Algebra Morphism Research Articles

Structure Theorems for Bicomodule Algebras Over Quasi-Hopf Algebras, Weak Hopf Algebras, and Braided Hopf Algebras

Let H be a quasi-Hopf algebra, a weak Hopf algebra, or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v: H → B. Then we can define an object Bco(H), which is a left-left Yetter–Drinfeld module over H, having extra properties that allow to make a smash product Bco(H)#H, which is an H-bicomodule algebra, isomorphic to B.

Additive maps onto matrix spaces compressing the spectrum

We prove that given a unital C⁎-algebra A and an additive and surjective map T:A→Mn such that the spectrum of T(x) is a subset of the spectrum of x for each x∈A, then T is either an algebra morphism, or an algebra anti-morphism. We arrive at the same conclusion for an arbitrary unital, complex Banach algebra A, by imposing an extra surjectivity condition on the map T.

On the Van Est homomorphism for Lie groupoids

The Van Est homomorphism for a Lie groupoid G \rightrightarrows M , as introduced by Weinstein–Xu, is a cochain map from the complex C^\infty (BG) of groupoid cochains to the Chevalley–Eilenberg complex \mathsf C(A) of the Lie algebroid A of G . It was generalized by Weinstein, Mehta, and Abad-Crainic to a morphism from the Bott–Shulman–Stasheff complex \Omega (BG) to a (suitably defined) Weil algebra \mathsf W(A) . In this paper, we will give an approach to the Van Est map in terms of the Perturbation Lemma of homological algebra. This approach is used to establish the basic properties of the Van Est map. In particular, we show that on the normalized subcomplex, the Van Est map restricts to an algebra morphism.

Obstruction theory for algebras over an operad

The goal of this paper is to set up an obstruction theory in the context of algebras over an operad and in the framework of differential graded modules over a field. Precisely, the problem we consider is the following: Suppose given two algebras A and B over an operad P and an algebra morphism from H * A to H * B. Can we realize this morphism as a morphism of P-algebras from A to B in the homotopy category? Also, if the realization exists, is it unique in the homotopy category?

Contact structures as Dirac structures and their associated Poisson algebras

We give a description of contact structures on smoothmanifolds as Dirac structures, and we compute their associated Poisson algebras of admissible functions. We also revisit the notion of moment map associated to smooth actions on Lie groups on contact manifolds, we show that a contact moment map induces a moment map in the Dirac sense as defined in [3], and this map induces a natural morphism of Lie algebras of vector fields, admissible functions and infinitesimal symmetries.

Quantitative BT-Theorem and automatic continuity for standard von Neumann algebras

We prove a general criterion for a von Neumann algebra M in order to be in standard form. It is formulated in terms of an everywhere defined, invertible, antilinear, a priori not necessarily bounded operator, intertwining M with its commutant M′ and acting as the ⁎-operation on the centre. We also prove a generalized version of the BT-Theorem which enables us to see that such an intertwiner must be necessarily bounded. It is shown that this extension of the BT-Theorem leads to the automatic boundedness of quite general operators which intertwine the identity map of a von Neumann algebra with a general bounded, real-linear, operator-valued map. We apply the last result to the automatic boundedness of linear operators implementing algebraic morphisms of a von Neumann algebra onto some Banach algebra, and to the structure of a W⁎-algebra M endowed with a normal, semi-finite, faithful weight φ, whose left ideal Nφ admits an algebraic complement in the GNS representation space Hφ, invariant under the canonical action of M.

Hochschild cohomology of relation extension algebras

Let B be the split extension of a finite dimensional algebra C by a C-C-bimodule E. We define a morphism of associative graded algebras φ⁎:HH⁎(B)→HH⁎(C) from the Hochschild cohomology of B to that of C, extending similar constructions for the first cohomology groups made and studied by Assem, Bustamante, Igusa, Redondo and Schiffler.In the case of a trivial extension B=C⋉E, we give necessary and sufficient conditions for each φn to be surjective. We prove the surjectivity of φ1 for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of φ1 for any trivial extension, and give a more precise description of this kernel in the case of relation extensions.

Infinitesimal and B∞-algebras, finite spaces, and quasi-symmetric functions

Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be put on recent topological and combinatorial Hopf algebra techniques. We will show that the linear span of finite spaces carries generalized Hopf algebraic structures that are closely connected with familiar constructions and structures in topology (such as the one of cogroups in the category of associative algebras that has appeared e.g. in the study of loop spaces of suspensions). The most striking results that we obtain are certainly that the linear span of finite spaces carries the structure of the enveloping algebra of a B∞-algebra, and that there are natural (Hopf algebraic) morphisms between finite spaces and quasi-symmetric functions. In the process, we introduce the notion of Schur–Weyl categories in order to describe rigidity theorems for cogroups in the category of associative algebras and related structures, as well as to account for the existence of natural operations (graded permutations) on them.

Complete Positivity and Natural Representation of Quantum Computations

We propose a new 'quantum domain theory' in which Scott-continuous functions are replaced by Scott-continuous natural transformations.Completely positive maps are widely accepted as a model of first-order quantum computation. We begin by establishing a categorical characterization of completely positive maps as natural families of positive maps. We explore this categorical characterization by building various representations of quantum computation based on different structures: affine maps between cones of positive elements, morphisms of algebras of effects, and affine maps of convex sets of states. By focusing on convex dcpos, we develop a quantum domain theory and show that it supports some important constructions such as tensor products by quantum data, and lifting.

Some notes on amenability and weak amenability of Lau product of Banach algebras defined by a Banach algebra morphism

Let A and B be Banach algebras and T : B → A be a continuous homomorphism. n- weak amenability of the Banach algebra A ×T B (defined in Bade, W. G., Curtis, P. C., Dales, H. G.: Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc., 55(2), 359-377 (1987)) is studied. The new version of a Banach algebra defined with a continuous homomorphism is introduced and Arens regularity and various notions of amenability of this algebra are studied.

A cohomological framework for homotopy moment maps

Given a Lie group acting on a manifold $M$ preserving a closed $n+1$-form $\omega$, the notion of homotopy moment map for this action was introduced in Callies-Fregier-Rogers-Zambon [6], in terms of $L_{\infty}$-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This description simplifies greatly computations, and we use it to study various properties of homotopy moment maps: their relation to equivariant cohomology, their obstruction theory, how they induce new ones on mapping spaces, and their equivalences. The results we obtain extend some of the results of [6].

A NOTE ON CYCLIC AMENABILITY OF THE LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

Let $T$ be a Banach algebra homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$ with $\Vert T\Vert \leq 1$. Recently, Bhatt and Dabhi [‘Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism’, Bull. Aust. Math. Soc.87 (2013), 195–206] showed that cyclic amenability of ${\mathcal{A}}\times _{T}{\mathcal{B}}$ is stable with respect to $T$, for the case where ${\mathcal{A}}$ is commutative. In this note, we address a gap in the proof of this stability result and extend it to an arbitrary Banach algebra ${\mathcal{A}}$.

The Hopf algebra of finite topologies and T-partitions

A noncommutative and noncocommutative Hopf algebra on finite topologies HT is introduced and studied (freeness, cofreeness, self-duality…). Generalizing Stanley's definition of P-partitions associated to a special poset, we define the notion of T-partitions associated to a finite topology, and deduce a Hopf algebra morphism from HT to the Hopf algebra of packed words WQSym. Generalizing Stanley's decomposition by linear extensions, we deduce a factorization of this morphism, which induces a combinatorial isomorphism from the shuffle product to the quasi-shuffle product of WQSym. It is strongly related to a partial order on packed words, here described and studied.

Modelling algebraic structures and morphisms in ACL2

In this paper, we present how algebraic structures and morphisms can be modelled in the ACL2 theorem prover. Namely, we illustrate a methodology for implementing a set of tools that facilitates the formalisations related to algebraic structures—as a result, an algebraic hierarchy ranging from setoids to vector spaces has been developed. The resultant tools can be used to simplify the development of generic theories about algebraic structures. In particular, the benefits of using the tools presented in this paper, compared to a from-scratch approach, are especially relevant when working with complex mathematical structures; for example, the structures employed in Algebraic Topology. This work shows that ACL2 can be a suitable tool for formalising algebraic concepts coming, for instance, from computer algebra systems.

Partial representations of Hopf algebras

In this work, the notion of a partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra H, one can associate to it a Hopf algebroid Hpar which has the universal property that each partial representation of H can be factorized by an algebra morphism from Hpar. We define also the category of partial modules over a Hopf algebra H, which is the category of modules over its associated Hopf algebroid Hpar. The Hopf algebroid structure of Hpar enables us to enhance the category of partial H-modules with a monoidal structure and such that the algebra objects in this category are the symmetric partial actions. Some examples of categories of partial H-modules are explored. In particular we can describe fully the category of partially Z2-graded modules.

Non-Commutative Ternary Nambu-Poisson Algebras and Ternary Hom-Nambu-Poisson Algebras

The main purpose of this paper is to study non-commutative ternary Nambu-Poisson algebras and their Homtype version. We provide construction results dealing with tensor product and direct sums of two (non- commutative) ternary (Hom-) Nambu-Poisson algebras. Moreover, we explore twisting principle of (non-commutative) ternary Nambu-Poisson algebras along with algebra morphism that lead to construct (non-commutative) ternary Hom- Nambu-Poisson algebras. Furthermore, we provide examples and a 3-dimensional classification of non-commutative ternary Nambu-Poisson algebras.

Higgs bundles and representation spaces associated to morphisms

Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\,\subset\, G$ be a maximal compact subgroup. Let $X$, $Y$ be irreducible smooth complex projective varieties and $f\colon X\to Y$ an algebraic morphism, such that $\pi_1(Y)$ is virtually nilpotent and the homomorphism $f_*\colon \pi_1(X)\to\pi_1(Y)$ is surjective. Define \begin{align*} {\mathcal R }^f\big(\pi_1(X), G\big)&= \{\rho \in \operatorname{Hom}\big(\pi_1(X), G\big) \mid A\circ\rho \ \text{ factors through }~ f_*\}\,,\\[6pt] {\mathcal R }^f\big(\pi_1(X), K\big)&= \{\rho \in \operatorname{Hom}\big(\pi_1(X), K\big) \mid A\circ\rho \ \text{ factors through }~ f_*\}\,, \end{align*} where $A\colon G\to \operatorname{GL}(\operatorname{Lie}(G))$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${\mathcal R }^f(\pi_1(X, x_0),\, G)/\!\!/G$ admits a deformation retraction to ${\mathcal R }^f(\pi_1(X, x_0),\, K)/K$. We also show that the space of conjugacy classes of $n$ almost commuting elements in $G$ admits a deformation retraction to the space of conjugacy classes of $n$ almost commuting elements in $K$.

Some remarks on representations of Yang-Mills algebras

In this article, we present some new properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra 𝔶𝔪(n) on n generators, for n ≥ 2m. We derive from this that any semisimple Lie algebra and even any affine Kac-Moody algebra is a quotient of 𝔶𝔪(n) for n ≥ 4. Combining this with previous results on representations of Yang-Mills algebras given in [Herscovich and Solotar, Ann. Math. 173(2), 1043–1080 (2011)], one may obtain solutions to the Yang-Mills equations by differential operators acting on sections of twisted vector bundles on the affine space of dimension n ≥ 4 associated to representations of any semisimple Lie algebra. We also show that this quotient property does not hold for n = 3, since any morphism of Lie algebras from 𝔶𝔪(3) to 𝔰𝔩(2, k) has in fact solvable image.

On the cohomology of locally finite $$A\#_HC$$ A # H C -modules

Let \(k\) be a field, \(H\) a Hopf algebra with bijective antipode, \(A\) a left \(H\)-module algebra and \(C\) a left \(H\)-comodule algebra. The categories of \(C\)-locally finite \(C\)-modules and \(C\)-locally finite \(A\#_HC\)-modules are abelian with enough injectives. Assuming that there are algebra morphisms \(\phi : H \rightarrow C\) and \(\epsilon _C : C \rightarrow k\) subjected to some conditions, we study the right derived functors of the associated Hom functors, and of the \(C\)-invariants functor. Then we derive spectral sequences that connect them. We also discuss when the \(C\)-invariants functor preserves injectives. In an appendix we establish some of these results in the category of \(C\)-locally finite Doi–Hopf modules.

E1–formality of complex algebraic varieties

Let X be a smooth complex algebraic variety. Morgan [Mor78] showed that the rational homotopy type of X is a formal consequence of the differential graded algebra defined by the first term of its weight spectral sequence. In the present work we generalize this result to arbitrary nilpotent complex algebraic varieties (possibly singular and/or non-compact) and to algebraic morphisms between them. The result for algebraic morphisms generalizes the Formality Theorem of [DGMS75] for compact K\"ahler varieties, filling a gap in Morgan's theory concerning functoriality over the rational numbers. As an application, we study the Hopf invariant of certain algebraic morphisms using intersection theory.