Leibniz Algebras with Invariant Bilinear Forms and Related Lie Algebras

In ([11 Benayadi, S., Hidri, S. (2014). Quadratic Leibniz algebras. Journal of Lie Theory 24:737–759.[Web of Science ®] , [Google Scholar]]), we have studied quadratic Leibniz algebras that are Leibniz algebras endowed with symmetric, nondegenerate, and associative (or invariant) bilinear forms. The nonanticommutativity of the Leibniz product gives rise to other types of invariance for a bilinear form defined on a Leibniz algebra: the left invariance, the right invariance. In this article, we study the structure of Leibniz algebras endowed with nondegenerate, symmetric, and left (resp. right) invariant bilinear forms. In particular, the existence of such a bilinear form on a Leibniz algebra 𝔏 gives rise to a new algebra structure ☆ on the underlying vector space 𝔏. In this article, we study this new algebra, and we give information on the structure of this type of algebras by using some extensions introduced in [11 Benayadi, S., Hidri, S. (2014). Quadratic Leibniz algebras. Journal of Lie Theory 24:737–759.[Web of Science ®] , [Google Scholar]]. In particular, we improve the results obtained in [22 Lin, J., Chen, Z. (2010). Leibniz algebras with pseudo-Riemannian bilinear forms. Front. Math. China 5(1):103–115.[Crossref], [Web of Science ®] , [Google Scholar]].     

The minimal non-Koszul A(Γ)

The algebras A(Γ), where Γ is a directed layered graph, were first constructed by Gelfand et al. [5 Gelfand, I., Serconek, S., Retakh, V., Wilson, R. L. (2005). On a class of algebras associated to directed graphs. Selecta Math. (N.S.) 11(2):281–295.[Crossref], [Web of Science ®] , [Google Scholar]]. These algebras are generalizations of the algebras Q_n, which are related to factorizations of non-commutative polynomials. It was originally conjectured that these algebras were Koszul. In 2008, Cassidy and Shelton found a counterexample to this claim, a non-Koszul A(Γ) corresponding to a graph Γ with 18 edges and 11 vertices. We produce an example of a directed layered graph Γ with 13 edges and 9 vertices, which produces a non-Koszul A(Γ). We also show this is the minimal example with this property.     

Breadth and characteristic sequence of nilpotent Lie algebras

The notion of breadth of a nilpotent Lie algebra was introduced and used to approach problems of classification up to isomorphism in [5 Khuhirun, B., Misra, K. C., Stitzinger, E. (2015). On nilpotent Lie algebras of small breadth. J. Algebra 444:328–338.[Crossref], [Web of Science ®] , [Google Scholar]]. In the present paper, we study this invariant in terms of characteristic sequence, another invariant, introduced by Goze and Ancochea in [1 Ancochea-Bermúdez, J. M., Goze, M. (1986). Sur la classification des algèbres de Lie nilpotentes de dimension 7. C. R. Acad. Sci. Paris 302:611–613. [Google Scholar]]. This permits to complete the determination of Lie algebras of breadth 2 studied in [5 Khuhirun, B., Misra, K. C., Stitzinger, E. (2015). On nilpotent Lie algebras of small breadth. J. Algebra 444:328–338.[Crossref], [Web of Science ®] , [Google Scholar]] and to begin the work for Lie algebras with breadth greater than 2.     

Splitting of operations for alternative and Malcev structures

In this paper, we define pre-Malcev algebras and alternative quadri-algebras and prove that they generalize pre-Lie algebras and quadri-algebras, respectively, to the alternative setting. We use the results and techniques from [4 Bai, C., Bellier, O., Guo, L., Ni, X. (2013). Splitting of operations, Manin products, and Rota-Baxter operators. Int. Math. Res. Not. 2013(3):485–524. [Google Scholar], 14 Gubarev, V. Y., Kolesnikov, P. S. (2013). Embedding of dendriform algebras into Rota-Baxter algebras. Cent. Eur. J. Math. 11(2):226–245.[Crossref], [Web of Science ®] , [Google Scholar]] to discuss and give explicit computations of different constructions in terms of bimodules, splitting of operations, and Rota–Baxter operators.     

Quasi-Deformations of 𝔰𝔩2(𝔽) Using Twisted Derivations

In this article we apply a method devised in Hartwig, Larsson, and Silvestrov (2006 Hartwig , J. T. , Larsson , D. , Silvestrov , S. D. ( 2006 ). Deformations of Lie algebras using σ-derivations . J. Algebra 295 : 314 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]) and Larsson and Silvestrov (2005a Larsson , D. , Silvestrov , S. D. (2005a). Quasi-hom-Lie algebras, Central extensions and 2-cocycle-like identities. J. Algebra 288:321–344.[Crossref], [Web of Science ®] , [Google Scholar]) to the simple 3-dimensional Lie algebra 𝔰𝔩₂(𝔽). One of the main points of this deformation method is that the deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when our deformation scheme is applied to 𝔰𝔩₂(𝔽) we can, by choosing parameters suitably, deform 𝔰𝔩₂(𝔽) into the Heisenberg Lie algebra and some other 3-dimensional Lie algebras in addition to more exotic types of algebras, this being in stark contrast to the classical deformation schemes where 𝔰𝔩₂(𝔽) is rigid.     

The q-difference Noether problem for complex reflection groups and quantum OGZ algebras

For any complex reflection group G = G(m,p,n), we prove that the G-invariants of the division ring of fractions of the n:th tensor power of the quantum plane is a quantum Weyl field and give explicit parameters for this quantum Weyl field. This shows that the q-difference Noether problem has a positive solution for such groups, generalizing previous work by Futorny and the author [10 Futorny, V., Hartwig, J. T. (2014). Solution to a q-difference Noether problem and the quantum Gelfand–Kirillov conjecture for 𝔤𝔩_N. Math. Z. 276(1–2):1–37. [Google Scholar]]. Moreover, the new result is simultaneously a q-deformation of the classical commutative case and of the Weyl algebra case recently obtained by Eshmatov et al. [8 Eshmatov, F., Futorny, V., Ovsienko, S., Fernando Schwarz, J. (2015). Noncommutative Noether’s Problem for Complex Reflection Groups. Available at: http://arxiv.org/abs/1505.05626 [Google Scholar]].

Second, we introduce a new family of algebras called quantum OGZ algebras. They are natural quantizations of the OGZ algebras introduced by Mazorchuk [18 Mazorchuk, V. (1999). Orthogonal Gelfand-Zetlin algebras, I. Beiträge Algebra Geom. 40(2):399–415. [Google Scholar]] originating in the classical Gelfand–Tsetlin formulas. Special cases of quantum OGZ algebras include the quantized enveloping algebra of 𝔤𝔩_n and quantized Heisenberg algebras. We show that any quantum OGZ algebra can be naturally realized as a Galois ring in the sense of Futorny-Ovsienko [11 Futorny, V., Ovsienko, S. (2010). Galois orders in skew monoid rings. J. Algebra 324:598–630.[Crossref], [Web of Science ®] , [Google Scholar]], with symmetry group being a direct product of complex reflection groups G(m,p,r_k).

Finally, using these results, we prove that the quantum OGZ algebras satisfy the quantum Gelfand–Kirillov conjecture by explicitly computing their division ring of fractions.     

Rank Properties of the Semigroup of Singular Transformations on a Finite Set

It is known that the semigroup Sing _n of all singular self-maps of X _n  = {1,2,…, n} has rank n(n ? 1)/2. The idempotent rank, defined as the smallest number of idempotents generating Sing _n , has the same value as the rank. (See Gomes and Howie, 1987 Gomes , G. M. S. , Howie , J. M. ( 1987 ). On the rank of certain finite semigroups of transformations . Math. Proc. Cambridge Phil. Soc. 101 : 395 – 303 .[Crossref], [Web of Science ®] , [Google Scholar].) Idempotents generating Sing _n can be seen as special cases (with m = r = 2) of (m, r)-path-cycles, as defined in Ay\i k et al. (2005 Ay?k , G. , Ay?k , H. , Howie , J. M. ( 2005 ). On factorisations and generators in transformation semigroups . Semigroup Forum 70 : 225 – 237 .[Crossref], [Web of Science ®] , [Google Scholar]). The object of this article is to show that, for fixed m and r, the (m, r)-rank of Sing _n , defined as the smallest number of (m, r)-path-cycles generating Sing _n , is once again n(n ? 1)/2.     

On computing the integral closure

In Hai and Thin [1 Hai , B. X. , Thin , N. V. On locally nilpotent subgroups of GL ₁(D). Communications in Algebra 37 ( 2 ): 712 – 718 . [Google Scholar]], there is a theorem, stating that every locally nilpotent subnormal subgroup in a division ring D is central (see [1 Hai , B. X. , Thin , N. V. On locally nilpotent subgroups of GL ₁(D). Communications in Algebra 37 ( 2 ): 712 – 718 . [Google Scholar], Theoerem 2.2]). Unfortunately, there is some mistake in the proof of this theorem. In this note, we give the another proof of this theorem.     

Examples of Commutative Right-Nilalgebras Over Small Fields

Correa et al. (2003 Correa , I. , Hentzel , I. R. , Labra , A. ( 2003 ). On nilpotence of commutative right nilalgebras of low dimension . Int. J. Math. Game Theory Algebra 13 ( 3 ): 199 – 202 . [Google Scholar]) proved that any commutative right-nilalgebra of nilindex 4 and dimension 4 is nilpotent in characteristic ≠ 2,3. They did not assume power-associativity. In this article we will further investigate these algebras without the assumption on the dimension and providing examples in those cases that are not covered in the classification concentrating mostly on algebras generated by one element.     

Construction of Nilpotent Jordan Algebras Over any Arbitrary Field

The paper is devoted to the study of annihilator extensions of Jordan algebras and suggests new approach to classify nilpotent Jordan algebras, which is analogous to the Skjelbred–Sund method for classifying nilpotent Lie algebras [2 de Graaf, W. (2007). Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. J. Algebra 309:640–653.[Crossref], [Web of Science ®] , [Google Scholar], 4 Gong, M.-P. (1998). Clasification of Nilpotent Lie Algebras of Dimension 7 [Ph.D]. Ontario, Canada: University of Waterloo. [Google Scholar], 15 Skjelbred, T., Sund, T. (1978). Sur la classification des algèbres de Lie nilpotentes. C. R. Acad. Sci. Paris Sér. A-B 286:241–242. [Google Scholar]]. Subsequently, we have classified nilpotent Jordan algebras of dimension up to four.     

Construction of CPs-Stratified Algebras

The results of [7 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras. In: Tachikawa, H., Brenner, S. eds. Representations of Algebras and Related Topics, London Math. Society Lecture Note Series 168:200–224 . [Google Scholar]] and [2 Ágoston , I. , Dlab , V. , Lukács , E. ( 2011 ). Constructions of stratified algebras . Comm. Algebra 39 : 2545 – 2553 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] gave a recursive construction for all quasi-hereditary and standardly stratified algebras starting with local algebras and suitable bimodules. Using the notion of stratifying pairs of subcategories, introduced in [3 Ágoston , I. , Lukács , E. Stratifying pairs of subcategories for CPS-stratified algebras . To appear in Journal of Algebra and Its Applications , p. 11 . [Google Scholar]], we generalize these earlier results to construct recursively all CPS-stratified algebras.     

The infinitesimal and the little q-Schur superalgebras

In this paper, based on the results in [8 Du, J., Gu, H.-X. (2014). A realization of the quantum supergroup U(𝔤𝔩_m|n). J. Algebra 404:60–99.[Web of Science ®] , [Google Scholar]] we give a monomial basis for q-Schur superalgebra and then a presentation for it. The presentation is different from that in [12 El Turkey, H., Kujawa, J. (2012). Presenting Schur superalgebras. Pacific J. Math., 262(2):285–316.[Crossref], [Web of Science ®] , [Google Scholar]]. Imitating [3 Cox, A. G. (1997). On some applications of infinitesimal methods to quantum groups and related algebras. Ph.D. Thesis. University of London. [Google Scholar]] and [7 Du, J., Fu, Q., Wang, J.-P. (2005). Infinitesimal quantum 𝔤𝔩_n and little q-Schur algebras. J. Algebra 287:199–233.[Crossref], [Web of Science ®] , [Google Scholar]], we define the infinitesimal and the little q-Schur superalgebras. We give a “weight idempotent presentation” for infinitesimal q-Schur superalgebras. The BLM bases and monomial bases of little q-Schur superalgebras are obtained, and dimension formulas of infinitesimal and little q-Schur superalgebras are deduced.     

Support Varieties for Modules Over Stacked Monomial Algebras

Let Λ be a finite-dimensional (D, A)-stacked monomial algebra. In this article, we give necessary and sufficient conditions for the variety of a simple Λ-module to be nontrivial. This is then used to give structural information on the algebra Λ, as it is shown that if the variety of every simple module is nontrivial, then Λ is a D-Koszul monomial algebra. We also provide examples of (D, A)-stacked monomial algebras which are not self-injective but nevertheless satisfy the finite generation conditions (Fg1) and (Fg2) of [4 Erdmann , K. , Holloway , M. , Snashall , N. , Solberg , Ø. , Taillefer , R. ( 2004 ). Support varieties for selfinjective algebras . K-Theory 33 : 67 – 87 .[Crossref] , [Google Scholar]], from which we can characterize all modules with trivial variety.     

On an Ingelstam's Theorem

A well-known Ingelstam's Theorem asserts that every real Hilbert space A with an associative unital product satisfying ‖ xy‖ ≤ ‖ x‖ ‖ y‖ and ‖ 1‖ = 1 is isomorphic to the reals ?, or the complex numbers ?, or the quaternions ?. This note deals with a nonunital and nonassociative extension of the Ingelstam Theorem. So the assumptions about associativity and existence of unity are weakened to the existence of a nonzero central idempotent e such that ‖ ex‖ = ‖e‖ ‖ x‖ for all x, and that in A holds a determined kind of algebraic identity strictly weaker that alternativeness. We prove that, up to isomorphisms, there are only seven algebras satisfying these assumptions, even without the requirement of completeness. On the other hand, Section 3 presents another characterization of the obtained algebras with the flavor of one of the main theorems in Bhatt et al. (1998 Bhatt , S. J. , Karia , D. J. , Kulkarni , S. H. , Shimpi , M. E. ( 1998 ). A note on the Gelfand-Mazur theorem . Proc. Amer. Math. Soc. 126 ( 10 ): 2999 – 3005 .[Crossref], [Web of Science ®] , [Google Scholar]).     

GRADINGS OF SIMPLE JORDAN ALGEBRAS AND THEIR RELATION TO THE GRADINGS OF SIMPLE ASSOCIATIVE ALGEBRAS

In this paper we describe all group gradings of the simple Jordan algebra of a non-degenerate symmetric form on a vector space over a field of characteristic different from 2. If we use the notion of the Clifford algebra, then we are able to recover some of the gradings on matrix algebras obtained in an entirely different way in [BSZ] Bahturin, Y., Seghal, S. and Zaicev, M. in press. Group Gradings of Associative Algebras. J. Algebra, [Web of Science ®] , [Google Scholar].

     

Application of Full Quivers of Representations of Algebras,to Polynomial Identities

In [7 Belov , A. , Rowen , L. H. , Vishne , U. Full quivers of representations of algebras. To appear in Trans. Amer. Math. Soc.  [Google Scholar]] we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras, and better suited for algebras over finite fields. Here, we consider full quivers as a combinatorial tool in order to describe PI-varieties of algebras. We apply the theory to clarify the proofs of diverse topics in the literature: Determining which relatively free algebras are weakly Noetherian, determining when relatively free algebras are finitely presented, presenting a quick proof for the rationality of the Hilbert series of a relatively free PI-algebra, and explaining counterexamples to Specht's conjecture for varieties of Lie algebras.