In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra gl(1|n), one is faced with determining the eigenvalues and eigenvectors for an arbitrary selfadjoint odd element of gl(1|n) in any unitary irreducible representation W . We show that the eigenvalue problem can be solved by the decomposition of W with respect to the branching gl(1|n) → gl(1|1)⊕gl(n−1). The eig...

Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebras and root vectors and which make it possible to construct representations by operators acting according to Gel’fand–Tsetlin-type formulas. Unitary representations of the q-deformed algebras Uq(son,1) are found. AMS subject classifications (1980). 16...

On the eigenvalue problem for arbitrary odd elements of the Lie superalgebra gl(1|n) and applications Abstract In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra gl(1|n), one is faced with determining the eigenvalues and eigenvectors for an arbitrary self-adjoint odd element of gl(1|n) in any unitary irreducible representation W. We show that the eigenvalue p...

When $G$ is a real reductive group and $G_0$ its Cartan motion group, the Mackey-Higson bijection natural one-to-one correspondence between all irreducible tempered representations of unitary $G_0$. In this short note, we collect some known facts about topology dual $\widetilde{G}$ that $\widehat{G_0}$, then verify $\widetilde{G} \to \widehat{G_0}$ continuous.

Introduction. In this note, we will announce a characterization of a connected, simply connected Type I solvable Lie group, G, and present a complete description of the set of all unitary equivalence classes of irreducible unitary representations of G together with a construction of an irreducible representation in each equivalence class. This result subsumes the results previously obtained on ...

In this paper, we investigate some mathematical structures underlying the physics of linear optical passive (LOP) devices. We show, in particular, that with the class of LOP transformations on N optical modes one can associate a unitary representation of U(N) in the N -mode Fock space, representation which can be decomposed into irreducible sub-representations living in the subspaces characteri...

A large class of representations of the quantum Lorentz group QLG (the one admitting Iwasawa decomposition) is found and described in detail. In a sense the class contains all irreducible unitary representations of QLG. Parabolic subgroup P of the group QLG is introduced. It is a smooth deformation of the subgroup of SL(2,C) consisting of the upper-triangular matrices. A description of the set ...

We study the irreducible unitary highest weight representations, which are obtained from free field realizations, of W infinity algebras (W∞, W1+∞, W 1,1 ∞ , W M ∞ , W N 1+∞, W M,N ∞ ) with central charges (2, 1, 3, 2M , N , 2M +N). The characters of these representations are computed. We construct a new extended superalgebra W ∞ , whose bosonic sector is W ∞ ⊕ W N 1+∞. Its representations obta...

Using the theta correspondence, we extend classification of irreducible representations quasi-split unitary groups (the so-called local Langlands which is due to Mok) non groups. We also prove that our satisfies some good properties, characterize it uniquely. In particular, this paper provides an alternative approach works Kaletha-Mínguez-Shin-White and Mœglin-Renard.