generators with the degree of each $t_i$ being one. Let $GL_h$ be the usual general linear group of rank $h$ over $\mathbb Z/2.$ The algebra $P^{\otimes h}$ admits a left action of $\mathcal A$ as well as a right action of $GL_h.$ A central problem of homotopy theory is to determine the structure of the space of $GL_h$-coinvariants, $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}H_n(BV_h; \mathbb Z/2) ,$ where ${\rm Ann}_{\overline{\mathcal A}}H_n(BV_h; \mathbb Z/2) ={\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ denotes the space of primitive homology classes, considered as a representation of $GL_h$ for all $n.$ Solving this problem is very difficult and still unresolved for $h\geq 4.$ The aim of this Note is of studying the dimension of $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ for the case $h = 4$ and the "generic" degrees $n$ of the form $k(2^{s} - 1) + r.2^{s},$ where $k,\, r,\, s$ are positive integers. Applying the results, we investigate the behaviour of the Singer cohomological "transfer" of rank $4$, which is a homomorphism from a certain subquotient of the divided power algebra $\Gamma(a_1^{(1)}, \ldots, a_4^{(1)})$ to mod-2 cohomology groups ${\rm Ext}_{\mathcal A}^{4, 4+n}(\mathbb Z/2, \mathbb Z/2)$ of the algebra $\mathcal A.$ Singer's algebraic transfer is one of the relatively efficient tools in determining the cohomology of the Steenrod algebra.

Hyperrigidity of C*-Correspondences

We show that hyperrigidity for a C*-correspondence (A, X) is equivalent to non-degeneracy of the left action of the Katsura ideal $$\mathcal {J}_X$$ J X on X. This extends the work of Kakariadis (Bull Lond Math Soc 45(6):1119–1130, 2013, Theorem 3.3) and Dor-On and Salomon (J Lond Math Soc 98(2):416–438, 2018, Theorem 3.5) who establish this equivalence for discrete graphs as well as the work of Katsoulis and Ramsey (The non-selfadjoint approach to the Hao–Ng isomorphism, arXiv preprint arXiv:1807.11425, 2018, Theorem 3.1), who establish one direction of this equivalence.

Fundamental Vector Fields

This chapter addresses fundamental vector fields. The concept of a connection on a principal bundle is essential in the construction of the Cartan model. To define a connection on a principal bundle, one first needs to define the fundamental vector fields. When a Lie group acts smoothly on a manifold, every element of the Lie algebra of the Lie group generates a vector field on the manifold called a fundamental vector field. On a principal bundle, the fundamental vectors are precisely the vertical tangent vectors. In general, there is a relation between zeros of fundamental vector fields and fixed points of the group action. Unless specified otherwise (such as on a principal bundle), a group action is assumed to be a left action.

Matrix semigroups over semirings

We study properties determined by idempotents in the following families of matrix semigroups over a semiring [Formula: see text]: the full matrix semigroup [Formula: see text], the semigroup [Formula: see text] consisting of upper triangular matrices, and the semigroup [Formula: see text] consisting of all unitriangular matrices. Il’in has shown that (for [Formula: see text]) the semigroup [Formula: see text] is regular if and only if [Formula: see text] is a regular ring. We show that [Formula: see text] is regular if and only if [Formula: see text] and the multiplicative semigroup of [Formula: see text] is regular. The notions of being abundant or Fountain (formerly, weakly abundant) are weaker than being regular but are also defined in terms of idempotents, namely, every class of certain equivalence relations must contain an idempotent. Each of [Formula: see text], [Formula: see text] and [Formula: see text] admits a natural anti-isomorphism allowing us to characterise abundance and Fountainicity in terms of the left action of idempotent matrices upon column spaces. In the case where the semiring is exact, we show that [Formula: see text] is abundant if and only if it is regular. Our main interest is in the case where [Formula: see text] is an idempotent semifield, our motivating example being that of the tropical semiring [Formula: see text]. We prove that certain subsemigroups of [Formula: see text], including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We also consider the subsemigroups [Formula: see text] and [Formula: see text] consisting of those matrices of [Formula: see text] and [Formula: see text] having all elements on and above the leading diagonal non-zero. We prove the idempotent generated subsemigroup of [Formula: see text] is [Formula: see text]. Further, [Formula: see text] and [Formula: see text] are families of Fountain semigroups with interesting and unusual properties. In particular, every [Formula: see text]-class and [Formula: see text]-class contains a unique idempotent, where [Formula: see text] and [Formula: see text] are the relations used to define Fountainicity, but yet the idempotents do not form a semilattice.

On Weak Graded Rings II

The G-weak graded rings are rings graded by a set G of left coset representatives for the left action of a subgroup H of a finite group X. The main aim of this article is to study the concept of G-weak graded rings and continue the investigation of their properties. Moreover, some results concerning G-weak graded rings of fractions are derived. Finally, some additional examples of G-weak graded rings are provided.

The Primitive Spectrum and Category for the Periplectic Lie Superalgebra

We solve two problems in representation theory for the periplectic Lie superalgebra $\mathfrak{p}\mathfrak{e}(n)$, namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category ${\mathcal{O}}$ into indecomposable blocks.To solve the first problem, we establish a new type of equivalence between category ${\mathcal{O}}$ for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.

ADJOINT FUNCTORS BETWEEN CATEGORIES OF HILBERT -MODULES

Let$E$be a (right) Hilbert module over a$C^{\ast }$-algebra$A$. If$E$is equipped with a left action of a second$C^{\ast }$-algebra$B$, then tensor product with$E$gives rise to a functor from the category of Hilbert$B$-modules to the category of Hilbert$A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clareet al.[Parabolic induction and restriction via$C^{\ast }$-algebras and Hilbert$C^{\ast }$-modules,Compos. Math.FirstView(2016), 1–33, 2].

On the Density of Certain Languages with $p^2$ Letters

The sequence $(x_n)_{n\in\mathbb N} = (2,5,15,51,187,\ldots)$ given by the rule $x_n=(2^n+1)(2^{n-1}+1)/3$ appears in several seemingly unrelated areas of mathematics. For example, $x_n$ is the density of a language of words of length $n$ with four different letters. It is also the cardinality of the quotient of $(\mathbb Z_2\times \mathbb Z_2)^n$ under the left action of the special linear group $\mathrm{SL}(2,\mathbb Z)$. In this paper we show how these two interpretations of $x_n$ are related to each other. More generally, for prime numbers $p$ we show a correspondence between a quotient of $(\mathbb Z_p\times\mathbb Z_p)^n$ and a language with $p^2$ letters and words of length $n$.

Permanent Weak Module Amenability of Semigroup Algebras

We employ the fact that L1(G) is n-weakly amenable for each n ≥ 1 to show that for an inverse semigroup S with the set of idempotents E, ℓ1(S) is n- weakly module amenable as an ℓ1(E)-module with trivial left action. We study module amenability and weak module amenability of the module projective tensor products of Banach algebras.