Effect Algebras as Presheaves on Finite Boolean Algebras

For an effect algebra A, we examine the category of all morphisms from finite Boolean algebras into A. This category can be described as a category of elements of a presheaf R(A) on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra A can be characterized in terms of some properties of the category of elements of the presheaf R(A). We prove that the tensor product of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. The tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.     

Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain \(\hat \otimes\)-algebras. We use well-developed homological techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a continuous morphism ?: x → y of complexes of complete nuclear DF-spaces, the isomorphism of cohomology groups H ⁿ(?): H ⁿ (x) → H ⁿ (y) is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective \(\hat \otimes\)-algebras: the tensor algebra E \(\hat \otimes\) F generated by the duality (E,F,<·,·>) for nuclear Fréchet spaces E and F or for nuclear DF-spaces E and F; nuclear biprojective Köthe algebras λ(P) which are Fréchet spaces or DF-spaces; the algebra of distributions ε^*(G) on a compact Lie group G.     

Modules for Yokonuma-type Hecke Algebras

This paper describes the module categories for a family of generic Hecke algebras, called Yokonuma-type Hecke algebras. Yokonuma-type Hecke algebras specialize both to the group algebras of the complex reflection groups G(r,1,n) and to the convolution algebras of (B \(^{\prime }\),B \(^{\prime }\))-double cosets in the group algebras of finite general linear groups, for certain subgroups B \(^{\prime }\) consisting of upper triangular matrices. In particular, complete sets of inequivalent, irreducible modules for semisimple specializations of Yokonuma-type Hecke algebras are constructed.     

Dualities and the duality density theorems

Dualities \({\langle,\, \rangle:S \times T \rightarrow K}\) for modules S =  _R S, T = T _D and bimodule K =  _R K _D over rings R, D are non-degenerate left dense K-pairings of S, T intertwined per adjoint, classification, and Galois correspondence theorems. Dualities are abundant per density theorems inspired by those of Jacobson and Chevalley. Duality theory generalizes classical duality theory and leads to a theory of duality semi-simplicity of rings R and R-modules S. The finite dimensional duality semi-simple algebras R are classified in terms of semi-simple algebras and bipolar algebras.     

Span of a DL-algebra

Unless otherswise specified, all objects are defined over a field k of characteristic 0. Let K be a field. The unessentialness of an extension of the algebra Der K by means of a splittable semisimple Lie algebra is established. Let D _K be the category of differential Lie algebras (DL-algebras) (g;K). In this paper for an extension L/K the functor η:D _K → D _L , defining the tensor product L ? _K of vector spaces and the homomorphism of Lie algebras, is constructed. If the extension L/K is algebraic, then η is unique. The results will be required for strengthening the progress on Gelfand–Kirillov problem and weakened conjecture [1, 2].     

Lexicographic effect algebras

We investigate a class of effect algebras that can be represented in the form \({\Gamma (H \overrightarrow{\times} G}\), (u, 0)), where \({H \overrightarrow{\times} G}\) means the lexicographic product of an Abelian unital po-group (H, u) and an Abelian directed po-group G. We study conditions when an effect algebra is of this form. Fixing a unital po-group (H, u), the category of strongly (H, u)-perfect effect algebras is introduced and it is shown that it is categorically equivalent to the category of directed po-groups with interpolation. We prove some representation theorems of lexicographic effect algebras, including a subdirect product representation by antilattice lexicographic effect algebras.     

Lattice subordinations and Priestley duality

There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. We show that the correspondence between Heyting algebras and S4-algebras extends naturally to distributive lattices and Boolean algebras with a lattice subordination. We also introduce Heyting lattice subordinations and prove that the category of Boolean algebras with a Heyting lattice subordination is isomorphic to the category of S4-algebras, thus obtaining the correspondence between Heyting algebras and S4-algebras as a particular case of our approach. In addition, we provide a uniform approach to dualities for these classes of algebras. Namely, we generalize Priestley spaces to quasi-ordered Priestley spaces and show that lattice subordinations on a Boolean algebra B correspond to Priestley quasiorders on the Stone space of B. This results in a duality between the category of Boolean algebras with a lattice subordination and the category of quasi-ordered Priestley spaces that restricts to Priestley duality for distributive lattices. We also prove that Heyting lattice subordinations on B correspond to Esakia quasi-orders on the Stone space of B. This yields Esakia duality for S4-algebras, which restricts to Esakia duality for Heyting algebras.     

Irreducible <Emphasis Type="Italic">A</Emphasis><Stack><Subscript>1</Subscript><Superscript>(1)</Superscript></Stack>-modules from modules over two-dimensional non-abelian Lie algebra

For any module V over the two-dimensional non-abelian Lie algebra b and scalar α ∈ C, we define a class of weight modules F ^α (V) with zero central charge over the affine Lie algebra A ₁ ⁽¹⁾ . These weight modules have infinitedimensional weight spaces if and only if V is infinite dimensional. In this paper, we will determine necessary and sufficient conditions for these modules F ^α(V) to be irreducible. In this way, we obtain a lot of irreducible weight A ₁ ⁽¹⁾ -modules with infinite-dimensional weight spaces.     

Numerical constructions involving Chebyshev polynomials

We propose a new algorithm for the character expansion of tensor products of finite-dimensional irreducible representations of simple Lie algebras. The algorithm produces valid results for the algebras B ₃, C ₃, and D ₃. We use the direct correspondence between Weyl anti-invariant functions and multivariate second-kind Chebyshev polynomials. We construct the triangular trigonometric polynomials for the algebra D ₃.     

Path <Emphasis Type="Italic">A</Emphasis><Subscript>∞</Subscript> algebras of positively graded quivers

Let A be a path A_∞-algebra over a positively graded quiver Q: We prove that the derived category of A is triangulated equivalent to the derived category of kQ; which is viewed as a DG algebra with trivial differential. The main technique used in the proof is Koszul duality for DG algebras.     

Multiplier Hopf Monoids

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele’s definition of multiplier Hopf algebra is re-obtained. It is shown that the key features of multiplier Hopf algebras (over fields) remain valid in this more general context. Namely, for a multiplier Hopf monoid A, the existence of a unique antipode is proved — in an appropriate, multiplier-valued sense — which is shown to be a morphism of multiplier bimonoids from a twisted version of A to A. For a regular multiplier Hopf monoid (whose twisted versions are multiplier Hopf monoids as well) the antipode is proved to factorize through a proper automorphism of the object A. Under mild further assumptions, duals in the base category are shown to lift to the monoidal categories of modules and of comodules over a regular multiplier Hopf monoid. Finally, the so-called Fundamental Theorem of Hopf modules is proved — which states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid.     

Orlicz Algebras on Homogeneous Spaces of Compact Groups and Their Abstract Linear Representations

Let G be a compact group, H a closed subgroup of G and let m be the normalized G-invariant measure on the homogeneous space G / H obtained from Weil’s formula. In this article, for a given Young function \(\varphi \), we give a new class of Banach convolution algebras on homogeneous spaces of compact groups by introducing a convolution and an involution on the Orlicz space \(L^\varphi (G/H, m)\). Finally, a class of linear representations of this class of Banach convolution algebras is presented.     

On 2-blocks with {k(B) - l(B) = 1}

We investigate some properties of Cartan matrices of symmetric algebras. In particular, we study the Cartan matrices of p-blocks B of finite groups for the cases that \({k(B) - l(B) = 1}\) and that \({k(B) = 3}\) where k(B) and l(B) are the numbers of irreducible ordinary and Brauer characters associated to B, respectively.     

A Representation Stability Theorem for VI-modules

Let VI be the category whose objects are the finite dimensional vector spaces over a finite field of order q and whose morphisms are the injective linear maps. A VI-module over a ring is a functor from the category VI to the category of modules over the ring. A VI-module gives rise to a sequence of representations of the finite general linear groups. We prove that the sequence obtained from any finitely generated VI-module over an algebraically closed field of characteristic zero is representation stable - in particular, the multiplicities which appear in the irreducible decompositions eventually stabilize. We deduce as a consequence that the dimension of the representations in the sequence {V _n } obtained from a finitely generated VI-module V over a field of characteristic zero is eventually a polynomial in q ⁿ . Our results are analogs of corresponding results on representation stability and polynomial growth of dimension for FI-modules (which give rise to sequences of representations of the symmetric groups) proved by Church, Ellenberg, and Farb.     

On Schur Superfunctors

We introduce super-analogues of the Schur functors defined by Akin, Buchsbaum and Weyman. These Schur superfunctors may be viewed as characteristic-free analogues of finite dimensional irreducible polynomial representations of the Lie superalgebra ????(m|n) studied by Berele and Regev. Our construction realizes Schur superfunctors as objects of a certain category of strict polynomial superfunctors. We show that Schur superfunctors are indecomposable objects of this category. In characteristic zero, these correspond to the set of all simple supermodules for the Schur superalgebra, S(m|n, d), for any m, n, d ? 0. We also provide decompositions of Schur bisuperfunctors in terms of tensor products of skew Schur superfunctors.     

<Emphasis Type="Italic">C</Emphasis>*-Algebras Generated by Mappings. Classification of Invariant Subspaces

We continue the study of an operator algebra associated with a self-mapping ? on a countable setX which can be represented as a directed graph. This C*-algebra belongs to a class of operator algebras, generated by a family of partial isometries satisfying some relations on their source and range projections. Earlier we have formulated the irreducibility criterion of such algebras, which give us a possibility to examine the structure of the corresponding Hilbert space. We will show that for reducible algebras the underlying Hilbert space can be represented either as an infinite sum of invariant subspaces or as a tensor product of a finite-dimensional Hilbert space with l²(Z). In the first case we present a conditions under which the studied algebra has an irreducible representation into a C*-algebra generated by a weighted shift operator. In the second case, the algebra has the irreducible finite-dimensional representations indexed by the unit circle.     

<Emphasis Type="Italic">γ</Emphasis>-Frames,GΓ-Algebras and Measurable Spaces

The category of γ -Frm of γ-frames, which is isomorphic to the category GΓ-Alg of \(\mathbb D\)-algebras satisfying certain identities, and the category γ -Top of γ-topological spaces provide the background for the category γ -Mbl of γ-measurable spaces. As in the category of frames, the functor \(\Omega_\gamma: \gamma {\text{ - }}Top \ni (X,\Omega_X) \mapsto \Omega_\S \in \gamma {\text{ - }}Frm^{op}\) has a right adjoint \(Pt_\gamma: G\Gamma {\text{ - }}Alg^{op} \to \gamma {\text{ - }}Top\).     

On <Emphasis Type="Italic">n</Emphasis>-cubic Pyramid Algebras

In this paper we study a class of algebras having n-dimensional pyramid shaped quiver with n-cubic cells, which we called n-cubic pyramid algebras. This class of algebras includes the quadratic dual of the basic n-Auslander absolutely n-complete algebras introduced by Iyama. We show that the projective resolutions of the simples of n-cubic pyramid algebras can be characterized by n-cuboids, and prove that they are periodic. So these algebras are almost Koszul and (n?1)-translation algebras. We also recover Iyama’s cone construction for n-Auslander absolutely n-complete algebras using n-cubic pyramid algebras and the theory of n-translation algebras.     

The varieties of Heisenberg vertex operator algebras

For a vertex operator algebra V with conformal vector ω,we consider a class of vertex operator subalgebras and their conformal vectors.They are called semi-conformal vertex operator subalgebras and semiconformal vectors of(V,ω),respectively,and were used to study duality theory of vertex operator algebras via coset constructions.Using these objects attached to(V,ω),we shall understand the structure of the vertex operator algebra(V,ω).At first,we define the set Sc(V,ω)of semi-conformal vectors of V, then we prove that Sc(V,ω)is an affine algebraic variety with a partial ordering and an involution map.Corresponding to each semi-conformal vector,there is a unique maximal semi-conformal vertex operator subalgebra containing it.The properties of these subalgebras are invariants of vertex operator algebras.As an example,we describe the corresponding varieties of semi-conformal vectors for Heisenberg vertex operator algebras.As an application,we give two characterizations of Heisenberg vertex operator algebras using the properties of these varieties.