Classification Of Irreducible Representations Research Articles

Jacobi–Trudi type formula for a class of irreducible representations of 𝔤𝔩(m|n)

We prove a determinantal type formula to compute the characters of a class of finite-dimensional irreducible representations of the general Lie super-algebra [Formula: see text] in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula, originally conjectured by van der Jeugt and Moens, can be regarded as a generalization of the well-known Jacobi–Trudi formula.

Group Theoretical Derivation of Consistent Free Particle Theories

The difficulties of relativistic particle theories formulated by means of canonical quantization, such as those of Klein–Gordon and Dirac, ultimately led theoretical physicists to turn to quantum field theory to model elementary particle physics. In order to overcome these difficulties, the theories of the present approach are developed deductively from the physical principles that specify the system, without making use of canonical quantization. For a free particle these starting assumptions are invariance of the theory and covariance of position with respect to Poincaré transformations. In pursuing the approach, the effectiveness of group theoretical methods is exploited. The coherent development of our program has shown that robust classes of representations of the Poincaré group, discarded by the known particle theories, can in fact be taken as bases for perfectly consistent theories. For massive spin zero particles, six inequivalent theories have been determined, two of which do not correspond to any of the current ones; all of these theories overcome the difficulties of Klein–Gordon one. The present lack of the explicit transformation properties of position with respect to boosts prevents the complete determination of non zero spin particle theories. In the past a particular form of these transformation properties was adopted by Jordan and Mukunda. We check its consistency within the present approach and find that for spin frac{1}{2} particles there is only one consistent theory, which is unitarily related to Dirac’s; yet, once again, it requires classes of irreducible representations previously discarded.

Representation Theory of Disconnected Reductive Groups

We study three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dual Weyl modules; and (iii) the decomposition map relating representations in characteristic $0$ and those in characteristic $p$ (for groups defined over discrete valuation rings). For each of these topics, we obtain natural generalizations of the well-known results for connected reductive groups.

Ergodic invariant states and irreducible representations of crossed product $C^∗$-algebras

Given a C∗-dynamical system consisting of a unital C∗-algebra A and a discrete group Γ acting on A as automorphisms, we show that the space of (ergodic) Γ-invariant states on A is homeomorphic to a subspace of (pure) state space of A o Γ. It follows that classification of ergodic Γ-invariant regular Borel probability measures on a compact Hausdorff space X is equivalent to classification of irreducible representations of C(X) o Γ whose restriction to Γ contains the trivial representation of Γ. As an application, a reformulation of Furstenberg’s conjecture via representation theory of the semidirect product group Z[ 1 pq ] o Z 2 is obtained.

Hypergroup structures arising from certain dual objects of a hypergroup

In the present paper hypergroup structures are investigated on distinguished dual objects related to a given hypergroup $K$, especially to a semi-direct product hypergroup $K = H \rtimes_\alpha G$ defined by an action $\alpha$ of a locally compact group $G$ on a commutative hypergroup $H$. Typical dual objects are the sets of equivalence classes of irreducible representations of $K$, of infinite-dimensional irreducible representations of type I hypergroups $K$, and of quasi-equivalence classes of type $\text{II}_1$ factor representations of non-type I hypergroups $K$. The method of proof relies on the notion of a character of a representation of $K = H \rtimes_\alpha G$.

Affinely prime dynamical systems

This paper deals with representations of groups by “affine” automorphisms of compact, convex spaces, with special focus on “irreducible” representations: equivalently “minimal” actions. When the group in question is PSL(2, R), the authors exhibit a one-one correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called “linear Stone-Weierstrass” for group actions on compact spaces. If it holds for the “universal strongly proximal space” of the group (to be defined), then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.

Characters of induced representations of a compact hypergroup

Let \(K\) be a compact hypergroup and \(K_0\) a closed subhypergroup of \(K\). For a state \(\varphi \) of the Banach \(*\)-algebra \(M^b(K_0)\) the induced state \(\mathrm{ind}_{K_0}^K \varphi \) of \(M^b(K)\) is introduced. Applying the notion of a character mapping \(ch\) from the space \(\mathrm{Rep}^\mathrm{f}(K)\) of finite-dimensional representations of \(K\) into the space \(M^1(\hat{K})\) of probability measures on the space \(\hat{K}\) of equivalence classes of irreducible representations of \(K\). For certain classes of hypergroups \(K\) it is shown that, whenever \(\pi \in \mathrm{Rep}^\mathrm{f}(K_0)\), the formula \(ch(\mathrm{ind}_{K_0}^K\pi ) = \mathrm{ind}_{K_0}^Kch(\pi )\) holds.

On the spectrum of the algebra of singular integral operators with discontinuities in symbols in momenta and coordinates

We study the C*-algebra generated in $ {L^2}\left( \mathbb{R} \right) $ by operators of multiplication by functions with finitely many discontinuities of the first kind and by convolution operators with the Fourier transforms of such functions. The algebra is represented as the restricted direct sum . We express the spectrum of the restricted direct sum in terms of the spectra of its summands. This result is used to express the spectrum of the algebra in terms of the spectra of and . We describe all equivalence classes of irreducible representations of the algebra , the topology on the spectrum of this algebra, and solving composition series. We discuss the abstract index group of the quotient algebra by the ideal of compact operators and by the ideal com generated by the commutators of elements of the algebra . Bibliography: 14 titles.

Mixed pairing symmetry inκ−(BEDT-TTF)2Xorganic superconductors from ultrasonic velocity measurements

Discontinuities in elastic constants are detected at the superconducting transition of layered organic conductors \kappa-(BEDT-TTF)_{2}X by longitudinal and transverse ultrasonic velocity measurements. Symmetry arguments show that discontinuities in shear elastic constants can be explained in the orthorhombic compound only if the superconducting order parameter has a mixed character that can be of two types, either A_{1g}+B_{1g} or B_{2g}+B_{3g} in the classification of irreducible representations of the orthorhombic point group D_{2h}. Consistency with other measurements suggests that the A_{1g}+B_{1g} (d_{xy}+d_{z(x+y)}) possibility is realized. Such clear symmetry-imposed signatures of mixed order parameters have not been observed in other superconducting compounds.

Class of representations of skew derivation Lie algebra over quantum torus

Let L be the skew derivation Lie algebra of the quantum torus ℂq. In this paper, we give a class of irreducible representations for L with infinite dimensional weight spaces.

Realizations of the Lie superalgebraq(2) and applications

The Lie superalgebra q(2) and its class of irreducible representations V_p of dimension 2p (p being a positive integer) are considered. The action of the q(2) generators on a basis of V_p is given explicitly, and from here two realizations of q(2) are determined. The q(2) generators are realized as differential operators in one variable x, and the basis vectors of V_p as 2-arrays of polynomials in x. Following such realizations, it is observed that the Hamiltonian of certain physical models can be written in terms of the q(2) generators. In particular, the models given here as an example are the sphaleron model, the Moszkowski model and the Jaynes-Cummings model. For each of these, it is shown how the q(2) realization of the Hamiltonian is helpful in determining the spectrum.

On the projective representations of the Bondi-Metzner-Sachs group

With the Bondi‐Metzner‐Sachs (BMS) group in general relativity as the main motivation and example, a theorem is proved which may be described as follows. Let G be a complex semisimple, connected and simply connected Lie group with compact real form K , and let A be a metrizable, complete and locally convex real topological vector space on which there is a continuous G action. Consider the semidirect product topological group Gx s A (which is, in general, infinite‐dimensional) constructed naturally out of G and A . If the set of equivalence classes of irreducible representations of K in A satisfies certain hypotheses, then the second cohomology group of G x s A in the sense of continuous group cohomology is trivial. When G = SL (2, C ) and A is an appropriate function space of real‐valued functions of the 2‐sphere endowed with a specific G action (e.g. A may consist of C k , k ⩾ 3, real‐valued functions defined on the 2‐sphere), the semidirect product group is the universal cover of the BMS group. The theorem implies the existence of lifting of the projective unitary representations of the BMS group to the linear unitary representations of its universal cover. In the quantum context when we consider massless quantum fields at null infinity of a non‐stationary, asymptotically Minkowskian space‐time, in place of the projective unitary representations of the BMS group, there is no loss of generality in considering the linear unitary representations of its universal cover instead.

C*-Algebras Associated with $$F_{2^n }$$ Unimodal Dynamical Systems

We consider C*-algebras associated with simple unimodal one-dimensional ambiguous dynamical systems \(\left( {f,\mathbb{R}} \right)\) with certain special restrictions. For these algebras, we present a complete classification of irreducible representations in Hilbert spaces and describe the dual space. As an example, we consider the one-parameter family fμ(x) = μx(1 − x).

Casson-Lin's Invariant and Floer Homology

Casson defined an invariant which can be thought of as the number of conjugacy classes of irreducible representations of π1(Y) into SU(2) counted with signs, where Y is an oriented integral homology 3-sphere. Lin defined a similar invariant (the signature of a knot) for a braid representative of a knot in S3. In this paper, we give a natural generalization of Casson-Lin's invariant. Our invariant is the symplectic Floer homology for the representation space of π1(S3 \ K) into SU(2) with trace-zero along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its Euler number is the negative of Casson-Lin's invariant.

Classical and quantum Heisenberg groups, their representations and applications

This paper is a survey on classical Heisenberg groups and algebras, q-deformed Heisenberg algebras, q-oscillator algebras, their representations and applications. Describing them, we tried, for the reader's convenience, to explain where the q-deformed case is close to the classical one, and where there are principal differences. Different realizations of classical Heisenberg groups, their geometrical aspects, and their representations are given. Moreover, relations of Heisenberg groups to other linear groups are described. Intertwining operators for different (Schrodinger, Fock, compact) realizations of unitary irreducible representations of Heisenberg groups are given in explicit form. Classification of irreducible representations and representations of the q-oscillator algebra is derived for the cases when q is not a root of unity and when q is a root of unity. The Fock representation of the q-oscillator algebra is studied in detail. In particular, q-coherent states are described. Spectral properties of some operators of the Fock representations of q-oscillator algebras are given. Some of applications of Heisenberg groups and algebras, q-Heisenberg algebras and q-oscillator algebras are briefly described.

Finite-dimensional representations of the quantum superalgebraU q[gl(n/m)] and relatedq-identities

Explicit expressions for the generators of the quantum superalgebra $U_q[gl(n/m)]$ acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a Gel'fand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set of $q$-number identities.

Representations of the q-deformed algebras U q(so2,1) and U q(so3,1)

Representations of the algebra Uq(so2,1) are studied. This algebra is a q-deformation of the universal enveloping algebra U(so2,1) of the Lie algebra of the group SO0(2,1) and differs from the quantum algebra Uq(su1,1). Classifications of irreducible representations and of infinitesimally unitary irreducible representations of Uq(so2,1), with simple and discrete spectrum of one of generators are derived. The set of infinitesimally unitary representations of Uq(so2,1) does not coincide with that for the algebra Uq(su1,1). The sets of irreducible representations and of infinitesimally unitary irreducible representations of the algebra Uq(so3,1) are given. We also consider representations of Uq(son,1) which are of class 1 with respect to the subalgebra Uq(son).

Locally compact groups whose conjugation representations satisfy a Kazhdan type property or have countable support

For a locally compact group G with left Haar measure and modular function δ the conjugation representation γG of G on L2(G) is defined byf ∈ L2(G), x, y ∈ G. γG has been investigated recently (see [19, 20, 21, 24, 32, 35]). For semi-simple Lie groups, a related representation has been studied in [25]. γG is of interest not least because of its connection to questions on inner invariant means on L∞(G). In what follows suppγG denotes the support of γG in the dual space Ĝ, that is the closed subset of all equivalence classes of irreducible representations which are weakly contained in γG. The purpose of this paper is to establish relations between properties such as a variant of Kazhdan's property and discreteness or countability of supp γG and the structure of G.

Etude d'un module de verma sur l'algebre de contact

In this paper we study a certain class of irreducible representations of the Lie algebra K of all polynomial vector fields preserving a contact structure. The Lie algebra K admits a triangular decomposition and we can associate to any character τ of Ko a Verma module V(τ). Using a method of caracteristic variety, we prove that if τ is not trivial then V(τ) is simple. Then we deduce a formula for the dimension of homogenous components of V(τ)

Generalized FS×W versus Dyson’s classification of irreducible representations

We study the characterization of the 13 cases obtained in the classification of the irreducible linear–antilinear representations of semigroups in a finite-dimensional vector space X over an algebraically closed field K with a conjugation j (generalized Frobenius–Schur–Wigner, or FS×W, classification). It has already been shown that each case can be characterized by various equivalent properties, some of which can be endowed with a physical interpretation. We show here that, whenever K is the complex field C, each case can be characterized by the structure (in the sense specified by the Weyl theorem on the structure of the matrix algebras and their commutators) of a pair of suitable operator algebras over the real field R. This characterization coincides with the one given by Dyson for each case of his classification of symmetry groups. Thus, the latter classification is recovered under more general assumptions and in a generalized framework, and its one-to-one correspondence with the generalized FS×W classification is shown. In the process, we obtain a classification of the algebras over R generated by complex semigroup representations of the aforesaid kind, and inquire into the connections between some properties of the representations in X over C and the structure (in the Weyl sense) of the representations in the space XR over R obtained by decomplexification of the space X.