Quotient Algebra Research Articles

Stone duality, topological algebra, and recognition

Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion of any abstract algebra is the extended Stone dual space of the Boolean algebra of recognisable subsets of the abstract algebra endowed with certain residuation operations. These results identify a connection between topological algebra as applied in algebra and Stone duality as applied in logic, and show that the notion of recognition originating in computer science is intrinsic to profinite completion in mathematics in general. This connection underlies a number of recent results in automata theory including a generalisation of Eilenberg–Reiterman theory for regular languages and a new notion of compact recognition applicable beyond the setting of regular languages. The purpose of this paper is to give the underlying duality theoretic result in its general form. Further we illustrate the power of the result by providing a few applications in topological algebra and language theory. In particular, we give a simple proof of the fact that any topological algebra quotient of a profinite algebra which is again based on a Boolean space is again profinite and we derive the conditions dual to the ones of the original Eilenberg theorem in a fully modular manner. We cast our results in the setting of extended Priestley duality for distributive lattices with additional operations as some classes of languages of interest in automata theory fail to be closed under complementation.

Idempotents of the Green algebras of finite dimensionalpointed rank one Hopf algebras of nilpotent type

In this paper, we intend to study idempotents of the Green algebra (complexied Green ring) of anynite dimensional pointed rank one Hopf algebra of nilpotent type over the complex numbereld. Werst determine all one dimensional representations of the quotient algebra of the Green algebra modulo its Jacobson radical. This gives rise to all primitive idempotents of the quotient algebra. Then we present explicitly primitive idempotents of the Green algebra by lifting the ones of the quotient algebra. Finally, as an example, we describe all primitive idempotents of the Green algebra of the Taft algebra T3 .

M-Koszul Artin–Schelter regular algebras

This paper studies the homological determinants and Nakayama automorphisms of not-necessarily-noetherian m-Koszul twisted Calabi–Yau or, equivalently, m-Koszul Artin–Schelter regular, algebras. Dubois-Violette showed that such an algebra is isomorphic to a derivation quotient algebra D(w,i) for a unique-up-to-scalar-multiples twisted superpotential w. By definition, D(w,i) is the quotient of the tensor algebra TV, where V=D(w,i)1, by (∂iw), the ideal generated by all i-th-order left partial derivatives of w. The restriction map σ↦σ|V is used to identify the group of graded algebra automorphisms of D(w,i) with a subgroup of GL(V). We show that the homological determinant of a graded algebra automorphism σ of an m-Koszul Artin–Schelter regular algebra D(w,i) is given by the formula hdet(σ)w=σ⊗(m+i)(w). It follows from this that the homological determinant of the Nakayama automorphism of an m-Koszul Artin–Schelter regular algebra is 1. As an application, we prove that the homological determinant and the usual determinant coincide for most quadratic noetherian Artin–Schelter regular algebras of dimension 3.

On the first Hochschild cohomology of admissible algebras

The aim of this paper is to investigate the first Hochschild cohomology of admissible algebras which can be regarded as a generalization of basic algebras. For this purpose, the authors study differential operators on an admissible algebra. Firstly, differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the k-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field k of characteristic 0.

On the ideals of a C*-algebra generated by a family of partial isometries and multipliers

The C*-subalgebra of the algebra of all bounded operators on the Hilbert space l2 generated by the multiplier algebra and a family of partial isometries satisfying certain relations is considered. The ideals of the algebra under examination and of its quotient by the algebra of compact operators are studied. It is shown that the quotient algebra can be represented as a direct sum of two principal ideals and has nontrivial center.

Braid equivalences in 3-manifolds with rational surgery description

In this paper we provide algebraic mixed braid classification of links in any c.c.o. 3-manifold M obtained by rational surgery along a framed link in S3. We do this by representing M by a closed framed braid in S3 and links in M by closed mixed braids in S3. We first prove an analogue of the Reidemeister theorem for links in M. We then give geometric formulations of the mixed braid equivalence using the L-moves and the braid band moves. Finally we formulate the algebraic braid equivalence in terms of the mixed braid groups Bm,n, using cabling and the parting and combing techniques for mixed braids. Our results set a homogeneous algebraic ground for studying links in 3-manifolds and in families of 3-manifolds using computational tools. We provide concrete formuli of the braid equivalences in lens spaces, in Seifert manifolds, in homology spheres obtained from the trefoil and in manifolds obtained from torus knots.Our setting is appropriate for constructing Jones-type invariants for links in families of 3-manifolds via quotient algebras of the mixed braid groups Bm,n, as well as for studying skein modules of 3-manifolds, since they provide a controlled algebraic framework and much of the diagrammatic complexity has been absorbed into the proofs. Further, our moves can be used in a braid analogue of Rolfsen's rational calculus and potentially in computing Witten invariants.

Commutative rings over which algebras generated by idempotents are quotients of group algebras

We study the relationship between algebras generated by idempotents over a commutative ring $R$ with identity and algebras that are quotient rings of group algebras $RG$ for torsion abelian groups $G$ without an element whose order is a zero-divisor in $R$. The main purpose is to seek conditions for $R$ to hold the equality between these two kinds of algebras.

Leibniz algebras associated with representations of filiform Lie algebras

In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra $n_{n,1}.$ We introduce a Fock module for the algebra $n_{n,1}$ and provide classification of Leibniz algebras $L$ whose corresponding Lie algebra $L/I$ is the algebra $n_{n,1}$ with condition that the ideal $I$ is a Fock $n_{n,1}$-module, where $I$ is the ideal generated by squares of elements from $L$.

Quotient Amenability of Banach Algebras

Let A be a Banach algebra and I be a non-zero closed two-sided ideal of A. We say that the Banach algebra A is I-quotient amenable if the quotient Banach algebra A / I is amenable. In this paper we study this notion and give a sufficient condition for I-quotient amenability. Also, we provide a characterization of I-quotient amenability whenever I has a bounded approximate identity. We prove that this notion may be coincide with amenability, then apply this result to give a new characterization for amenability of C *-algebras. Finally, we give some results over the Fourier algebra

A note on the restricted universal enveloping algebra of a restricted Lie-Rinehart Algebra

Lie-Rinehart algebras, also known as Lie algebroids, give rise to Hopf algebroids by a universal enveloping algebra construction, much as the universal enveloping algebra of an ordinary Lie algebra gives a Hopf algebra, of infinite dimension. In finite characteristic, the universal enveloping algebra of a restricted Lie algebra admits a quotient Hopf algebra which is finite-dimensional if the Lie algebra is. Rumynin has shown that suitably defined restricted Lie algebroids allow to define restricted universal enveloping algebras that are finitely generated projective if the Lie algebroid is. This note presents an alternative proof and possibly fills a gap that might, however, only be a gap in the author's understanding.

Prime Quotients of Jordan Systems and Lie Algebras

We show that, unlike alternative algebras, prime quotients of a nondegenerate Jordan system or a Lie algebra need not be nondegenerate, even if the original Jordan system is primitive, or the Lie algebra is strongly prime, both with nonzero simple hearts. Nevertheless, for Jordan systems and Lie algebras directly linked to associative systems, we prove that even semiprime quotients are necessarily nondegenerate.

Simple toroidal vertex algebras and their irreducible modules

In this paper, we continue the study on toroidal vertex algebras initiated in [15], to study concrete toroidal vertex algebras associated to toroidal Lie algebra Lr(gˆ)=gˆ⊗Lr, where gˆ is an untwisted affine Lie algebra and Lr=C[t1±1,…,tr±1]. We first construct an (r+1)-toroidal vertex algebra V(T,0) and show that the category of restricted Lr(gˆ)-modules is canonically isomorphic to that of V(T,0)-modules. Let c denote the standard central element of gˆ and set Sc=U(Lr(Cc)). We furthermore study a distinguished subalgebra of V(T,0), denoted by V(Sc,0). We show that (graded) simple quotient toroidal vertex algebras of V(Sc,0) are parametrized by a Zr-graded ring homomorphism ψ:Sc→Lr such that Imψ is a Zr-graded simple Sc-module. Denote by L(ψ,0) the simple quotient (r+1)-toroidal vertex algebra of V(Sc,0) associated to ψ. We determine for which ψ, L(ψ,0) is an integrable Lr(gˆ)-module and we then classify irreducible L(ψ,0)-modules for such a ψ. For our need, we also obtain various general results.

EI, involutory and EQI-ideal in bounded BCK-algebras

The aim of this work is to investigate the relationship between ideals in bounded BCK-algebras so we introduce the concepts of involutory and EI-ideals in bounded BCK-algebras and characterise their properties. Also we introduce the concepts of EQI-algebras and EQI-ideals in bounded BCK-algebras and show that EQI-algebras include some important BCK structures such as involutory BCK-algebras, commutative and PC-lattices. The relationships between these ideals and quotient algebras that are constructed via these ideals are described. We clarify that EI, involutory and commutative ideals coincide in PC-lattices, whereas they are not the same in bounded BCK-algebras in general. It is proved that EQI-ideals contain some current ideals such as involutory, commutative, positive implicative and implicative ideals.

Free nilpotent and H-type Lie algebras. Combinatorial and orthogonal designs

The aim of our paper is to construct pseudo H-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a byproduct result, we recover the existence of a rational structure on pseudo H-type algebras, which implies the existence of lattices on the corresponding pseudo H-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo H-type algebras with combinatorial and orthogonal designs. One of the key tools is the family of Hurwitz–Radon orthogonal matrices.

THE COMPOSITION SERIES OF IDEALS OF THE PARTIAL-ISOMETRIC CROSSED PRODUCT BY SEMIGROUP OF ENDOMORPHISMS

Let + be the positive cone in a totally ordered abelian group , andan action of + by extendible endomorphisms of a C ∗ -algebra A. Suppose I is an extendible �-invariant ideal of A. We prove that the partial-isometric crossed product I := I × piso + embeds naturally as an ideal of A× piso + , such that the quotient is the partial-isometric crossed product of the quotient algebra. We claim that this ideal I together with the kernel of a natural homomorphism � : A× piso + ! A × iso + gives a composition series of ideals of A × piso + studied by Lindiarni and Raeburn.

Finite-dimensional quotients of Hecke algebras

Let W be a complex reflection group. We prove that there is the maximal finite dimensional quotient of the Hecke algebra H_q(W) of W and that the dimension of this quotient coincides with |W|. This is a weak version of a Brou\'e-Malle-Rouquier conjecture from 1998. The proof is based on categories O for Rational Cherednik algebras.

Stratifications of Finite Directed Categories and Generalized APR Tilting Modules

A finite directed category is a k-linear category with finitely many objects and an underlying poset structure, where k is an algebraically closed field. This concept unifies structures such as k-linerizations of posets and finite EI categories, quotient algebras of finite-dimensional hereditary algebras, triangular matrix algebras, etc. In this article, we study representations of finite directed categories and discuss their stratification properties. In particular, we show the existence of generalized Auslander-Platzeck-Reiten tilting modules for triangular matrix algebras under some assumptions.

Hopf algebra structure on packed square matrices

We construct a new bigraded Hopf algebra whose bases are indexed by square matrices with entries in the alphabet {0,1,…,k}, k⩾1, without null rows or columns. This Hopf algebra generalizes the one of permutations of Malvenuto and Reutenauer, the one of k-colored permutations of Novelli and Thibon, and the one of uniform block permutations of Aguiar and Orellana. We study the algebraic structure of our Hopf algebra and show, by exhibiting multiplicative bases, that it is free. We moreover show that it is self-dual and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra, we obtain a new one indexed by alternating sign matrices. We study some of its properties and algebraic quotients defined through alternating sign matrices statistics.

Another characterization of tilted algebras

We give a new characterization of tilted algebras by the existence of certain special subquivers in their Auslander-Reiten quiver. This result includes the existent characterizations of this kind and yields a way to obtain more tilted quotient algebras from a given algebra.

Representations of Hopf-Ore Extensions of Group Algebras and Pointed Hopf Algebras of Rank One

In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field $k$. Let $H=kG(\chi, a,\d)$ be a Hopf-Ore extension of $kG$ and $H'$ a rank one quotient Hopf algebra of $H$, where $k$ is a field, $G$ is a group, $a$ is a central element of $G$ and $\chi$ is a $k$-valued character for $G$ with $\chi(a)\neq 1$. We first show that the simple weight modules over $H$ and $H'$ are finite dimensional. Then we describe the structures of all simple weight modules over $H$ and $H'$, and classify them. We also consider the decomposition of the tensor product of two simple weight modules over $H'$ into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over $H$ and $H'$, and classify them. Finally, when $\chi(a)$ is a primitive $n$-th root of unity for some $n>2$, we determine all finite dimensional indecomposable projective objects in the category of weight modules over $H'$.