Abstract:
Several advances have extended the power and versatility of coherent state theory to the extent that it has become a vital tool in the representation theory of Lie groups and their Lie algebras. Representative applications are reviewed and some new developments are introduced. The examples given are chosen to illustrate special features of the scalar and vector coherent state constructions and how they work in practical situations. Comparisons are made with Mackey's theory of induced representations. For simplicity, we focus on square integrable (discrete series) unitary representations although many of the techniques apply more generally, with minor adjustment.

Abstract:
The aim of this work is to study finite dimensional representations of the Lie superalgebra psl(2|2) and their tensor products. In particular, we shall decompose all tensor products involving typical (long) and atypical (short) representations as well as their so-called projective covers. While tensor products of long multiplets and projective covers close among themselves, we shall find an infinite family of new indecomposables in the tensor products of two short multiplets. Our note concludes with a few remarks on possible applications to the construction of AdS_3 backgrounds in string theory.

Abstract:
We analyze the decomposition of tensor products between infinite dimensional (unitary) and finite-dimensional (non-unitary) representations of SL(2,R). Using classical results on indefinite inner product spaces, we derive explicit decomposition formulae, true modulo a natural cohomological reduction, for the tensor products.

Abstract:
We study multiplicity space signatures in tensor products of representations of $\mathfrak{sl}_2$ and $U_q(\mathfrak{sl}_2)$, and give some applications. We completely classify definite multiplicity spaces for generic tensor products of $\mathfrak{sl}_2$ Verma modules. This provides a classification of a family of unitary representations of a basic quantized quiver variety, one of the first such classifications for any quantized quiver variety. We use multiplicity space signatures to provide the first real critical point lower bound for generic $\mathfrak{sl}_2$ master functions. As a corollary of this bound, we obtain a simple and asymptotically correct approximation for the number of real critical points of a generic $\mathfrak{sl}_2$ master function. We obtain a formula for multiplicity space signatures in tensor products of finite dimensional simple $U_q(\mathfrak{sl}_2)$ representations. Our formula also gives multiplicity space signatures in generic tensor products of $\mathfrak{sl}_2$ Verma modules and generic tensor products of real $U_q(\mathfrak{sl}_2)$ Verma modules. Our results have relations with knot theory, statistical mechanics, quantum physics, and geometric representation theory.

Abstract:
It is shown that if H is a Hilbert space for a representation of a group G, then there are triplets of spaces F_H, H, F^H, in which F^H is a space of coherent state or vector coherent state wave functions and F_H is its dual relative to a conveniently defined measure. It is shown also that there is a sequence of maps F_H -> H -> F^H which facilitates the construction of the corresponding inner products. After completion if necessary, the F_H, H, and F^H, become isomorphic Hilbert spaces. It is shown that the inner product for H is often easier to evaluate in F_H than F^H. Thus, we obtain integral expressions for the inner products of coherent state and vector coherent state representations. These expressions are equivalent to the algebraic expressions of K-matrix theory, but they are frequently more efficient to apply. The construction is illustrated by many examples.

Abstract:
We introduce Brauer characters for representations of the bismash products of groups in characteristic p > 0, p not 2 and study their properties analogous to the classical case of finite groups. We then use our results to extend to bismash products a theorem of Thompson on lifting Frobenius-Schur indicators from characteristic p to characteristic 0

Abstract:
We obtain a faithful representation of the twisted tensor product $B\otimes_{\chi} A$ of unital associative algebras, when $B$ is finite dimensional. This generalizes the representations of [C] where $B=K[X]/$, [GGV] where $B=K[X]/$ and [JLNS] where $B=K^n$. Furthermore, we establish conditions to extend twisted tensor products $B\otimes_{\chi} A$ and $C\otimes_{\psi} A$ to a twisted tensor product $(B\times C) \otimes_{\varphi} A$.

Abstract:
We develop the deformation theory of A_\infty algebras together with \infty inner products and identify a differential graded Lie algebra that controls the theory. This generalizes the deformation theories of associative algebras, A_\infty algebras, associative algebras with inner products, and A_\infty algebras with inner products.

Abstract:
How to represent the genetic code? Despite the fact that it is extensively known, the DNA mapping into proteins remains as one of the relevant discoveries of genetics. However, modern genomic signal processing usually requires converting symbolic-DNA strings into complex-valued signals in order to take full advantage of a broad variety of digital processing techniques. The genetic code is revisited in this paper, addressing alternative representations for it, which can be worthy for genomic signal processing. Three original representations are discussed. The inner-to-outer map builds on the unbalanced role of nucleotides of a 'codon' and it seems to be suitable for handling information-theory-based matter. The two-dimensional-Gray map representation is offered as a mathematically structured map that can help interpreting spectrograms or scalograms. Finally, the world-map representation for the genetic code is investigated, which can particularly be valuable for educational purposes -besides furnishing plenty of room for application of distance-based algorithms.

Abstract:
We initiate a study of infinite tensor products of projective unitary representations of a discrete group G. Special attention is given to regular representations twisted by 2-cocycles and to projective representations associated with CCR-representations of bilinear maps. Detailed computations are presented in the case where G is a finitely generated free abelian group. We also discuss an extension problem about product type actions of G, where the projective representation theory of G plays a central role.