Projective representations of SU(2)ΛR4

We consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recovering for the phase-retrieval problem. For a given multiplier $$\mu $$ of a finite abelian group G, we show that the representation dimension of any irreducible $$\mu $$ -projective representation of G is exactly the rank of the symmetric multiplier matrix associated with $$\mu $$ . With the help of this result we are able to prove that every irreducible $$\mu $$ -projective representation of a finite abelian group G admits a frame vector with the maximal span property, and obtain a complete characterization for all such frame vectors. Consequently the complement of the set of all the maximal span frame vectors for any projective unitary representation of any finite abelian group is Zariski-closed. These generalize some of the recent results about phase-retrieval with Gabor (or STFT) measurements.

Projective unitary positive-energy representations of diff ( S1)

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.

We give an elementary analysis of the multiplicator group of the Galilei group in 1+2 dimensions 𝒢+↑. For a nontrivial multiplicator we give a list of all the corresponding projective unitary irreducible representations of 𝒢+↑.

Creation and annihilation operators are defined which are Wigner operators (tensor shift operators) for SU(3). While the annihilation operators are simply boson operators, the creation operators are cubic polynomials in boson operators. Together they generate under commutation the Lie algebra of SO(6,2). The vector space generated from a vacuum vector by repeated application of the creation operators carries an irreducible representation of the SO(6,2) algebra, equivalent to an hermitian representation, and also carries in direct sum every different irreducible representation of SU(3)<SO(6,2) exactly once. A model for SU(3), in the sense of Bernstein, Gel'fand and Gel'fand, is therefore defined. The different SU(3) irreducible representations appear explicitly as manifestly covariant, irreducible tensors, whose orthogonality and normalisation properties are examined. Other Wigner operators for SU(3) can be constructed simply as products of the new creation and annihilation operators, or sums of such products.

The theory of induced projective representations is applied to finite wreath products, yielding algorithms which add to the collection of groups for which projective representations can be computed systematically. For finite Abelian and Abelian-wreath-cyclic groups, the factor sets are determined explicitly by establishing a one-to-one correspondence between certain lower triangular matrices and the inequivalent factor sets of these two classes of groups. This correspondence is used to determine the number and degrees of the inequivalent, irreducible projective representations.

This note is aimed at presenting a new algebraic approach to momentum-space correlators in conformal field theory. As an illustration we present a new Lie-algebraic method to compute frequency-space two-point functions for charged scalar operators of CFT$_{1}$ dual to AdS$_{2}$ black hole with constant background electric field. Our method is based on the real-time prescription of AdS/CFT correspondence, Euclideanization of AdS$_{2}$ black hole and projective unitary representations of the Lie algebra $\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{sl}(2,\mathbb{R})$. We derive novel recurrence relations for Euclidean CFT$_{1}$ two-point functions, which are exactly solvable and completely determine the frequency- and charge-dependences of two-point functions. Wick-rotating back to Lorentzian signature, we obtain retarded and advanced CFT$_{1}$ two-point functions that are consistent with the known results.

Integrating Unitary Representations of Infinite-Dimensional Lie Groups

String junctions for arbitrary Lie-algebra representations

For every finite-dimensional nilpotent complex Lie algebra or superalgebran, we offer three algorithms for realizing it in terms of creation and annihilation operators. We use these algorithms to realize Lie algebras with a maximal subalgebra of finite codimension. For a simple finite-dimensionalg whose maximal nilpotent subalgebra isn, this gives its realization in terms of first-order differential operators on the big open cell of the flag manifold corresponding to the negative roots ofg. For several examples, we executed the algorithms using the MATHEMATICA-based package SUPERLie. These realizations form a preparatory step in an explicit construction and description of an interesting new class of simple Lie (super) algebras of polynomial growth, generalizations of the Lie algebra of matrices of complex size.

Previous discussions of the bases and projection operators for projective representations are analyzed and alternatives are proposed. Detailed proofs are provided for a result which is often assumed or proved within unacceptable limitations, namely that the form of the projection operators which is standard for vector representations is also valid for unitary projective representations. These proofs provide necessary conditions for this result, and they are constructed in terms of the definition given for the bases of projective representations. The calculation of Clebsch–Gordan coefficients for projective representations by means of the projection operators is discussed. Whereas in the method of Dirl the work is carried out entirely in terms of the matrix representations, and the symmetrization of the bases has to be considered in a second step, all the work of this paper is conducted starting from the symmetrization of the bases of the projective representations, so that those two steps are carried out simultaneously.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

In this chapter, we establish some basic facts on Clifford theory for induced projective representations. We follow quite closely the exposition for the case of ordinary representations in Chap. 2 and we give the first applications of the theory. Another fundamental application is in Sect. 10.2 on irreducible projective representations of finite Abelian groups.

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of [Get09, Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of \pi_2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integrate to a central extension of Lie groups. This obstruction comes from non-trivial \pi_2 2 for general Lie groups. We show that this obstruction may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions of étale Lie 2-groups. As an application, we obtain a generalization of Lie’s Third Theorem to infinite-dimensional Lie algebras.

1. Let Γn be the representation group or spin group (4, 9) of Sn. Then the irreducible representations of Γn are of two distinct types. These are (a) ordinary representations, which are the irreducible representations of the symmetric group and (b) spin or projective representations. Corresponding to every partition (λ) = (λ1, λ2, …, λm) of n with λ1 &gt; λ2 &gt; … &gt; λm &gt; 0 there is an irreducible spin representation 〈λ〉 of Γn.