Annihilation Theorem and Separation Theorem for basic classical Lie superalgebras

In this article we prove that for a basic classical Lie superalgebra the annihilator of a strongly typical Verma module is a centrally generated ideal. For a basic classical Lie superalgebra of type I we prove that the localization of the enveloping algebra by a certain central element is free over its centre.

     

Special Lie superalgebras with maximality condition for subalgebras

We consider finitely generated Lie superalgebras over a field of characteristic zero satisfying Capelli identities. We prove that any such an algebra with the maximality condition for abelian subalgebras is finite dimensional. In particular, any special Lie superalgebra with the maximality condition for its subalgebras has a finite dimension. We also prove that the universal enveloping algebra U(L) of special Lie superalgebra L is Noetherian if and only if $\dim L<\infty$ .     

The Zhang Transformation and U q (osp(1,2l))-Verma Modules Annihilators

R. B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l+1) and the Lie superalgebra osp(1,2l). The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebras à la Drinfeld and Jimbo and to show how this construction can explain the main theorem of Gorelik and Lanzmann: the annihilator of a Verma module over the Lie superalgebra osp(1,2l) is generated by its intersection with the centralizer of the even part of the enveloping algebra.     

Lie superalgebras whose enveloping algebras satisfy a non-matrix polynomial identity

Let L be a Lie superalgebra with its enveloping algebra U(L) over a field F. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2×2 matrices over F. We characterize L when U(L) satisfies a non-matrix polynomial identity. We also characterize L when U(L) is Lie solvable, Lie nilpotent, or Lie super-nilpotent.     

量子群Uq(osp(1, 2))的中心

本文讨论了单李超代数osp(1,2)的量子化包络代数Uq(osp(1,2))的中心,利用Uq(osp(1,2))的表示的已知结果,证明了量子群Uq(osp(1,2))的中心的刻画,证明了该量子群的中心是由一个元素生成的多项式代数.     

A note on the homogenized lie superalgebra osp(l, 2)

The homogenization of a superenveloping algebra is Auslander regular iff this enveloping algebra is a domain. This note determines points and lines in the quantum space of the homogenized envelope of the classical simple Lie superalgebra osp(l,2). It is also concerned with homogenized sl(2), which is graded embedded in homogenized osp(l,2), and describes the conies in the associated quantum space.     

李超代数的泛包络代数同构问题

李超代数的一个性质P叫做关于泛包络代数的不变量,如果对于任意李超代数L,H,只要L具有性质P,并且泛包络代数U(L)和U(H)作为结合超代数是同构的,那么H亦具有性质P.通过讨论李超代数关于泛包络代数的不变量证明了:如果L的幂零长度不超过2,那么L和H是同构的.     

The annihilation theorem for the completely reducible Lie superalgebras

A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1,2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not generated by its intersection with the centre of universal enveloping algebra. More generally, Musson produced in [Mu1] a family of such “singular” Verma modules for osp(1,2l) cases. In this article we give a necessary and sufficient condition on the highest weight of a osp(1,2l)-Verma module for its annihilator to be generated by its intersection with the centre. This answers a question of Musson. The classical proof of the Duflo theorem is based on a deep result of Kostant which uses some delicate algebraic geometry reasonings. Unfortunately these arguments can not be reproduced in the quantum and super cases. This obstruction forced Joseph and Letzter, in their work on the quantum case (see [JL]), to find an alternativeapproach to the Duflo theorem. Following their ideas, we compute the factorization of the Parthasarathy–Ranga-Rao–Varadarajan (PRV) determinants. Comparing it with the factorization of Shapovalov determinants we find, unlike to the classical and quantum cases, that the PRV determinant contains some extrafactors. The set of zeroes of these extrafactors is precisely the set of highest weights of Verma modules whose annihilators are not generated by their intersection with the centre. We also find an analogue of Hesselink formula (see [He]) giving the multiplicity of every simple finite dimensional module in the graded component of the harmonic space in the symmetric algebra. Oblatum 1-IX-1998 & 4-XII-1998 / Published online: 10 May 1999     

关于李三代数的幂零性

In this paper, we mainly concerned about the nilpotence of Lie triple algebras. We give the definition of nilpotence of the Lie triple algebra and obtained that if Lie triple algebra is nilpotent, then its standard enveloping Lie algebra is nilpotent.     

Symmetric nonself-adjoint operators in an enveloping algebra

Nelson and Stinespring proved that in any unitary representation of a Lie group with compact Lie algebra the representation of Hermitian elements in the enveloping algebra are essentially self-adjoint. If the Lie algebra is noncompact, we construct in its enveloping algebra a Hermitian element u such that in any locally faithful unitary representation the representative of u has no self-adjoint extension.     

Malcev superalgebras with trival nucleus

The nucleus of a Malcev superalgebra M measures how far it is from being a Lie superalgebraM being a Lie superalgebra if and only if its nucleus is the whole M. This paper is devoted to study Malcev superalgebras in the opposite direction, that is, with trivial nucleus. The odd part of any finite-dimensional Malcev superalgebra with trivial nucleus is shown to be contained in the solvable radical. For algebraically closed fields, any such superalgebra splits as the sum of its solvable radical and a semisimple Malcev algebra contained in the even part, which is a direct sum of copies of sl(2, F) and the seven-dimensional simple non-Lie Malcev algebra, obtained from the Cayley-Dickson algebra.     

Omni-Lie superalgebras and Lie 2-superalgebras

We introduce the notion of omni-Lie superalgebras as a super version of an omni-Lie algebra introduced by Weinstein. This algebraic structure gives a nontrivial example of Leibniz superalgebras and Lie 2-superalgebras. We prove that there is a one-to-one correspondence between Dirac structures of the omni-Lie superalgebra and Lie superalgebra structures on a subspace of a super vector space,     

Integral Bases for the Universal Enveloping Algebras of Cartan Map Superalgebras

Let ?? be a finite dimensional complex simple Lie superalgebra of Cartan type and A be a commutative, associative algebra with unity over ?. We refer to the Lie superalgebras of the form ?? ? A as Cartan map superalgebras. In this paper, following Bagci and Chamberlin (J. of Pure and Applied Algebra 218(8), 1563–1576, 2014), we define an integral form for the universal enveloping algebra of the Cartan map superalgebras, and exhibit an explicit integral basis for this integral form.     

无限维奇Hamilton模李超代数的导子

本文首先确定了无限维奇Hamilton模李超代数的生成元集,然后确定了奇Hamilton模李超代数到广义Witt模李超代数的导子空间,进而确定了无限维奇Hamilton模李超代数的导子代数.     

Pointed Hopf Algebras—from Enveloping Algebras to Quantum Groups and Beyond

A fundamental example of a pointed Hopf algebra is the universal enveloping algebra of a Lie algebra. This example finds its generalization in quantum groups. The class of pointed Hopf algebras is rather extensive and has been the subject of intense study. We recount some of the basic ideas in the development of the theory. A fundamental structure in the general theory is an analog of the enveloping algebra in a certain category, the Nichols algebra.     

Cartan matrices for restricted Lie algebras

Consider a finite dimensional restricted Lie algebra over a field of prime characteristic. Each linear form on this Lie algebra defines a finite dimensional quotient of its universal enveloping algebra, called a reduced enveloping algebra. This leads to a Cartan matrix recording the multiplicities as composition factors of the simple modules in the projective indecomposable modules for such a reduced enveloping algebra. In this paper we show how to compare such Cartan matrices belonging to distinct linear forms. As an application we rederive and generalise the reciprocity formula first discovered by Humphreys for Lie algebras of reductive groups. For simple Lie algebras of Cartan type we see, for example, that the Cartan matrices for linear forms of non-positive height are submatrices of the Cartan matrix for the zero linear form.     

How to realize a Lie algebra by vector fields

We describe an algorithm for embedding a finite-dimensional Lie algebra (superalgebra) into a Lie algebra (superalgebra) of vector fields that is suitable for a ground field of any characteristic and also a way to select the Cartan, complete, and partial prolongations of the Lie algebra of vector fields using differential equations. We illustrate the algorithm with the example of Cartan’s interpretation of the exceptional simple Lie algebra (2) as the Lie algebra preserving a certain nonintegrable distribution and also several other examples. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 450–469, June, 2006.     

Formal power series, operator calculus, and duality on Lie algebras

This paper presents an operator calculus approach to computing with non-commutative variables. First, we recall the product formulation of formal exponential series. Then we show how to formulate canonical boson calculus on formal series. This calculus is used to represent the action of a Lie algebra on its universal enveloping algebra. As applications, Hamilton's equations for a general Hamiltonian, given as a formal series, are found using a double-dual representation, and a formulation of the exponential of the adjoint representation is given. With these techniques one can represent the Volterra product acting on the enveloping algebra. We illustrate with a three-step nilpotent Lie algebra.     

Twisted affine Lie superalgebra of type Q and quantization of its enveloping superalgebra

We introduce a new quantum group which is a quantization of the enveloping superalgebra of a twisted affine Lie superalgebra of type Q. We study generators and relations for superalgebras in the finite and twisted affine cases, and also universal central extensions. Afterwards, we apply the FRT formalism to a certain solution of the quantum Yang–Baxter equation to define that new quantum group and we study some of its properties. We construct a functor of Schur–Weyl type which connects it to affine Hecke–Clifford algebras and prove that it provides an equivalence between two categories of modules.     

Generalised nilpotence conditions in n-engel lie algebras

A Lie algebra is said to satisfy the Baer condition provided each of its 1-dimensional subalgebras is a subideal; if the defect is bounded then the Lie algebra is bounded Baer. We first characterise restricted enveloping algebras (of odd characteristic p) that satisfy the bounded Baer condition. Using this characterisation, we are then able to construct a Lie algebra satisfying the bounded Engel condition, in each odd characteristic p, that does not satisfy the Baer condition. Such a bounded Engel Lie algebra must contain a non-nilpotent 1-generated ideal