Semisimple Algebra Research Articles

Certain properties of Jordan homomorphisms, n-Jordan homomorphisms and n-homomorphisms on rings and Banach algebras

We investigate under what conditions n-Jordan homomorphisms between rings are n-homomorphism, or homomorphism; and under what conditions, n-Jordan homomorphisms are continuous. One of the main goals in this work is to show that every n-Jordan homomorphism f : A → B, from a unital ring A into a ring B with characteristic greater than n, is a multiple of a Jordan homomorphism and hence, it is an n-homomorphism if every Jordan homomorphism from A into B is a homomorphism. In particular, if B is an integral domain whose characteristic is greater than n, then every n-Jordan homomorphism f : A → B is an n-homomorphism. Along with some other results, we show that if A and B are unital rings such that the characteristic of B is greater than n, then every unital n-Jordan homomorphism f : A → B is a Jordan homomorphism and hence, it is an m-Jordan homomorphism for any positive integer m ≥ 2. We also investigate the automatic continuity of n-Jordan homomorphisms from a unital Banach algebra either into a semisimple commutative Banach algebra, onto a semisimple Banach algebra, or into a strongly semisimple Banach algebra whenever the n-Jordan homomorphism has dense range.

Good codes from metacyclic groups II

In this paper, we consider semisimple group algebras [Formula: see text] of split metacyclic groups over finite fields. We construct left codes in [Formula: see text] in the case when the order [Formula: see text] is [Formula: see text], where [Formula: see text] and [Formula: see text] are different primes such that [Formula: see text] extend the construction described in a previous paper, determine their dual codes and find some good codes.

Ext groups in the category of bimodules over a simple Leibniz algebra

In this article, we generalize Loday and Pirashvili's [11] computation of the Ext -category of Leibniz bimodules for a simple Lie algebra to the case of a simple (non Lie) Leibniz algebra. Most of the arguments generalize easily, while the main new ingredient is the Feldvoss-Wagemann's cohomology vanishing theorem for semi-simple Leibniz algebras.

On the dimension of ideals in group algebras, and group codes

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of semisimple group algebras, to compute, for any abelian code, an element with Hamming weight equal to its dimension. Finally, to get bounds on the minimum distance of certain MDS group codes. A relation between a class of group codes and MDS codes is presented. Examples illustrating the main results are provided.

The Terwilliger Algebra of the Twisted Grassmann Graph: the Thin Case

The Terwilliger algebra $T(x)$ of a finite connected simple graph $\Gamma$ with respect to a vertex $x$ is the complex semisimple matrix algebra generated by the adjacency matrix $A$ of $\Gamma$ and the diagonal matrices $E_i^*(x)=\operatorname{diag}(v_i)$ $(i=0,1,2,\dots)$, where $v_i$ denotes the characteristic vector of the set of vertices at distance $i$ from $x$. The twisted Grassmann graph $\tilde{J}_q(2D+1,D)$ discovered by Van Dam and Koolen in 2005 has two orbits of the automorphism group on its vertex set, and it is known that one of the orbits has the property that $T(x)$ is thin whenever $x$ is chosen from it, i.e., every irreducible $T(x)$-module $W$ satisfies $\dim E_i^*(x)W\leqslant 1$ for all $i$. In this paper, we determine all the irreducible $T(x)$-modules of $\tilde{J}_q(2D+1,D)$ for this "thin" case.

About the joint spectrum of vector-valued functions

We generalize the results of Abtahi and Farhangi about the joint spectrum and A-valued spectrum of a vector-valued map from the class of unital commutative semisimple Banach algebras to the case of unital topological algebras.

On realizable Galois module classes by the inverse different

Let k be a number field and Ok its ring of integers. Let Γ be a finite group. Let be a maximal Ok -order in the semi-simple algebra containing Let (resp. ) be the locally free classgroup of (resp. ). We denote by (resp. ) the set of classes c in (resp. ) such that there exists a tamely ramified extension N/k, with Galois group isomorphic to Γ (Γ-extension) and the class of (resp. ) is equal to c, where is the different of N/k and its inverse different. We say that (resp. ) is the set of realizable Galois module classes by the inverse different. In the present article, combining some of our published results, and a result due to A. Fröhlich giving a link between the Galois module class of the ring of integers of a tamely ramified -extension and that of its inverse different, we explicitly determine (resp. ) for several groups and show that it is a subgroup of (resp. ). In addition, we determine the set of the Steinitz classes of N/k runs through the set of tamely ramified Γ-extension of k, and prove that is a subgroup of Cl(k), also for several groups

-algebra structure on certain Banach algebra products

AbstractLet $\mathcal A$ and $\mathcal B$ be commutative and semisimple Banach algebras and let $\theta \in \Delta (\mathcal B)$ . In this paper, we prove that $\mathcal A\times _{\theta }\mathcal B$ is a type I-BSE algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are so. As a main application of this result, we prove that $\mathcal A\times _{\theta }\mathcal B$ is isomorphic with a $C^*$ -algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are isomorphic with $C^* $ -algebras. Moreover, we derive related results for the case where $\mathcal A$ is unital.

Centrally generated primitive ideals of [formula omitted] for exceptional types

Let g be a complex semisimple Lie algebra, let b be a Borel subalgebra of g, let n be the nilradical of b, and let U(n) be the universal enveloping algebra of n. We study primitive ideals of U(n). Almost all primitive ideals are centrally generated, i.e., are generated by their intersections with the center Z(n) of U(n). We present an explicit characterization of the centrally generated primitive ideals of U(n) in terms of the Dixmier map and the Kostant cascade in the case when g is a simple algebra of exceptional type. (For classical simple Lie algebras, a similar characterization was obtained by Ivan Penkov and the first author.) As a corollary, we establish a classification of centrally generated primitive ideals of U(n) for an arbitrary semisimple algebra g.

Finite-dimensional Hopf algebras over the Hopf algebra H:−1,1 of Kashina

Let H be the 16-dimensional non-trivial semisimple Hopf algebra H d : − 1 , 1 in the classification work of Kashina [28] . Actually, H ≅ H 8 ⊗ k C 2 , where H 8 is the Kac-Paljutkin algebra [26] . Let V = ⨁ i ∈ I V i , where V i is a simple object in YD H H . All finite-dimensional Nichols algebras satisfying B ( V ) ≅ ⨂ i ∈ I B ( V i ) are determined completely. Under this assumption, we consider and classify all those Hopf algebras of finite-dimensional growth from the semisimple Hopf algebra H d : − 1 , 1 via the relevant Nichols algebras B ( V ) .

Poisson-Lie T-duality of WZW model via current algebra deformation

Poisson-Lie T-duality of the Wess-Zumino-Witten (WZW) model having the group manifold of SU(2) as target space is investigated. The whole construction relies on the deformation of the affine current algebra of the model, the semi-direct sum mathfrak{su}(2)left(mathrm{mathbb{R}}right)overset{cdot }{oplus}mathfrak{a} , to the fully semisimple Kac-Moody algebra mathfrak{sl}left(2,mathrm{mathbb{C}}right)left(mathrm{mathbb{R}}right) . A two-parameter family of models with SL(2, ℂ) as target phase space is obtained so that Poisson-Lie T-duality is realised as an O(3, 3) rotation in the phase space. The dual family shares the same phase space but its configuration space is SB(2, ℂ), the Poisson-Lie dual of the group SU(2). A parent action with doubled degrees of freedom on SL(2, ℂ) is defined, together with its Hamiltonian description.

The method of virtual copies and contractions of simple Lie algebras

Contractions of simple Lie algebras onto semidirect sums are analyzed in the context of the method of virtual copies, and various existence conditions are obtained in terms of the branching rules of representations and the properties of the characteristic representation associated to an embedding of semisimple algebras. Contractions of the Lie algebra with respect to nonmaximal symplectic subalgebras are studied in some detail. In particular, a realization of the subalgebra in terms of cubic polynomial operators is constructed.

The applications of probability groups on Hopf algebras

In this work, we use probability groups, introduced by Harrison in 1979, as a tool to study a semisimple Hopf algebra H with a commutative character ring and prove that the algebra generalized by the dual probability group is the center of H and the product of two class sums is an integral combination up to a factor of of the class sums of H. We classify all the 2-integral probability groups with two or three elements.

2-LOCAL DERIVATIONS ON SEMISIMPLE BANACH ALGEBRAS WITH MINIMAL LEFT IDEALS

AbstractLet${\mathcal{A}}$be a semisimple Banach algebra with minimal left ideals and$\text{soc}({\mathcal{A}})$be the socle of${\mathcal{A}}$. We prove that if$\text{soc}({\mathcal{A}})$is an essential ideal of${\mathcal{A}}$, then every 2-local derivation on${\mathcal{A}}$is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and${\mathcal{J}}$-subspace lattice algebras, is a derivation.

Categories of Modules over Semisimple Finite-Dimensional HOPF Algebras

There is given a construction of a series of semisimple finite-dimensional Hopf algebras having a single irreducible representation of dimension greater than 1. This dimension is equal to the number of one-dimensional representations.

Higher-order commutators with power central values on rings and algebras involving generalized derivations

Abstract Let ℜ {\mathfrak{R}} be a ring with center Z ⁢ ( ℜ ) {Z(\mathfrak{R})} . In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation 𝔉 {\mathfrak{F}} such that ( [ 𝔉 ⁢ ( s m ) ⁢ s n + s n ⁢ 𝔉 ⁢ ( s m ) , s r ] k ) l ∈ Z ⁢ ( ℜ ) {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every s ∈ ℜ {s\in\mathfrak{R}} , then either 𝔉 ⁢ ( s ) = p ⁢ s {\mathfrak{F}(s)=ps} for every s ∈ ℜ {s\in\mathfrak{R}} or ℜ {\mathfrak{R}} satisfies s 4 {s_{4}} and 𝔉 ⁢ ( s ) = s ⁢ p {\mathfrak{F}(s)=sp} for every s ∈ ℜ {s\in\mathfrak{R}} and p ∈ 𝔘 {p\in\mathfrak{U}} , the Utumi quotient ring of ℜ {\mathfrak{R}} . As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.

New results on common properties of the products AC and BA, II

AbstractIn this note, we continue to investigate common properties of the products in the setting of rings, bounded linear operators, or Banach algebras. We prove: (i) If are elements in a unital associative ring satisfying , then von Neumann regularity (resp. generalized Fredholmness relative to an ideal of ) of is converted into that of . (ii) If are bounded linear operators satisfying , then and share common complementability of kernels and ranges. (iii) If are elements in a unital semisimple Banach algebra satisfying and is a trace ideal of such that soc kh(soc, then and share common Fredholmness relative to and have the same abstract index. A similar result holds for B‐Fredholmness in primitive Banach algebra.

Cocycle deformations and Galois objects of semisimple Hopf algebras of dimension 16

In this article, we determine cocycle deformations and Galois objects of non-commutative and non-cocommutative semisimple Hopf algebras of dimension 16. We show that these Hopf algebras are pairwise twist inequivalent mainly by calculating their higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of them admit non-trivial cocycle deformations.

Rota-Baxter paired modules and their constructions from Hopf algebras

In this paper, we introduce the concept of a Rota-Baxter paired module to study Rota-Baxter modules without necessarily a Rota-Baxter operator. We obtain two characterizations of Rota-Baxter paired modules, and give some basic properties of Rota-Baxter paired modules. Beginning with the connection between the notion of integrals in the representations of Hopf algebra and of the notion of an integral algebra as a motivation and special case of Rota-Baxter algebra of weight zero, we obtain a large number of Rota-Baxter paired modules from Hopf related algebras and modules, including semisimple Hopf algebras, weak Hopf algebras, Long bialgebras, quasitriangular Hopf algebras, weak Hopf modules, dimodules and Doi-Hopf modules. Some of them give new examples of Rota-Baxter algebras.

Representations and Conjugacy Classes of Semisimple Quasitriangular Hopf Algebras

In this paper we give two general formulae for the M\"uger centralizers in the category of representations of a semisimple quasitriangular Hopf algebra. The first formula is given in the terms of the Drinfeld map associated to the quasitriangular Hopf algebra. The second formula for the M\"uger centralizer is given in the terms of the conjugacy classes introduced by Cohen and Westreich in [J. Algebra 283 (2005), 42-62]. In the case of a factorizable Hopf algebra these formulae extend some particular cases obtained by the author in [Math. Z. 279 (2015), 227-240].