V. K. Dobrev

Characters of the Unitarizable Highest Weight Modules Over the

The characters of the unitarizable highest weight modules over the N=2 superconformal algebras are presented. This is a slightly extended version of an Encyclopedia entry.

N=2

Let G be a real linear connected semisimple Lie group. We present a canonical construction of the differential operators intertwining elementary (≡ generalized principal series) representations of G. The results are easily extended to real linear reductive Lie groups.

Superconformal Algebras

The author proposes a procedure for q-deformations of the real forms G of complex Lie (super-)algebras associated with (generalized) Cartan matrices. The procedure gives different q-deformations for the nonconjugate Cartan subalgebras of G. The author gives several illustrations, e.g., q-deformed Lorentz and conformal (super-)algebras. The q-deformed conformal algebra contains as a subalgebra a q- deformed Poincare algebra and as Hopf subalgebras two conjugate 11-generator q-deformed Weyl algebras. The q-deformed Lorentz algebra is a Hopf subalgebra of both Weyl algebras.

Canonical construction of differential operators intertwining representations of real semisimple Lie groups

We give a constructive classification of the positive energy (lowest weight) unitary irreducible representations of the D = 6 superconformal algebras osp(8*/2N). Our results confirm all but one of the conjectures of Minwalla (for N = 1, 2) on this classification. Our main tool is the explicit construction of the norms of the states that have to be checked for positivity. We also give the reduction of the four exceptional unitary irreducible representations.

Canonical q-deformations of noncompact Lie (super-)algebras

It is shown that the algebra Up,q dual to GLp,q(2,C) is isomorphic to U(pq)1/2(sl(2,C)) ⊗ Z as a commutation algebra, where Z is a subalgebra central in Up,q. The subalgebra Z is a Hopf subalgebra of Up,q, while the commutation subalgebra U(pq)1/2(sl(2,C)) is not a Hopf subalgebra.

Positive energy unitary irreducible representations of D = 6 conformal supersymmetry

Abstract Global realizations of all elementary induced representations (EIR) of the group SU ∗ (4), which is the double covering group of SO↑(5,1), are given. The Knapp-Stein intertwining operators are constructed and their harmonic analysis carried out. The invariant subspaces of the reducible EIR are introduced and the differential intertwining operators between partially equivalent EIR are defined. Invariant sequilinear forms on pairs of invariant subspaces are constructed. Differential identities between invariant sesquilinear forms on pairs of irreducible components of the reducible representations are derived. The results will be applied elsewhere to the nonpertubative analysis of Euclidean conformal invariant quantum field theory with fields of arbitrary spin.

Duality for the matrix quantum group GLp,q(2,C)

The structure of the group SU(2, 2) and of its Lie algebra is studied in detail. The results will be applied in subsequent parts devoted to the explicit construction of elementary representations of SU(2, 2) induced from different parabolic subgroups and of the intertwining operators between these representations. A summary of some results of Parts II and III is given.

Elementary representations and intertwining operators for the group SU∗(4)

We show that the Hopf algebra Uuq dual to the multiparameter matrix quantum group GLuq(n) may be realized a la Sudbery(1990), i.e. tangent vectors at the identity. Furthermore, we give the Cartan-Weyl basis of Uuq and show that this is consistent with the duality. We show that as a commutation algebra Uuq equivalent to Uu(sl(n, C)) (X) Uu(Z), where Z is one-dimensional and Uu(Z) is a central algebra in Uuq. However, as a co-algebra Uuq cannot be split in this way and depends on all parameters.

Elementary representations and intertwining operators for SU(2, 2). I

We give explicit formulae for singular vectors of Verma modules over Uq(G), where G is any complex simple Lie algebra. The vectors we present correspond exhaustively to a class of positive roots of G which we call straight roots. In some special cases, we give singular vectors corresponding to arbitrary positive roots. For our vectors we use a special basis of Uq(G-), where G- is the negative roots subalgebra of G, which was introducted in our earlier work in the case q=1. This basis seems more economical than the Poincaré-Birkhoff-Witt type of basis used by Malikov, Feigin, and Fuchs for the construction of singular vectors of Verma modules in the case q=1. Furthermore, this basis turns out to be part of a general basis recently introduced for other reasons by Lusztig for Uq(ℬ-), where ℬ- is a Borel subalgebra of G.

Duality for multiparametric quantum GL(n)

We give a systematic discussion of the relation between subsingular vectors of Verma modules over semisimple Lie algebras G and differential equations which are conditionally G-invariant. This is extended to the Drinfeld-Jimbo q-deformation Uq(G) of G. We treat in detail the conformal algebra su(2,2), its complexification sl(4) and their q-deformations. The conditionally invariant equations are the d`Alembert equation and a new equation arising from a subsingular vector proposed by Bernstein-Gel`fand-Gel`fand (1971). We also give the q-difference analogues of these equations.