On Central Elements in the Universal Enveloping Algebras of the Orthogonal Lie Algebras

We present an analogy of the famous formula that the square of the Pfaffian is equal to the determinant for an alternating matrix for the case where the entries are the generators of the orthogonal Lie algebras. This identity clarifies the relation between the two sets of central elements in the enveloping algebra of the orthogonal Lie algebras. We employ systematically the exterior calculus for the proof.     

李超代数与Frobenius代数

利用限制李超代数的新定义,给出了李超代数的p-包络的一些相关结果,并将李代数中表示理论的一些结果推广到李超代数上,进而研究了限制李超代数与Frobenius代数的关系．     

An Equivalence of Categories for Lie Color Algebras

Abstract

Let k be a field with char k = p > 0 and G an abelian group with a bicharacter λ on G. For each p-(G,λ)-Lie color algebra L over k the p-universal enveloping algebra u(L) is a G-graded Hopf algebra,i.e.,a Hopf algebra in the category ^kG ? of kG-comodules. In this paper we describe a subcategory of ^kG ? which is equivalent to the category of the finite dimensional p-(G,λ)-Lie color algebras over k.     

Free Akivis Algebras, Primitive Elements, and Hyperalgebras

Free Akivis algebras and primitive elements in their universal enveloping algebras are investigated. It is proved that subalgebras of free Akivis algebras are free and that finitely generated subalgebras are finitely residual. Decidability of the word problem for the variety of Akivis algebras is also proved.The conjecture of K. H. Hofmann and K. Strambach (Problem 6.15 in [Topological and analytic loops, in “Quasigroups and Loops Theory and Applications,” Series in Pure Mathematics (O. Chein, H. O. Pflugfelder, and J. D. H. Smith, Eds.), Vol. 8, pp. 205–262, Heldermann Verlag, Berlin, 1990]) on the structure of primitive elements is proved to be not valid, and a full system of primitive elements in free nonassociative algebra is constructed.Finally, it is proved that every algebra B can be considered as a hyperalgebra, that is, a system with a series of multilinear operations that plays a role of a tangent algebra for a local analytic loop, where the hyperalgebra operations on B are interpreted by certain primitive elements.     

The Finite Dimension Property of the Irreducible Representations of Noetherian p-Lie Algebras

We prove that the irreducible representations are finite-dimensional of the almost solvable restricted Lie algebras with the ascending chain condition for p-subalgebras over a perfect field.     

关于Malcev代数的幂零性

We investigate the nilpotence of a Malcev algebra M and of its standard enveloping Lie algebra L(M)=M D(M, M). The main result shows that an ideal A of M is nilpotent in M if and only if the corresponding ideal Ⅰ(A) = A D(A, M)is nilpotent in L(M).     

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.     

Realization of Poisson enveloping algebra

For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping algebra is an associative one. In this article, for a Poisson structure on a polynomial algebra S, we first construct a Poisson algebra R, then prove that the Poisson enveloping algebra of S is isomorphic to the specialization of the quantized universal enveloping algebra of R, and therefore, is a deformation quantization of R.     

An Embedding Theorem for Reduced Albert Algebras Over Arbitrary Fields

Extending two classical embedding theorems of Albert and Jacobson and Jacobson for Albert (exceptional simple Jordan) algebra over fields of characteristic not two to base fields of arbitrary characteristic, we show that any element of a reduced Albert algebra can be embedded into a reduced absolutely simple subalgebra of degree 3 and dimension 9 which may be chosen to be split if the Albert algebra was split to begin with.     

Presenting Schur Algebras as Quotients of the Universal Enveloping Algebra of gl2

We give a presentation of the Schur algebras S _Q (2,d) by generators and relations, in fact a presentation which is compatible with Serre's presentation of the universal enveloping algebra of a simple Lie algebra. In the process we find a new basis for S _Q (2,d), a truncated form of the usual PBW basis. We also locate the integral Schur algebra within the presented algebra as the analogue of Kostant's Z-form, and show that it has an integral basis which is a truncated version of Kostant's basis.     

A Comment on the Integration of Leibniz Algebras

In this note we point out that the definition of the universal enveloping dialgebra for a Leibniz algebra is consistent with the interpretation of a Leibniz algebra as a generalization not of a Lie algebra, but of the adjoint representation of a Lie algebra. From this point of view, the formal integration problem of Leibniz algebras is, essentially, trivial.     

The Isomorphism Problem for Universal Enveloping Algebras of Lie Algebras

Let L be a Lie algebra with universal enveloping algebra U(L). We prove that if H is another Lie algebra with the property that U(L) ≅ U(H) then certain invariants of L are inherited by H. For example, we prove that if L is nilpotent then H is nilpotent with the same class as L. We also prove that if L is nilpotent of class at most two then L is isomorphic to H. Presented by D. Passman     

Non-Existence of Coassociative Quantized Universal Enveloping Algebras of the Traceless Octonions

The purpose of this brief note is to prove that any coassociative bialgebra deformation of the universal enveloping algebra of the seven dimensional central simple exceptional Malcev algebra over a field of characteristic zero is cocommutative.     

The Magnus embedding as a special case of a homological construction

Let E be a group extension with Abelian kernel. Then it can be assigned an extension E′ of modules over the group ring of the quotient group. As a consequence, an embedding of the initial extension in some splitting extension arises. We prove that the celebrated Magnus embedding is a special case of this general construction.     

Every Akivis Algebra is Linear

An Akivis algebra is a vector space V endowed with a skew-symmetric bilinear product [x,y] and a trilinear product A(x,y,z) that satisfy the identity

Realizing enveloping algebras via varieties of modules

By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra L（A） generated by indecomposable constructible sets in the varieties of modules for any finite- dimensional C-algebra A. We obtain a geometric realization of the universal enveloping algebra R（A） of L（A）, this generalizes the main result of Riedtmann. We also obtain Green＇s formula in a geometric form for any finite-dimensional C-algebra A and use it to give the comultiplication formula in R（A）.     

Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let \(\mathfrak{h}\) be an algebraic subalgebra of the tangent Lie algebra \(\mathfrak{g}\) of G. We find all subalgebras \(\mathfrak{h}\) that have no nontrivial characters and whose centralizers \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) and \(P(\mathfrak{g})^\mathfrak{h} \) in the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g})\) and in the associated graded algebra \(P(\mathfrak{g})\), respectively, are commutative. For all these subalgebras, we prove that \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g} \) and \(P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g} \). Furthermore, we obtain a criterion for the commutativity of \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) in terms of representation theory.     

Newton's formula for

This paper presents an explicit relation between the two sets which are well-known generators of the center of the universal enveloping algebra of the Lie algebra : one by Capelli (1890) and the other by Gelfand (1950). Our formula is motivated to give an exact analogy for the classical Newton's formula connecting the elementary symmetric functions and the power sum symmetric functions. The formula itself can be deduced from a more general result on Yangians obtained by Nazarov. Our proof is elementary and has an advantage in its direct accessibility.

     

Improved High Order Integrators Based on the Magnus Expansion

We build high order efficient numerical integration methods for solving the linear differential equation = A(t)X based on the Magnus expansion. These methods preserve qualitative geometric properties of the exact solution and involve the use of single integrals and fewer commutators than previously published schemes. Sixth- and eighth-order numerical algorithms with automatic step size control are constructed explicitly. The analysis is carried out by using the theory of free Lie algebras.