Ian M. Musson

Lie Superalgebras and Enveloping Algebras

Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. This book develops the theory of Lie superalgebras, their enveloping algebras, and their representations. The book begins with five chapters on the basic properties of Lie superalgebras, including explicit constructions for all the classical simple Lie superalgebras. Borel subalgebras, which are more subtle in this setting, are studied and described. Contragredient Lie superalgebras are introduced, allowing a unified approach to several results, in particular to the existence of an invariant bilinear form on

Noetherian down-up algebras

\mathfrak{g}

Differential operators on toric varieties

. The enveloping algebra of a finite dimensional Lie superalgebra is studied as an extension of the enveloping algebra of the even part of the superalgebra. By developing general methods for studying such extensions, important information on the algebraic structure is obtained, particularly with regard to primitive ideals. Fundamental results, such as the Poincare-Birkhoff-Witt Theorem, are established. Representations of Lie superalgebras provide valuable tools for understanding the algebras themselves, as well as being of primary interest in applications to other fields. Two important classes of representations are the Verma modules and the finite dimensional representations. The fundamental results here include the Jantzen filtration, the Harish-Chandra homomorphism, the Sapovalov determinant, supersymmetric polynomials, and Schur-Weyl duality. Using these tools, the center can be explicitly described in the general linear and orthosymplectic cases. In an effort to make the presentation as self-contained as possible, some background material is included on Lie theory, ring theory, Hopf algebras, and combinatorics.

Multiparameter quantum enveloping algebras

Down-up algebras A = A(α, β, γ) were introduced by G. Benkart and T. Roby to better understand the structure of certain posets. In this paper, we prove that β 6= 0 is equivalent to A being right (or left) Noetherian, and also to A being a domain. Furthermore, when this occurs, we show that A is Auslander-regular and has global dimension 3.

A classification of primitive ideals in the enveloping algebra of a classical simple Lie superalgebra

If X is a toric variety we show that X is isomorphic to a quotient Y G where G is a torus acting on an affine space ks and Y is a G-invariant open subset of ks. We also show that any ring of differential operators on X twisted by an invertible sheaf is a factor ring of the fixed ring D(Y)G by an ideal generated by central elements.

Hopf down–up algebras

Let A = A(p, λ) be the multiparameter deformation of the coordinate algebra of n × n matrices as described by Artin, et al. (1991). Let U be the quantum enveloping algebra which is associated to A, in the sense of Faddeev, Reshetikhin and Takhtadzhyan. We prove a PBW theorem for U and establish a presentation by generators and relations, when λ is not a root of unity. Our approach depends on a cocycle twisting method which reduces many arguments to the standard one-parameter deformation.

Complete sets of representations of classical Lie superalgebras

For a Lie superalgebra 9 we denote the even and odd parts of 9 by go and g,, respectively. The simple Lie superalgebra 9 is called classical if f0 is reductive. For 9 classical simple we study primitive ideals in the enveloping algebra U(g). Our main result is that any graded primitive ideal is the annihilator of a graded simple quotient of a Verma module. This is an analogue of the well-known theorem of Duflo [D] on primitive ideals in the enveloping algebra of a semisimple Lie algebra. The proof is based on Duflo’s theorem and some work of E. Letzter [Ll, L2] on primitive ideals in finite ring extensions. The definition of a Verma module depends on the existence of a triangular decomposition in 9. This is dicussed in Section 1. A more precise statement of the main theorem is given Section 2. In Section 3 we discuss some corollaries, for example we show that if JZ # Q(H) then graded prime ideals are prime (Corollary 3.1), and if f # P(n), then any factor ring of U(g) has the same left and right Krull dimension (Corollary 3.3). Classical simple Lie superalgebras which are not Lie algebras have been classified by Kac [Kl, Theorem 2, p. 441 (see also [Sch, Theorem 1, p. 1401). In the notation of Kac these algebras are as follows. Scheunert’s notation, if different is given in parentheses. A(m,n)=sl(m+l,n+l), m#n,m,n~O(spl(m+1,n+1)) A(n, n) = son + 1, n + l)/(Zzn+z>, n>O(spl(n+1,n+1)/@rz,+2)

MONOLITHIC MODULES OVER NOETHERIAN RINGS

and hence we know it has a graded Hopf structure, but wedo not know if it has a Hopf structure.In the third section we show that a theorem of De Concini and Procesi providesa furthertechnique for proving that an algebra is not a Hopf algebra. We use this result to showthat under certain conditions a localization of a down–up algebra is not a Hopf algebra.Localizations of down–up algebras were considered by Jordan [13], and the algebras heconsiders include the algebra defined by Woronowicz [17].Throughout this paper let

Rings of differential operators and zero divisors

Descriptions of the complete sets of irreducible highest-weight modules over complex classical simple Lie superalgebras are recorded. It is further shown that the finite-dimensional irreducible modules over a (not necessarily classical simple) finite-dimensional complex Lie superalgebra form a complete set if and only if the even part of the Lie superalgebra is reductive and the universal enveloping superalgebra is semiprime.

Finite dimensional representations of invariant differential operators

We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantized Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained in [arXiv:0906.2930] for down-up algebras.