Compact Semi-simple Group Research Articles

Crystal limits of compact semisimple quantum groups as higher-rank graph algebras

Abstract Let O q ⁢ [ K ] \mathcal{O}_{q}[K] be the quantized coordinate ring over the field C ⁢ ( q ) \mathbb{C}(q) of rational functions corresponding to a compact semisimple Lie group 𝐾, equipped with its ∗-structure. Let A 0 ⊂ C ⁢ ( q ) {\mathbf{A}_{0}}\subset\mathbb{C}(q) denote the subring of regular functions at q = 0 q=0 . We introduce an A 0 \mathbf{A}_{0} -subalgebra O q A 0 ⁢ [ K ] ⊂ O q ⁢ [ K ] \mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K]\subset\mathcal{O}_{q}[K] which is stable with respect to the ∗-structure and which has the following properties with respect to the crystal limit q → 0 q\to 0 . The specialization of O q ⁢ [ K ] \mathcal{O}_{q}[K] at each q ∈ ( 0 , ∞ ) ∖ { 1 } q\in(0,\infty)\setminus\{1\} admits a faithful ∗-representation π q \pi_{q} on a fixed Hilbert space, a result due to Soibelman. We show that, for every element a ∈ O q A 0 ⁢ [ K ] a\in\mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K] , the family of operators π q ⁢ ( a ) \pi_{q}(a) admits a norm limit as q → 0 q\to 0 . These limits define a ∗-representation π 0 \pi_{0} of O q A 0 ⁢ [ K ] \mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K] . We show that the resulting ∗-algebra O ⁢ [ K 0 ] = π 0 ⁢ ( O q A 0 ⁢ [ K ] ) \mathcal{O}[K_{0}]=\pi_{0}(\mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K]) is a Kumjian–Pask algebra, in the sense of Aranda Pino, Clark, an Huef and Raeburn. We give an explicit description of the underlying higher-rank graph in terms of crystal basis theory. As a consequence, we obtain a continuous field of C * C^{*} -algebras ( C ⁢ ( K q ) ) q ∈ [ 0 , ∞ ] (C(K_{q}))_{q\in[0,\infty]} , where the fibres at q = 0 q=0 and ∞ are explicitly defined higher-rank graph algebras.

AUTOMATIC CONTINUITY OF ABSTRACT HOMOMORPHISMS BETWEEN LOCALLY COMPACT AND POLISH GROUPS

We prove results about automatic continuity and openness of abstract surjective group homomorphisms \( K\overset{\varphi }{\to }G, \) where G and K belong to a certain class К of topological groups, and where the kernel of φ satisies a certain topological countability condition. Our results apply in particular to the case where G is a semisimple Lie group or a semisimple compact group, and where К is either the class of all locally compact groups or the class of all Polish groups.

Boundedness control sets for linear systems on Lie groups

Abstract Let Σ be a linear system on a connected Lie group G and assume that the reachable set 𝓐 from the identity element e ∈ G is open. In this paper, we give an algebraic condition to warrant the boundedness of the existent control set with a nonempty interior containing e. We concentrate the search for the class of decomposable groups which includes any solvable group and also every compact semisimple group.

The number of invariant Einstein metrics on a homogeneous space, Newton polytopes and contractions of Lie algebras

To every homogeneous space of a Lie group with a compact isotropy group , where the isotropy representation consists of irreducible components of multiplicity , we assign a compact convex polytope in , namely, the Newton polytope of the rational function defined to be the scalar curvature of the invariant metric on . If is a compact semisimple group, then the ratio of the volume of to the volume of the standard -simplex is a positive integer . We note that in many cases, coincides with the number of isolated invariant holomorphic Einstein metrics (up to homothety) on . We deduce from results of Kushnirenko and Bernshtein that in all cases, . To every proper face of we assign a non-compact homogeneous space with Newton polytope that is a contraction of . The appearance of a “defect” is explained by the fact that there is a Ricci-flat holomorphic invariant metric on the complexification of at least one of the .

Deformations of WZW models

Current–current deformations for WZW models of semi-simple compact groups are discussed in a sigma model approach. We start with the Abelian rank one group U(1). Afterwards, we keep the rank one but allow for non-Abelian structures by considering SU(2). Finally, we present the general case of rank larger than one.

Some dichotomy results for central Gaussian convolution semigroups on compact groups

Consider a central Gaussian convolution semigroup (μt)t > 0 on a connected compact semisimple group. Then either the measure μt is singular with respect to Haar measure for all t > 0, or there exists a time t such that μt is absolutely continuous with respect to Haar measure and admits a continuous density.

A comparison between star products on regular orbits of compact Lie groups*

In this paper, an algebraic and a differential star product defined on a regular coadjoint orbit of a compact semisimple group are compared. It has been proved that there is an injective algebra homomorphism between the algebra of polynomials with the algebraic star product and the algebra of differential functions with the differential star product structure.

Furry representation for fermions interacting with an external gauge field

We have obtained a Furry representation for the fermion sector (spin 1/2) of any gauge theory based on a semisimple compact group with an external field breaking down the stability of the vacuum. We have found expressions for the generating functionals of the matrix elements of the processes and averages, and have determined all the required types of propagators.

Numerical knot invariants of finite type from Chern-Simons perturbation theory

Chern-Simons gauge theory for compact semisimple groups is analyzed from a perturbation theory point of view. The general form of the perturbative series expansion of a Wilson line is presented in terms of the Casimir operators of the gauge group. From this expansion new numerical knot invariants are obtained. These knot invariants turn out to be of finite type (Vassiliev invariants) and to possess an integral representation. Using known results about Jones, HOMFLY, Kauffman and Akutsu-Wadati polynomial invariants these new knot invariants are computed up to type six for all prime knots up to six crossings. Our results suggest that these knot invariants can be normalized in such a way that they are integer-valued.

The hidden group structure of quantum groups: Strong duality, rigidity and preferred deformations

A notion of well-behaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeld-type and their duals, algebras of coefficients of compact semi-simple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantum groups. Moyal-product-like deformations are naturally found for all FRT-models on coefficients andC ∞-functions. Strong rigidity (H bi 2 ={0}) under deformations in the category of bialgebras is proved and consequences are deduced.

Three-dimensional Chern-Simons theory as a theory of knots and links (III). Compact semi-simple group

Chern-Simons field theory based on a compact non-abelian gauge group is studied as a theory of knots and links in three dimensions. A method to obtain the invariants for links made from braids of upto four strands is developed. This generalizes our earlier work on SU(2) Chern-Simons theory.

What is special about the group of the standard model?

The standard model is based on the algebra of U 1×SU 2×SU 3. The systematics of charges of the fundamental fermions seems to suggest the importance of a particular group having this algebra, viz. S(U 2×U 3). This group is distinguished from all other connected compact non semisimple groups with dimensionality up to 12 by a characteristic property: it is very “skew”. By this we mean that the group has relatively few “generalised outer automorphisms”. One may speculate about physical reasons for this fact.

Irreducible embeddings and polynomial tensors

The problem of finding irreducible tensors which are polynomials in the components of an irreducible tensor (n) of a compact semisimple group H is simplified if the representation (n) is contained irreducibly in a representation (m) of a larger group G of which H is a subgroup. A number of examples are given for which the problem is completely solved by generating function methods. The authors discuss how one can use the computer to tackle the more difficult cases; the problem of the reduction of the SO(8) enveloping algebra is given as an example.

Classical Yang-Mills fields with non-compact symmetry and an arbitrary compact gauge group

The authors consider classical Yang-Mills equations on Minkowski space. When the gauge group is SU(2), they have shown previously that the only solutions invariant under a non-compact maximal subgroup of the conformal group are Abelian Maxwell fields. They now extend the analysis to the case where the gauge group is an arbitrary semisimple compact group, with the same negative result.

Exact monopole solutions in gauge theories for an arbitrary semisimple compact group

We construct an exact n-parametric monopole and dyon solutions for an arbitrary compact gauge group G of rank n by using the symmetry between cylindrically symmetric instanton equations in Euclidean space R4 and monopole equations in Minkowski space R3,1 (with Higgs scalar field in adjoint representation). The solutions are spherically symmetric with respect to the total momentum operator \( - \overrightarrow {ir} \times \overrightarrow \nabla + \overrightarrow T , where \overrightarrow T \) represents the minimal embedding of SU(2) in G. Explicit expressions for the monopole magnetic charge and mass matrices are obtained. The remarkable aspect of our results is the existence of discrete series of the monopole solutions, which are labelled by n ‘quantum’ numbers and degenerated in the latter ones at a fixed monopole mass matrix.

Generating functions for polynomial irreducible tensors

A general method for constructing a generating function for all irreducible polynomial tensors, with respect to a given compact semisimple group, out of a given set of tensors is derived. The method is applied to the construction of polynomial bases for the IR’s of a group reduced according to a subgroup and to the finding of subgroup scalars in the enveloping algebra of a group. A number of examples are worked out.

Spherically symmetric monopoles

The task of finding all spherically symmetric three-dimensional point monopoles in a gauge theory with arbitrary compact semisimple group is completely formulated in an Abelian gauge, where the problem is purely group-theoretical. The form of the gauge transformation to the spherically symmetric gauge is explicitly given in the general case. For $\mathrm{SU}(N)$ groups this result is reduced to a simple diagrammatic method which yields all such point monopoles by inspection. Also for $\mathrm{SU}(N)$ groups, a technique is given for efficient construction of the radial differential equations satisfied by the corresponding finite-energy solutions.

On angular momentum Helmholtz theorems and cohomology of Lie algebras

Helmholtz’ 2nd theorem (that every vector field on R 3 {{\mathbf {R}}^3} with vanishing curl is gradient of a function) can be viewed as a statement about the group of translations of R 3 {{\mathbf {R}}^3} . We prove similar theorems for other Lie transformation groups, in particular for semidirect products of abelian and compact semisimple groups. Using Hodge theory we also obtain results analogous to the 1st Helmholtz theorem, but only for compact Lie transformation groups.

Extension of Elliott's projection technique to compact semisimple groups

The SU(4) ⊃ SU(2) ⊗ SU(2) projection technique introduced by Draayer, is simplified in that all basis states of an irreducible representation can be projected from only one ‘intrinsic’ state. Furthermore a general projection method for compact semisimple Lie groups is derived, )ontaining SU(3) ⊃ R(3) and SU(2) ⊗ SU(2) as particular cases.

Symmetrized Kronecker Products of Group Representations

Certain phases are associated with the Kronecker squares and cubes of representations of the finite and of the compact semi-simple groups. These phases are important in giving the symmetry properties of the 1 — jm and 3 — jm symbols of the groups [4; 9]. It is our primary purpose to evaluate these phases. The Frobenius-Schur invariant [12, p. 142] for an irreducible representation of group G (1.1)