Sesquilinear Morita Equivalence and Orthogonal Sum of Algebras with Antiautomorphism

We define a notion of Morita equivalence between algebras with antiautomorphisms such that two equivalent algebras have the same category of sesquilinear forms. This generalizes the Morita equivalence of algebras with involutions defined by Fröhlich and Mc Evett [5 Fröhlich , A. , McEvett , A. M. ( 1969 ). Forms over rings with involution . J. Algebra 12 : 79 – 104 .[Crossref], [Web of Science ®] , [Google Scholar]], and their categories of ?-hermitian forms.

For two Morita equivalent algebras with involution, with an additional technical property (which is true for central simple algebras), we define a new algebra with antiautomorphism, called the orthogonal sum, which generalizes the usual notion of orthogonal sum of forms. We explore the invariants of this sum.     

Automorphisms of Generalized Down-Up Algebras

A generalization of down-up algebras was introduced by Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 ( 1 ): 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]), the so-called “generalized down-up algebras”. We describe the automorphism group of conformal Noetherian generalized down-up algebras L(f, r, s, γ) such that r is not a root of unity, listing explicitly the elements of the group. In the last section, we apply these results to Noetherian down-up algebras, thus obtaining a characterization of the automorphism group of Noetherian down-up algebras A(α, β, γ) for which the roots of the polynomial X ² ? α X ? β are not both roots of unity.     

Quotients of Koszul Algebras and 2-d-Determined Algebras

Vatne [13 Vatne , J. E. ( 2012 ). Quotients of Koszul algebras with almost linear resolution. Preprint, arXiv:1103.3572 . [Google Scholar]] and Green and Marcos [9 Green , E. L. , Marcos , E. N. (2011). d-Koszul algebras, 2-d-determined algebras and 2-d-Koszul algebras. J. Pure Appl. Algebra 215(4):439–449.[Crossref], [Web of Science ®] , [Google Scholar]] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green and Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.     

The Existence of Large E(R)-Algebras That Are Sharply Transitive Modules

The Bose–Mesner algebra of the association scheme of the ordinary n-gon has the following remarkable properties:

• (i) It has a P-polynomial structure with respect to every faithful basis element; and

• (ii) Any closed subset generated by a basis element has a P-polynomial structure with respect to this basis element.

C-algebras or table algebras that have these two properties are called perfect P-polynomial C-algebras or table algebras. By applying and extending some of the techniques developed in Xu (2006 Xu , B. ( 2006 ). Table algebras with multiple P-polynomial structures . J. Alg. Combin. 23 : 377 – 393 .[Crossref], [Web of Science ®] , [Google Scholar]), we will give a classification of perfect P-polynomial table algebras in terms of intersection matrices. As a direct consequence of this classification, we will prove that a standard real integral table algebra (A, B) with |B| ≥ 6 is a perfect P-polynomial table algebra if and only if it is exactly isomorphic to the Bose–Mesner algebra of the association scheme of the ordinary (2|B|?2)-gon or (2|B|?1)-gon. This result generalizes part of the main theorem in Xu (2006 Xu , B. ( 2006 ). Table algebras with multiple P-polynomial structures . J. Alg. Combin. 23 : 377 – 393 .[Crossref], [Web of Science ®] , [Google Scholar]). We will present examples revealing that this result is not true if |B| ≤ 5.     

Table algebras with exactly one irreducible character whose degree and multiplicity are not equal

Commutative standard table algebras with exactly one multiplicity not equal to 1 are characterized by the wreath product of some special table algebras in [1 Antonou, A. (2015). Commutative standard table algebras with at most one nontrivial multiplicity. Commun. Algebra 43:2516–2523.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. A natural and much more general question is the characterization of standard table algebras (not necessarily commutative) with exactly one irreducible character whose degree and multiplicity are not equal and the degree is 1. We will give a characterization of such table algebras, including the main result of [1 Antonou, A. (2015). Commutative standard table algebras with at most one nontrivial multiplicity. Commun. Algebra 43:2516–2523.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] as a special case. Applications to association schemes are also discussed.     

Freeness of Quasi-Hopf Algebras Over Right Coideal Subalgebras

We will generalize Skryabin's Freeness Theorem [11 Skryabin , S. ( 2007 ). Projectivity and freeness over comodule algebras . Trans. Amer. Math. Soc. 359 : 2597 – 2623 .[Crossref], [Web of Science ®] , [Google Scholar]]to quasi-Hopf algebras. We will show that for a finite dimensional quasi-Hopf algebra H and a right coideal subalgebra K ? H all (H, K)-quasi Hopf bimodules are free K-modules, in particular, H is a free right and left K-module.     

A Few Properties of Just Infinite Algebras

In this article we prove a few interesting properties of just infinite algebras. Bartholdi (2006 Bartholdi , L. ( 2006 ). Branch rings, thinned rings, tree enveloping rings . Israel J. Math. 154 : 93 – 139 .[Crossref], [Web of Science ®] , [Google Scholar]), defines a particular class of just infinite algebras and demonstrates various properties of these examples. One such property, which is tedious to prove for his specific examples, is primality. We prove here that, in fact, all just infinite algebras are prime. We then consider two corollaries of this theorem; one suggests a weaker definition of just infinite for finitely generated algebras and the other examines the specific case of just infinite algebras which also satisfy a polynomial identity.     

Lie algebras of vector fields on smooth affine varieties

We reprove the results of Jordan [18 Jordan, D. (1986). On the ideals of a Lie algebra of derivations. J. London Math. Soc. 33:33–39.[Crossref], [Web of Science ®] , [Google Scholar]] and Siebert [30 Siebert, T. (1996). Lie algebras of derivations and a?ne algebraic geometry over fields of characteristic 0. Math. Ann. 305:271–286.[Crossref], [Web of Science ®] , [Google Scholar]] and show that the Lie algebra of polynomial vector fields on an irreducible a?ne variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not depend on papers mentioned above. Besides, the structure of the module of polynomial functions on an irreducible smooth a?ne variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered.     

Simple Modules and Primitive Ideals of Non-Noetherian Generalized Down-Up Algebras

Generalized down-up algebras were first introduced in Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 : 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]). Their simple weight modules were classified in Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 : 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]) in the noetherian case, and in Praton (2007 Praton , I. ( 2007 ). Simple weight modules of non-noetherian generalized down-up algebras . Comm. Algebra 35 : 325 – 337 .[Taylor &; Francis Online] , [Google Scholar]) in the non-noetherian case. Here we concentrate on non-noetherian down-up algebras. We show that almost all simple modules are weight modules. We also classify the corresponding primitive ideals.     

Singular Solutions to the Quantum Yang–Baxter Equations

Let X and A be weak Hopf algebras in the sense of Li (1998 Li , F. ( 1998 ). Weak Hopf algebras and some new solutions of the quantum Yang–Baxter equation . J. Algebra 208 ( 1 ): 72 – 100 .[Crossref], [Web of Science ®] , [Google Scholar]). As in the case of Hopf algebras (Majid, 1990 Majid , S. ( 1990 ). Quasitriangular Hopf algebras and Yang–Baxter equations . Internat. J. Modern Phys. A 5 : 1 – 91 . [Google Scholar]), a weak bicrossed coproduct X∞ _R A is constructed by means of good regular R-matrices of the weak Hopf algebras X and A. Using this, we provide a new framework of obtaining singular solutions of the quantum Yang–Baxter equation by constructing weak quasitriangular structures over X∞ _R A when both X and A admit a weak quasitriangular structure. Finally, two explicit examples are given.     

Representations of Generalized Almost-Jordan Algebras

This paper deals with the variety of commutative algebras satisfying the identity β{(yx ²)x ? ((yx)x)x} + γ{yx ³ ? ((yx)x)x} = 0, where β, γ are scalars. These algebras appeared as one of the four families of degree four identities in Carini, Hentzel, and Piacentini-Cattaneo [6 Carini , L. , Hentzel , I. R. , Piacentini-Cattaneo , J. M. ( 1988 ). Degree four identities not implies by commutativity . Comm. in Algebra 16 ( 2 ): 339 – 357 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. We give a characterization of representations and irreducible modules on these algebras. Our results require that the characteristic of the ground field is different from 2, 3.     

On a Class of Finite-dimensional Semisimple Hopf Algebras

Let H be a finite-dimensional and semisimple Hopf algebra over an algebraically closed field of characteristic 0 such that H has exactly one isomorphism class of simple modules that have not dimension 1. These Hopf algebras were the object of study in, for instance, [1 Artamonov , V. A. ( 2007 ). Semisimple finite-dimensional Hopf algebras . Sbornik: Mathematics 198 ( 9 ): 1221 – 1245 .[Crossref], [Web of Science ®] , [Google Scholar]] and [9 Mukhatov , R. B. ( 2009 ). On semisimple finite-dimensional Hopf algebras . Fundamentalnaya i Prikladnaya Matematika 15 ( 2 ): 133 – 143 . [Google Scholar]]. In this paper we study this property in the context of certain abelian extensions of group algebras and give a group theoretical criterion for such Hopf algebras to be of the above type. We also give a classification result in a special case thereof.     

A Braided Version of Some Results of Skryabin

We extend the main result of Skryabin in [9 Skryabin , S. ( 2007 ). Projectivity and freeness over comodule algebras . Trans. Amer. Math. Soc. 359 : 2597 – 2623 .[Crossref], [Web of Science ®] , [Google Scholar]] to Yetter-Drinfeld Hopf algebras over an arbitrary Hopf algebra H with bijective antipode.     

NIVEAU HERMITIEN D'UNE ALGÈBRE DE QUATERNIONS À DIVISION MUNIE D'UNE INVOLUTION DE DEUXIÈME ESPÈCE

ABSTRACT

In this note we show that the hermitian level of a quaternion division algebra with involution of second kind, is always a power of 2, when it is finite. This result holds for a field with trivial or non-trivial involution, and quaternion division algebras with involution of first kind [6] Pfister, A. 1965. Darstellung von -1 als Summe Von Quadraten in Einem Körper. J. London Math. Soc., 40: 159–165. [Crossref], [Web of Science ®] , [Google Scholar], [5] Lewis, D.W. 1988. Sums of Hermitian Squares. Journal of Algebra, 115(2): 446–480.  [Google Scholar], [9] Serhir, A. 1997. Niveau Hermitien de Certaines Algèbres de Quaternions. Communications in Algebra, 25(8): 2531–2538. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar].     

GRADED IDENTITIES AND PI EQUIVALENCE OF ALGEBRAS IN POSITIVE CHARACTERISTIC

ABSTRACT

The algebras M _{a, b} (E) ? E and M _a+b (E) are PI equivalent over a field of characteristic 0 where E is the infinite-dimensional Grassmann algebra. This result is a part of the well-known tensor product theorem. It was first proved by Kemer in 1984–1987 (see Kemer 1991 Kemer , A. ( 1991 ). Ideals of identities of associative algebras . Translations Math. Monographs 87 . Providence , RI : Am. Math. Soc. .[Crossref] , [Google Scholar]); other proofs of it were given by Regev (1990 Regev , A. ( 1990 ). Tensor products of matrix algebras over the Grassmann algebra . J. Alg. 133 ( 2 ): 512 – 526 . [CROSSREF]  [Google Scholar]), and in several particular cases, by Di Vincenzo (1992 Di Vincenzo , O. M. ( 1992 ). On the graded identities of M_{1, 1}( E) . Israel J. Math. 80 ( 3 ): 323 – 335 .[Crossref], [Web of Science ®] , [Google Scholar]), and by the authors (2004). Using graded polynomial identities, we obtain a new elementary proof of this fact and show that it fails for the T-ideals of the algebras M _{1, 1}(E) ? E and M ₂(E) when the base field is infinite and of characteristic p > 2. The algebra M _{a, a} (E) ? E satisfies certain graded identities that are not satisfied by M _2a (E). In another paper we proved that the algebras M _{1, 1}(E) and E ? E are not PI equivalent in positive characteristic, while they do satisfy the same multilinear identities.     

The h-Vector of a Relatively Compressed Level Algebra

The purpose of this note is to supply an upper and a lower bound (which are in general sharp) for the h-vector of a level algebra which is relatively compressed with respect to any arbitrary level algebra A.

The useful concept of relatively compressed algebra was recently introduced in Migliore et al. (2005 Migliore , J. , Miró-Roig , R. , Nagel , U. ( 2005 ). Minimal resolution of relatively compressed level algebras . J. Algebra 284 ( 1 ): 337 – 370 .[Crossref] , [Google Scholar]) (whose investigations mainly focused on the particular case of A a complete intersection). The key idea of this note is the simple observation that the level algebras which are relatively compressed with respect to A coincide (after an obvious isomorphism) with the generic level quotients of suitable truncations of A. Therefore, we are able to apply to relatively compressed algebras the main result of our recent work, Zanello (2007 Zanello , F. ( 2007 ). Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type . Trans. Amer. Math. Soc. 359 ( 6 ): 2675 – 2686 .[Crossref], [Web of Science ®] , [Google Scholar]).     

The representations of nested composition algebras

In this paper, we investigate the basis graph of the monoid algebra of a submonoid of the monoid of mappings from N = {1,…,n} to itself, defined by a nested sequence of compositions of N. Each such monoid is a left regular band (LRB), that is, a semigroup S satisfying x² = x and xyx = xy for all x,y∈S. This class is su?ciently rich that every path algebra of an acyclic quiver can be embedded in such a monoid algebra. The multiplication in the monoid algebra has a particularly simple quasi-multiplicative form, allowing definition over the integers. Combining this with a formula for Ext-groups for LRBs due to Margolis et al. [6 Margolis, S., Saliola, F., Steinberg, B. (2015). Combinatorial topology and the global dimension of algebras arising in combinatorics. J. Eur. Math Soc. 17(12):3037–3080.[Crossref], [Web of Science ®] , [Google Scholar]], we get a simple criterion for the nested composition algebras to be hereditary.     

Liouville Extensions of Artinian Simple Module Algebras

In a previous article (Amano and Masuoka, 2005 Amano , K. , Masuoka , A. ( 2005 ). Picard–Vessiot extensions of Artinian simple module algebras . J. Algebra 285 : 743 – 767 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), the author and Masuoka developed a Picard–Vessiot theory for module algebras over a cocommutative pointed smooth Hopf algebra D. By using the notion of Artinian simple (AS)D-module algebras, it generalizes and unifies the standard Picard–Vessiot theories for linear differential and difference equations. The purpose of this article is to define the notion of Liouville extensions of AS D-module algebras and to characterize the corresponding Picard–Vessiot group schemes.