Algebra Homomorphism Research Articles

Spectral properties and stability of perturbed Cartesian product

Let $${\mathcal {A}}$$ and $${\mathcal {B}}$$ be commutative Banach algebras, and let $$T:{\mathcal {B}} \rightarrow {\mathcal {A}}$$ be an algebra homomorphism with $${\Vert T\Vert }\le 1$$ . Then T induces a Banach algebra product $$\times _T$$ perturbing the coordinatewise product on the Cartesian product space $${\mathcal {A}} \times {\mathcal {B}}$$ . We show that the spectral properties like spectral extension property, unique uniform norm property, regularity, weak regularity as well as Ditkin’s condition are stable with respect to this product.

Spectral-like duality for distributive Hilbert algebras with infimum

Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and $${\wedge}$$ -semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.

An algebra associated with a subspace lattice over a finite field and its relation to the quantum affine algebra [formula omitted

It is known that the incidence algebra of a subspace lattice over a finite field with q elements is a homomorphic image of the quantum algebra Uq1/2(sl2). In this paper, we extend this situation. For a fixed proper subspace (which is an object of the subspace lattice), we define naturally a new algebra H which contains the incidence algebra as a proper subalgebra, and show how it is related to the quantum affine algebra Uq1/2(slˆ2). We show that there is an algebra homomorphism from Uq1/2(slˆ2) to H, and that H is generated by its image together with the center. Moreover, we show that any irreducible H-module is also irreducible as a Uq1/2(slˆ2)-module and is isomorphic to the tensor product of two evaluation modules. We also obtain a small set of generators of the center of H.

On the regularity of homomorphisms between Riesz subalgebras of [formula omitted], II

We extend and improve our earlier results on automatic regularity of continuous algebra homomorphisms between Riesz algebras of regular operators.

The characteristic polynomial of an algebra and representations

We pose two new linear preserver problems. First, given a field k and a finite dimensional k-algebra B we show that the only linear maps ϕ:k×d→B which preserve the unit and the n-th roots of unity (for some n>2 coprime to char(k)) are the algebra homomorphisms. Second, we consider linear maps ϕ:A→B between finite dimensional k-algebras which preserve the Cayley–Hamilton relations. We show that if preservation of the Cayley–Hamilton property is understood in a certain non-commutative sense, and A=k×d, then the only such linear mappings are the algebra homomorphisms.

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].

Homomorphisms with small bound between Fourier algebras

Inspired by Kalton and Wood's work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier-Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism $T$ between the associated Fourier algebras (resp. Fourier-Stieltjes algebras) with completely bounded norm $\| T \|_{cb} < \sqrt {3/2}$ (resp. $ \| T \|_{cb} < \sqrt {5}/2$). We show similar results involving the norm distortion $\| T \| \| T ^{-1} \|$ with universal but non-explicit bound. Our results subsume Walter's well-known structural theorems and also Lau's theorem on second conjugate of Fourier algebras.

Whittaker Modules for the Insertion-Elimination Lie Algebra

This paper addresses the representation theory of the insertion-elimination Lie algebra, a Lie algebra that can be naturally realized in terms of tree-inserting and tree-eliminating operations on rooted trees. The insertion-elimination algebra admits a triangular decomposition in the sense of Moody and Pianzola, and thus it is natural to define Whittaker modules corresponding to a given algebra homomorphism. Among other results, we show that the standard Whittaker modules are simple under certain constraints on the corresponding algebra homomorphism.

The q-Onsager algebra and the positive part of [formula omitted

The positive part Uq+ of Uq(slˆ2) has a presentation by two generators X,Y that satisfy the q-Serre relations. The q-Onsager algebra Oq has a presentation by two generators A,B that satisfy the q-Dolan/Grady relations. We give two results that describe how Uq+ and Oq are related. First, we consider the filtration of Oq whose nth component is spanned by the products of at most n generators. We show that the associated graded algebra is isomorphic to Uq+. Second, we introduce an algebra □q and show how it is related to both Uq+ and Oq. The algebra □q is defined by generators and relations. The generators are {xi}i∈Z4 where Z4 is the cyclic group of order 4. For i∈Z4 the generators xi,xi+1 satisfy a q-Weyl relation, and xi,xi+2 satisfy the q-Serre relations. We show that □q is related to Uq+ in the following way. Let □qeven (resp. □qodd) denote the subalgebra of □q generated by x0,x2 (resp. x1,x3). We show that (i) there exists an algebra isomorphism Uq+→□qeven that sends X↦x0 and Y↦x2; (ii) there exists an algebra isomorphism Uq+→□qodd that sends X↦x1 and Y↦x3; (iii) the multiplication map □qeven⊗□qodd→□q, u⊗v↦uv is an isomorphism of vector spaces. We show that □q is related to Oq in the following way. For nonzero scalars a,b there exists an injective algebra homomorphism Oq→□q that sends A↦ax0+a−1x1 and B↦bx2+b−1x3.

English

We modify slightly the definition of $H$-partial functions given by Celani and Montangie (2012); these partial functions are the morphisms in the category of $H^\vee $-space and this category is the dual category of the category with objects the Hilbert algebras with supremum and morphisms, the algebraic homomorphisms. As an application we show that finite pure Hilbert algebras with supremum are determined by the monoid of their endomorphisms.

Algebras defined by homomorphisms

Let $mathcal{R}$ be a commutative ring with identity, let $A$ and $B$ be two $mathcal{R}$-algebras and $varphi:Blongrightarrow A$ be an $mathcal{R}$-additive algebra homomorphism. We introduce a new algebra $Atimes_varphi B$, and give some basic properties of this algebra. Generalized $2$-cocycle derivations on $Atimes_varphi B$ are studied. Accordingly, $Atimes_varphi B$ is considered from the perspective of Banach algebras.

A Hochschild–Kostant–Rosenberg theorem for cyclic homology

Let A be a commutative algebra over the field F2=Z/2. We show that there is a natural algebra homomorphism ℓ(A)→HC⁎−(A) which is an isomorphism when A is a smooth algebra. Thus, the functor ℓ can be viewed as an approximation of negative cyclic homology and ordinary cyclic homology HC⁎(A) is a natural ℓ(A)-module. In general, there is a spectral sequence E2=L⁎(ℓ)(A)⇒HC⁎−(A). We find associated approximation functors ℓ+ and ℓper for ordinary cyclic homology and periodic cyclic homology, and set up their spectral sequences. Finally, we discuss universality of the approximations.

On amalgamated Banach algebras

Let A and B be Banach algebras, $$\theta : A\rightarrow B$$ be a continuous Banach algebra homomorphism and I be a closed ideal in B. Then the direct sum of A and I with respect to $$\theta $$ , denoted $$A\bowtie ^{\, \theta }I$$ , with a special product becomes a Banach algebra which is called the amalgamated Banach algebra. In this paper, among other things, we compute the topological centre of $$A\bowtie ^{\, \theta }I$$ in terms of that of A and I. Using this, we provide a characterization of the Arens regularity of $$A\bowtie ^{\, \theta }I$$ . Then we determine the character space of $$A\bowtie ^{\, \theta }I$$ in terms of that of A and I. Moreover, we study the weak amenability of $$A\bowtie ^{\, \theta }I$$ .

Quantum families of quantum group homomorphisms

ABSTRACTThe notion of a quantum family of maps has been introduced in the framework of C*-algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a state). In this paper, we define a quantum family of homomorphisms of locally compact quantum groups. Roughly speaking, we show that such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras, is introduced and the algebraic counterpart of the aforementioned result is proved. Moreover, we show that a quantum family of homomorphisms of Hopf algebras is consistent with the counits and coinverses of the given Hopf algebras. We compare our concept with weak coactions introduced by Andruskiewitsch and we apply it to the analysis of adjoint coaction.

Frobenius induction for algebras

ABSTRACTLet B→A be a homomorphism of Hopf algebras and let C be an algebra. We consider the induction from B to A of C in two cases: when C is a B-interior algebra and when C is a B-module algebra. Our main results establish the connection between the two inductions. The inspiration comes from finite group representation theory, and some constructions work in even more general contexts.

Some remarks on Bridgeland’s Hall algebras

ABSTRACTWe study the functorial properties of Bridgeland’s Hall algebras. Specifically, let 𝒜 and ℬ be two categories satisfying certain conditions for the definitions of Bridgeland’s Hall algebras, and let F:𝒜→ℬ be a fully faithful exact functor, which preserves projectives, then F induces an embedding of algebras from the Bridgeland’s Hall algebra of 𝒜 to the one of ℬ. In addition, let A be a finite-dimensional algebra over a finite field and B some special quotient algebra of A, then the Bridgeland’s Hall algebra of B is the quotient algebra of the one of A. Moreover, we consider the BGP-reflection functors on the category of 2-cyclic complexes and obtain some homomorphisms of algebras among the subalgebras of Bridgeland’s Hall algebras.

Cesàro sums and algebra homomorphisms of bounded operators

Let X be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra l 1(N0) and fractional versions of Cesaro sums of a linear operator T ∈ B(X) is established. In particular, we show that every (C, α)-bounded operator T induces an algebra homomorphism — and it is in fact characterized by such an algebra homomorphism. Our method is based on some sequence kernels, Weyl fractional difference calculus and convolution Banach algebras that are introduced and deeply examined. To illustrate our results, improvements to bounds for Abel means, new insights on the (C, α)-boundedness of the resolvent operator for temperated a-times integrated semigroups, and examples of bounded homomorphisms are given in the last section.

Hom-structures on simple graded Lie algebras of finite growth

A Hom-structure on a Lie algebra [Formula: see text] is a linear map [Formula: see text] satisfying the Hom–Jacobi identity: [Formula: see text] for all [Formula: see text]. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. In this paper, using a classification theorem due to Mathieu, we determine explicitly all the Hom-structures on the simple graded Lie algebras of finite growth. As a direct consequence, all the Hom-structures on any simple graded Lie algebras of finite growth constitute a Jordan algebra in the usual way.

Multiplier projections on spaces of real analytic functions in several variables

Let be open with . We characterize the sets having the following property: for every real analytic function f on with Taylor expansion at zero, the series is also the Taylor expansion at zero of some real analytic function on . This result gives a characterization of the idempotents in the algebra of Hadamard-type operators on the space of all real analytic functions , i.e. operators with all monomials being eigenvectors. In many cases, we also describe the multiplicative functionals on and the (continuous) algebra homomorphisms . We show that the algebra is never locally m-convex and in many cases it is not a Q-algebra.

Bridgeland's Hall algebras of tilted algebras and Lusztig's symmetries

For each finite dimensional algebra A of finite global dimension over a finite field, Bridgeland considered the Ringel–Hall algebra of 2-cyclic complexes of projective A-modules, and achieved an algebra called Bridgeland's Hall algebra by taking some localization and reduction. In this paper, we show that there exist homomorphisms of algebras from quantum groups to Bridgeland's Hall algebras of tilted algebras. In particular, for APR-tilting, this homomorphism provides a realization of Lusztig's symmetries of quantum groups via Bridgeland's Hall algebras.