Integral bases for the universal enveloping algebras of map superalgebras

Let g$\mathfrak{g}$ be a finite dimensional complex simple classical Lie superalgebra and A   be a commutative, associative algebra with unity over C$\mathbb{C}$. In this paper we define an integral form for the universal enveloping algebra of the map superalgebra g⊗A$\mathfrak{g}\otimes A$, and exhibit an explicit integral basis for this integral form.     

Minimal prime ideals in enveloping algebras of Lie superalgebras

Let be a finite dimensional Lie superalgebra over a field of characteristic zero. Let be the enveloping algebra of . We show that when , then is not semiprime, but it has a unique minimal prime ideal; it follows then that when is classically simple, has a unique minimal prime ideal. We further show that when is a finite dimensional nilpotent Lie superalgebra, then has a unique minimal prime ideal.

     

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.     

On Borcherds type Lie algebras

For each even lattice \({\mathcal L}\), there is a canonical way to construct an infinite-dimensional Lie algebra via lattice vertex operator algebra theory, we call this Lie algebra and its subalgebras the Borcherds type Lie algebras associated to \({\mathcal L}\). In this paper, we apply this construction to even lattices arising from representation theory of finite-dimensional associative algebras. This is motivated by the different realizations of Kac-Moody algebras by Borcherds using lattice vertex operators and by Peng-Xiao using Ringel-Hall algebras respectively. For any finite-dimensional algebra \(A\) of finite global dimension, we associate a Borcherds type Lie algebra \(\mathfrak {BL}(A)\) to \(A\). In contrast to the Ringel-Hall Lie algebra approach, \(\mathfrak {BL}(A)\) only depends on the symmetric Euler form or Tits form but not the full representation theory of \(A\). However, our results show that for certain classes of finite-dimensional algebras whose representation theory is ’controlled’ by the Euler bilinear forms or Tits forms, their Borcherds type Lie algebras do have close relations with the representation theory of these algebras. Beyond the class of hereditary algebras, these algebras include canonical algebras, representation-directed algebras and incidence algebras of finite prinjective types.     

Beyond Borcherds Lie algebras and inside

We give a definition for a new class of Lie algebras by generators and relations which simultaneously generalize the Borcherds Lie algebras and the Slodowy G.I.M. Lie algebras. After proving these algebras are always subalgebras of Borcherds Lie algebras, as well as some other basic properties, we give a vertex operator representation for a factor of them. We need to develop a highly non-trivial generalization of the square length two cut off theorem of Goddard and Olive to do this.

     

Projective functors for quantized enveloping algebras

The object of this paper is to show that many of the known results concerning the structure of semiperfect FPF rings can be extended to a larger class of FPF rings. The main attributes of this larger class of rings are they have enough principle idempo-tents and idempotents lift modulo the Jacobson radical. We call these rings epi-semiperfect rings.     

A strict Positivstellensatz for enveloping algebras

Let G be a connected and simply connected real Lie group with Lie algebra . Semialgebraic subsets of the unitary dual of G are defined and a strict Positivstellensatz for positive elements of the universal enveloping algebra of is proved. An erratum to this article is available at .     

Universal enveloping conformal algebras

The main objective of this paper is to study embeddings of Lie conformal algebras into associative conformal algebras. We prove that not all Lie conformal algebras admit such embeddings. However, in many important cases, including semisimple Lie conformal algebras of finite type, embeddings of this form exist and sometimes we can even describe universal enveloping associative conformal algebras of Lie conformal algebras and prove an analogue of the classical Poincaré-Birkhoff-Witt theorem.     

Growth in enveloping algebras

It is proven that a finitely generated soluble-by-finite Lie algebra has a subexponential growth. This implies that in its universal envelope every subring is an Ore domain.     

The Beilinson-Bernstein correspondence for quantized enveloping algebras

Theory of the quantized flag manifold as a quasi-scheme (non-commutative scheme) has been developed by Lunts-Rosenberg [15]. They have formulated an analogue of the Beilinson-Bernstein correspondence using the q-differential operators introduced in their earlier paper [14]. In this paper we shall establish its modified version using a class of q-differential operators, which is (possibly) smaller than the one in [14].To Professor Noriaki Kawanaka in celebration of his sixtieth birthdayMathematics Subject Classification (2000): 20G42, 16S32, 17B37     

Uniform topologies on enveloping algebras

Let G be a Lie group with Lie algebra $\text{g}$ and $\text{E}$(G) the unversal enveloping algebra of $\text{g}$ realized as the algebra of left-invariant differential operators on G. It is proved that the uniform topology on $\text{E}$(G), i.e., the topology of uniform convergence on weakly bounded sets of vector states, coincides with the strongest locally convex topology on $\text{E}$(G). This implies that each linear functional on $\text{E}$(G) is a linear combination of strictly positive functionals.     

Centers of reduced enveloping algebras

We compute the inverse image of a functional in the Zassenhaus variety. We apply this computation to describe the category of representations for a regular functional. Received November 11, 1997; in final form February 9, 1998     

Extensions of quantum enveloping algebras

In this paper, we define a class of extended quantum enveloping algebras U _q (f(K, J)) and some new Hopf algebras, which are certain extensions of quantum enveloping algebras by a Hopf algebra H. This construction generalizes some well-known extensions of quantum enveloping algebras by a Hopf algebra and provides a large of new noncommutative and noncocommutative Hopf algebras.