Weyl Modules Associated to Kac–Moody Lie Algebras

Local Weyl modules were originally defined for affine Lie algebras by Chari and Pressley in [5 Chari, V., Pressley, A. (2001). Weyl modules for classical and quantum affine algebras. Represent. Theory 5:191–223 (electronic).[Crossref] , [Google Scholar]]. In this paper we extend the notion of local Weyl modules for a Lie algebra 𝔤 ?A, where 𝔤 is any Kac–Moody algebra and A is any finitely generated commutative associative algebra with unit over ?, and prove a tensor product decomposition theorem which generalizes result in [2 Chari, V., Fourier, G., Khandai, T. (2010). A categorical approach to Weyl modules. Transform. Groups 15(3):517–549.[Crossref], [Web of Science ®] , [Google Scholar], 5 Chari, V., Pressley, A. (2001). Weyl modules for classical and quantum affine algebras. Represent. Theory 5:191–223 (electronic).[Crossref] , [Google Scholar]].     

A Tensor Product Factorization for Certain Tilting Modules

Let G be a semisimple, simply connected linear algebraic group over an algebraically closed field k of characteristic p > 0. In a recent article [6 Doty , S. R. ( 2009 ). Factoring tilting modules for algebraic groups . Journal of Lie Theory 19 ( 3 ): 531 – 535 .[Web of Science ®] , [Google Scholar]], Doty introduces the notion of r-minuscule weight and exhibits a tensor product factorization of a corresponding tilting module under the assumption p ≥ 2h ? 2, where h is the Coxeter number. We remove this restriction and consider some variations involving the more general notion of (p, r)-minuscule weights.     

Continuous Cohomology of Permutation Groups on Profinite Modules

ABSTRACT

Model theorists have made use of low-dimensional continuous cohomology of infinite permutation groups on profinite modules, see Ahlbrandt and Ziegler (1991 Ahlbrandt , G. , Ziegler , M. ( 1991 ). What's so special about (?/4?)^ω? Archive for Math. Logic 31 : 115 – 132 . [CSA] [Crossref] , [Google Scholar]), Evans (1997b Evans , D. M. ( 1997b ). Computation of first cohomology groups of finite covers . J. Algebra 193 : 214 – 238 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), Evans et al. (1997 Evans , D. M. , Ivanov , A. A. , Macpherson , H. D. ( 1997 ). Finite covers . In: Evans , D. M. , ed. Model Theory of Groups and Automorphism Groups . London Mathematical Society Lecture Notes 244 . Cambridge : Cambridge Univ Press , pp. 1 – 72 .[Crossref] , [Google Scholar]), and Hodges and Pillay (1994 Hodges , W. , Pillay , A. ( 1994 ). Cohomology of structures and some problems of Ahlbrandt and Ziegler . J. London Math. Soc. 50 ( 2 ): 1 – 16 . [CSA] [Crossref] , [Google Scholar]), for example. We expand the module category in order to widen the cohomological toolkit. For an important class of groups we use these tools to establish criteria for finiteness of cohomology.     

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.     

On Semiprime Multiplication Modules over Pullback Rings

We classify all those indecomposable semiprime multiplication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given in [9 Ebrahimi Atani , S. , Farzalipour , F. ( 2009 ). Weak multiplication modules over a pullback of Dedekind domains . Colloquium Math. 114 : 99 – 112 .[Crossref] , [Google Scholar]] to a more general semiprime multiplication modules case.     

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.     

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.     

Homotopies for the Generalized Bar Complex Associated to Certain 3-Rowed Weyl Modules

In Buchsbaum and Rota (1994 Buchsbaum , D. , Rota , G.-C. ( 1994 ). A new construction in homological algebra . Proc. Nat. Acad. Sci. 91 : 4115 – 4119 . [CSA] [CROSSREF] [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), the authors presented a generalized bar complex associated to certain 3-rowed Weyl modules and proved that this complex is in fact a resolution via an induction on the number of overlaps between the second and third rows and a fundamental exact sequence (Akin and Buchsbaum, 1985 Akin , K. , Buchsbaum , D. ( 1985 ). Characteristic-free representation theory of the general linear group . Adv. Math 58 ( 2 ): 149 – 200 . [CSA]  [Google Scholar]). In this article we study the structure of this resolution by constructing a splitting contracting homotopy for the complexes corresponding to certain shapes.     

Some cases of a conjecture on L-functions of twisted Carlitz modules

We prove two polynomial identies which are particular cases of a conjecture arising in the theory of L-functions of twisted Carlitz modules. This conjecture is stated in [6 Grishkov, A., Logachev, D. (2016). Resultantal varieties related to zeroes of L-functions of Carlitz modules. Finite Fields Appl. 38:116–176. Available at: arxiv.org/pdf/1205.2900.pdf.[Crossref], [Web of Science ®] , [Google Scholar], p. 153, 9.3, 9.4] and [8 Logachev, D., Zobnin, A. (2017). L-functions of Carlitz Modules, Resultantal Varieties and Rooted Binary Trees, arXiv:1607.06147v3 [math.AG]. [Google Scholar], p. 5, 0.2.4].     

Simple Nil Lie Algops

A Lie algop is a pair (A, L) where A is a commutative algebra and L is a Lie algebra operating on A by derivations. Faithful simple Lie algops (A, L) are of interest because the corresponding Lie algebras AL are simple—with some rare exceptions at characteristic 2. The simplicity and representation theory of Jordan Lie algops is reduced in Winter (2005b Winter , D. J. ( 2005b ). Lie algops and simple Lie algebras . Comm. Algebra 33 : 3157 – 3178 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) to the simplicity theory of nil Lie algops and the simplicity and representation theory of toral Lie algops. This paper is devoted to building the first of these two theories, the simplicity theory of nil Lie algops, as a structure theory.     

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.     

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.