scholarly journals Unitary representations of some infinite dimensional groups

1981 ◽  
Vol 80 (3) ◽  
pp. 301-342 ◽  
Author(s):  
Graeme Segal
Keyword(s):  
Unitary Representations ◽  
Infinite Dimensional

MULTIPLICITY-FREE DECOMPOSITIONS OF THE MINIMAL REPRESENTATION OF THE INDEFINITE ORTHOGONAL GROUP

2008 ◽  
Vol 19 (10) ◽  
pp. 1187-1201 ◽  
Author(s):  
MASAYASU MORIWAKI
Keyword(s):  
Unitary Representation ◽  
Orthogonal Group ◽  
Irreducible Unitary Representation ◽  
Unitary Representations ◽  
Minimal Representation ◽  
Infinite Dimensional ◽  
Irreducible Unitary Representations ◽  
Direct Product Group ◽  
Multiplicity Free ◽  
Kirillov Dimension

Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation. We consider, for which subgroup G′ of G, the restriction π|G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p1) × U(p2) × U(q) for p1, p2 ≥ 1 with p1 + p2 = p is multiplicity-free, whereas the restriction to U(p1) × U(p2) × U(q1) × U(q2) for q1, q2 ≥ 1 with q1 + q2 = q has infinite multiplicities.

Elements of Group Theory

2021 ◽  
pp. 51-110
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Group Theory ◽  
Lie Groups ◽  
Lorentz Group ◽  
Unitary Representations ◽  
Compact Groups ◽  
Mathematical Language ◽  
Group Representations ◽  
Infinite Groups ◽  
Infinite Dimensional ◽  
Symmetry Properties

The mathematical language which encodes the symmetry properties in physics is group theory. In this chapter we recall the main results. We introduce the concepts of finite and infinite groups, that of group representations and the Clebsch–Gordan decomposition. We study, in particular, Lie groups and Lie algebras and give the Cartan classification. Some simple examples include the groups U(1), SU(2) – and its connection to O(3) – and SU(3). We use the method of Young tableaux in order to find the properties of products of irreducible representations. Among the non-compact groups we focus on the Lorentz group, its relation with O(4) and SL(2,C), and its representations. We construct the space of physical states using the infinite-dimensional unitary representations of the Poincaré group.

scholarly journals Branching laws for small unitary representations of GL(n, ℂ)

2014 ◽  
Vol 25 (06) ◽  
pp. 1450052
Author(s):  
Jan Möllers ◽  
Benjamin Schwarz
Keyword(s):  
Parabolic Subgroup ◽  
Principal Series ◽  
Unitary Representations ◽  
Symmetric Pair ◽  
Maximal Parabolic Subgroup ◽  
Infinite Dimensional ◽  
Branching Laws ◽  
Series Representations ◽  
Unitary Principal Series ◽  
Kirillov Dimension

The unitary principal series representations of G = GL (n, ℂ) induced from a character of the maximal parabolic subgroup P = ( GL (1, ℂ) × GL (n - 1, ℂ)) ⋉ ℂn-1 attain the minimal Gelfand–Kirillov dimension among all infinite-dimensional unitary representations of G. We find the explicit branching laws for the restriction of these representations to all reductive subgroups H of G such that (G, H) forms a symmetric pair.

Relativistic extension of non-compact symmetry groups

1966 ◽  
Vol 289 (1417) ◽  
pp. 177-191 ◽  
Keyword(s):  
Symmetry Group ◽  
Conventional Method ◽  
Space Time ◽  
Unitary Representations ◽  
Symmetry Groups ◽  
Infinite Dimensional ◽  
Induced Representations ◽  
Relativistic Extension ◽  
A Chain

The problem of relativistieally boosting the unitary representations of a non-compact spin-containing rest-symmetry group is solved by starting with non-unitary infinite-dimensional representations of a relativistic extension of this group, by adjoining to this extension four space-time translations and by the napplying Bargmann-Wigner equations to guarantee aunitary norm. The procedure has similarities to the conventional method of induced representations. The boosting problem considered here is the first step towards the solution of the problem of coupling of such infinite-dimensional representations which is also briefly investigated. Startin g from a rest-symmetry like U (6,6) a chain of subgroups GL (6), U (3,3), etc., is exhibited for collinear and coplanar processes, etc.

scholarly journals ON THE ANALYTIC PROPERTIES OF THE z-COLOURED JONES POLYNOMIAL

2005 ◽  
Vol 14 (04) ◽  
pp. 435-466 ◽  
Author(s):  
JOÃO FARIA MARTINS
Keyword(s):  
Jones Polynomial ◽  
Unitary Representations ◽  
Target Space ◽  
Analytic Extension ◽  
Torus Knots ◽  
Infinite Dimensional ◽  
Knot Invariants ◽  
Starting Point ◽  
Zero Radius ◽  
Formal Power

We analyse the possibility of defining ℂ-valued Knot invariants associated with infinite-dimensional unitary representations of SL(2,ℝ) and the Lorentz Group taking as starting point the Kontsevich integral and the notion of infinitesimal character. This yields a family of knot invariants whose target space is the set of formal power series in ℂ, which contained in the Melvin–Morton expansion of the coloured Jones polynomial. We verify that for some knots the series have zero radius of convergence and analyse the construction of functions of which this series are asymptotic expansions by means of Borel re-summation. Explicit calculations are done in the case of torus knots which realise an analytic extension of the values of the coloured Jones polynomial to complex spins. We present a partial answer in the general case.

su(2) -expansion of the Lorentz algebra so(3,1 )

2013 ◽  
Vol 91 (8) ◽  
pp. 589-598 ◽  
Author(s):  
Rutwig Campoamor-Stursberg ◽  
Hubert de Guise ◽  
Marc de Montigny
Keyword(s):  
Coherent State ◽  
Unitary Representations ◽  
Iwasawa Decomposition ◽  
Matrix Formalism ◽  
Matrix Elements ◽  
Infinite Dimensional ◽  
Lorentz Algebra ◽  
Finite Dimensional

We exploit the Iwasawa decomposition to construct coherent state representations of [Formula: see text], the Lorentz algebra in 3 + 1 dimensions, expanded on representations of the maximal compact subalgebra [Formula: see text]. Examples of matrix elements computation for finite dimensional and infinite-dimensional unitary representations are given. We also discuss different base vectors and the equivalence between these different choices. The use of the [Formula: see text]-matrix formalism to truncate the representation or to enforce unitarity is discussed.

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