Characterizing n-Isoclinism Classes of Lie Algebras

In this article, we introduce the notion of the equivalence relation, n-isoclinism, between Lie algebras, and obtain some criterions under which Lie algebras are n-isoclinic. In particular, we show that n-isoclinic Lie algebras can be isoclinically embedded into one Lie algebra. Also, we present the notion of an n-stem Lie algebra and prove its existence within an arbitrary n-isoclinism class. In addition, similar to a result of Hekster [6 Hekster , N. S. ( 1986 ). On the structure of n-isoclinam classes of groups . J. Pure Appl. Algebra 40 : 63 – 85 .[Crossref], [Web of Science ®] , [Google Scholar]] in the group case, we characterize the n-stem Lie algebras in the n-isoclinism classes which contains at least one finitely generated Lie algebra L with dim (L ⁿ⁺¹) finite.     

Simplicity of a Vertex Operator Algebra Whose Griess Algebra is the Jordan Algebra of Symmetric Matrices

Let r ∈ ? be a complex number, and d ∈ ?_≥2 a positive integer greater than or equal to 2. Ashihara and Miyamoto [4 Ashihara , T. , Miyamoto , M. ( 2009 ). Deformation of central charges, vertex operator algebras whose Griess algebras are Jordan algebras . Journal of Algebra 32 : 1593 – 1599 . [Google Scholar]] introduced a vertex operator algebra V _𝒥 of central charge dr, whose Griess algebra is isomorphic to the simple Jordan algebra of symmetric matrices of size d. In this article, we prove that the vertex operator algebra V _𝒥 is simple if and only if r is not an integer. Further, in the case that r is an integer (i.e., V _𝒥 is not simple), we give a generator system of the maximal proper ideal I _r of the VOA V _𝒥 explicitly.     

Schur Algebras of Classical Groups II

We study Schur algebras of classical groups over an algebraically closed field of characteristic different from 2. We prove that Schur algebras are generalized Schur algebras (in Donkin's sense) in types A, C, and D, while this does not hold in type B. Consequently Schur algebras of types A, C, and D are integral quasi-hereditary by Donkin [7 Donkin , S. ( 1986 ). Schur algebras and related algebras I . J. Algebra 104 : 310 – 328 . [Google Scholar], 9 Donkin , S. ( 1994 ). Schur algebras and related algebras III: integral representations . Math. Proc. Camb. Phil. Soc. 116 : 37 – 55 . [Google Scholar]]. By using the coalgebra approach we put Schur algebras of a fixed classical group into a certain inverse system. We find that the corresponding hyperalgebra is contained in the inverse limit as a subalgebra. Moreover in types A, C, and D, the surjections in the inverse systems are compatible with the integral quasi-hereditary structure of Schur algebras.     

Characterizations for Supersolvable Lie Algebras

In (2009), Towers [10 Towers , D. A. ( 2009 ). C-ideals of Lie algebras . Comm. Algebra 37 : 4366 – 4373 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] presented the notion of c-ideality of a subalgebra of a Lie algebra, and gave some characterizations of solvable and supersolvable Lie algebras. In this article, we further investigate the influence of c-ideality of some subalgebras on the structure of Lie algebras. We also obtain some equivalent conditions for supersolvability of a finite dimensional Lie algebra.     

Quadratic Lie Algebras

In this article, the notion of universal enveloping algebra introduced in Ardizzoni [4 Ardizzoni , A. On primitively generated braided bialgebras . Algebr. Represent. Theory , to appear. (doi:10.1007/s10468-010-9257-z)  [Google Scholar]] is specialized to the case of braided vector spaces whose Nichols algebra is quadratic as an algebra. In this setting, a classification of universal enveloping algebras for braided vector spaces of dimension not greater than 2 is handled. As an application, we investigate the structure of primitively generated connected braided bialgebras whose braided vector space of primitive elements forms a Nichols algebra, which is a quadratic algebra.     

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]).     

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.     

A Method to Construct Oriented Quantum Algebras

In this note, we mainly give a method to construct the oriented quantum algebra structure on the tensor product of two different oriented quantum algebras, generalizing Radford ([8 Radford , D. E. ( 2007 ). On the tensor product of an oriented quantum algebra with itself . J. Knot Theory Ramifications 16 : 929 – 957 . [Google Scholar]], Theorem 4.1).     

Lie Algops and Simple Lie Algebras

ABSTRACT

Pairs (A, L) with A a commutative algebra and L a Lie algebra acting on A by derivations, called Lie algops, are studied as algebraic structures over arbitrary fields of arbitrary characteristic. Lie algops possess modules and tensor products—and are considered with respect to a central simple theory.

The simplicity problem of determining the faithful unital simple Lie algops ( A, L ) is of interest since the corresponding Lie algebras AL are usually simple (Jordan, 2000 Jordan , D. A. ( 2000 ). On the simplicity of Lie algebras of derivations of commutative algebras . J. Algebra 228 : 580 – 585 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). For locally finite Lie algops, and up to purely inseparable descent, this problem reduces by way of closures to the closed central simplicity problem of determining those which are closed central simple.

The simplicity and representation theories for locally nilpotent separably triangulable unital Lie algops are of particular interest because they relate to the problems of classifying simple Lie algebras of Witt type and their representations. Of these, the simplicity theory reduces to that of Jordan Lie algops.

The main Theorems 7.3 and 7.4 reduce the simplicity and representation theories for Jordan Lie algops to the simplicity and representation theories for simple nil and toral Lie algops.     

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.     

On Plenary Train Algebras of Rank 4

The existence of idempotent elements in plenary train algebras of rank greater than 3, is an open problem to be solved. J. Carlos Gutierrez's results on plenary train algebras in Gutierrez (2000 Gutierrez , J. C. ( 2000 ). Principal and plenary train algebras . Comm. Algebra 28 ( 2 ): 653 – 667 . [Google Scholar]) are based on the underlying assumption of the existence of an idempotent. In this article we study conditions on the scalars defining a plenary train algebra of rank 4 to assure the existence of such an idempotent.     

New Directions in Representation Theory

The Iwahori?CHecke algebra H(G, B) of a finite Chevalley group G with respect to a Borel subgroup B is described as a deformation of the group algebra of the Weyl group of G Similarly, the +-part of the quantized enveloping algebra ${{U^+_v (\mathfrak{g})}}$ associated with a semisimple Lie algebra ${{\mathfrak{g}}}$ can be viewed as a deformation of the +-part of the universal enveloping algebra ${{U(\mathfrak{g})}}$ . In both cases it is shown how information concerning the deformed algebras H(G, B) and ${{U^+_v (\mathfrak{g})}}$ can be used to obtain results about the representation theory of the Chevalley group G and the semisimple Lie algebra ${{\mathfrak{g}}}$ .     

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.     

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.     

The Antipode of a Braided Dual Quasi-Hopf Algebra is Bijective

We show that the antipode of a braided dual quasi-Hopf algebra is inner, and, a fortiori, bijective. This improves a result of Li [10 Li , J. Q. ( 2006 ). Dual quasi-Hopf algebras and antipodes . Algebra Colloquium 13 : 111 – 118 .[Crossref] , [Google Scholar]].     

Right and left solvable extensions of an associative Leibniz algebra

A sequence of nilpotent Leibniz algebras denoted by N_n,18 is introduced. Here n denotes the dimension of the algebra defined for n≥4; the first term in the sequence is ?₁₈ in the list of four-dimensional nilpotent Leibniz algebras introduced by Albeverio et al. [4 Albeverio, S., Omirov, B. A., Rakhimov, I. S. (2006). Classification of 4-dimensional nilpotent complex Leibniz algebras. Extr. Math. 21(3):197–210. [Google Scholar]]. Then all possible right and left solvable indecomposable extensions over the field ? are constructed so that N_n,18 serves as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

Irreducible Modules for the Lie Algebra of Divergence Zero Vector Fields on a Torus

This article investigates the irreducibility of certain representations for the Lie algebra of divergence zero vector fields on a torus. In [2 Eswara Rao, S. (1996). Irreducible representations of the Lie-algebra of the diffeomorphisms of a d-dimensional torus. J. Algebra 182(2):401–421.[Crossref], [Web of Science ®] , [Google Scholar]] Rao constructs modules for the Lie algebra of polynomial vector fields on an N-dimensional torus, and determines the conditions for irreducibility. The current article considers the restriction of these modules to the subalgebra of divergence zero vector fields. It is shown here that Rao's results transfer to similar irreducibility conditions for the Lie algebra of divergence zero vector fields.