Hans Plesner Jakobsen

Wave and Dirac Operators, and Representations of the Conformal Group

Abstract Let M be the flat Minkowski space. The solutions of the wave equation, the Dirac equations, the Maxwell equations, or more generally the mass 0, spin s equations are invariant under a multiplier representation U s , of the conformal group. We provide the space of distributions solutions of the mass 0, spin s equations with a Hilbert space structure H s , such that the representation U s , will act unitarily on H s . We prove that the mass 0 equations give intertwining operators between representations of principal series. We relate these representations to the Segal-Shale-Weil (or “ladder”) representation of U (2, 2).

Hermitian symmetric spaces and their unitary highest weight modules

Abstract The purpose of this article is to determine the set of unitarizable highest weight modules corresponding to Hermitian symmetric spaces of the noncompact type. The major step is that of proving unitarity at the “last possible place.” With this established the description of the full set of unitarizable highest weight modules follows by a straightforward tensor product argument combined with the main ingredients of the proof of the key theorem: Bernstein-Gelfand-Gelfand, and a diagramatic representation of the set of positive noncompact roots.

Restrictions and Expansions of Holomorphic Representations

We compute tensor products of representations of the holomorphic discrete series of a Lie group G, or restrictions to some subgroup G′. A detailed study is done for the case of the conformal group O(4, 2).

Topological quantum field theories from generalized 6j symbols

Given an associative algebra with a distinguished finite set of representations that is closed under a (deformed) tensor product, and satisfies some technical assumptions, we define generalized 6j-symbols, and show that they can be associated, in a natural way, with certain labeled tetrahedra. Given a 3-dimensional compact oriented manifold M with boundary ∂M = Σ we choose an arbitrary triangulation of M and exploit the above correspondence between 6j-symbols and labeled tetrahedra to construct a vectorspace UΣ and a vector Z(M) ∈ UΣ, independent of , and fulfilling the axioms of a topological quantum field theory as formulated by Atiyah [11]. Examples covered by our approach are quantum groups corresponding to the classical simple Lie algebras as well as, expectedly, chiral algebras of 2-dimensional rational conformal field theories.

On singular holomorphic representations

unitary irreducible representations on vector valued holomorphic functions have been obtained by Gross and Kunze [1] from the decomposition of tensor products of the harmonic (Segal-Shale-Weil) repre- sentation L. Later the complete description of these tensor products for the groups

Tensor products, reproducing kernels, and power series

Abstract Tensor products of holomorphic discrete series representations in reproducing kernel Hilbert spaces are decomposed by considering power series expansions of functions in the direction perpendicular to the diagonal in D × D .

A class of quadratic matrix algebras arising from the quantized enveloping algebra Uq(A2n−1)

A natural family of quantized matrix algebras is introduced. It includes the two best studied such. Located inside Uq(A2n−1), it consists of quadratic algebras with the same Hilbert series as polynomials in n2 variables. We discuss their general properties and investigate some members of the family in great detail with respect to associated varieties, degrees, centers, and symplectic leaves. Finally, the space of rank r matrices becomes a Poisson submanifold, and there is an associated tensor category of rank ⩽r matrices.

QUANTIZED MATRIX ALGEBRAS AND QUANTUM SEEDS

We determine explicitly quantum seeds for classes of quantized matrix algebras. Furthermore, we obtain results on centres and block diagonal forms of these algebras. In the case where is an arbitrary root of unity, this further determines the degrees.

Quantized Heisenberg space

We investigate the algebra Fq(N) introduced by Faddeev, Reshetikhin and Takhadjian. In the case where q is a primitive root of unity, the degree, the center, and the set of irreducible representations are found. The Poisson structure is determined and the De Concini–Kac–Procesi Conjecture is proved for this case.

Indecomposable Finite-Dimensional Representations of a Class of Lie Algebras and Lie Superalgebras

The topic of indecomposable finite-dimensional representations of the Poincare group was first studied in a systematic way by Paneitz [5, 6]. In these investigations only representations with one source were considered, though by duality, one representation with two sources was implicitly present.