Classical Compact Lie Groups Research Articles

Random stochastic matrices from classical compact Lie groups and symmetric spaces

We consider random stochastic matrices M with elements given by Mij = |Uij|2, with U being uniformly distributed on one of the classical compact Lie groups or some of the associated symmetric spaces. We observe numerically that, for large dimensions, the spectral statistics of M, discarding the Perron-Frobenius eigenvalue 1, are similar to those of the Gaussian orthogonal ensemble for symmetric matrices and to those of the real Ginibre ensemble for nonsymmetric matrices. We compute some spectral statistics using Weingarten functions and establish connections with some difficult enumerative problems involving permutations.

SU(2) and SU(1,1) group parametrizations via qubit induced correlated probabilities

A group is an algebraic structure equipped with binary associative multiplication, identity element and inverse operation with respect to identity. Lie group is a smooth manifold with group multiplication and the inverse being smooth functions. Therefore as one chooses manifold coordinate charts, various parametrizations of multiplication and inverse operations arise. Recently, the simple probability representation of qubit states and observables and an associated triangle geometrical picture called ‘quantum suprematism’ were introduced. At its core lies the representation of the Bloch sphere by correlated dichotomic probability distribution. The aim of the present paper is the introduction of probability parametrization of two classical compact and non-compact Lie groups, SU(2) and SU(1,1) respectively.

Invariant Einstein metrics on certain compact semisimple Lie groups

In this paper, we study left invariant Einstein metrics on compact semisimple Lie groups. A new method to construct holonomy irreducible non-naturally reductive Einstein metrics on certain compact semisimple (non-simple) Lie groups is presented. In particular, we show that if G is a classical compact simple Lie group and H is a closed subgroup such that G/H is a standard homogeneous Einstein manifold, then there exist holonomy irreducible non-naturally reductive Einstein metrics on H×G, except for some very special cases. A further interesting result of this paper is that for any compact simple Lie group G, there always exist holonomy irreducible non-naturally reductive Einstein metrics on the compact semisimple Lie groups Gn, for any n≥4.

Integration formulas for Brownian motion on classical compact Lie groups

Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. Śniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of invariant theory and of classical Schur–Weyl dualities based on stochastic calculus.

Biharmonic Functions on the Classical Compact Simple Lie Groups

The main aim of this work is to construct several new families of proper biharmonic functions defined on open subsets of the classical compact simple Lie groups $$\mathbf{SU}(n)$$ , $$\mathbf{SO}(n)$$ and $$\mathbf{Sp}(n)$$ . We work in a geometric setting which connects our study with the theory of submersive harmonic morphisms. We develop a general duality principle and use this to interpret our new examples on the Euclidean sphere $${\mathbb S}^3$$ and on the hyperbolic space $${\mathbb H}^3$$ .

Adams operations on classical compact Lie groups

Let G G be U ( n ) U(n) , S U ( n ) SU(n) , S p ( n ) Sp(n) or S p i n ( n ) Spin(n) . In this short note we give explicit general formulas for Adams operations on K ∗ ( G ) K^*(G) and eigenvectors of Adams operations on K ∗ ( U ( n ) ) K^*(U(n)) .

Harmonic maps of finite uniton number and their canonical elements

We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group $$G$$ , whether $$G$$ has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group $$\Omega _\mathrm {alg}{G}$$ of algebraic loops in $$G$$ and corresponding canonical elements. This will allow us to give estimations for the minimal uniton number of the corresponding harmonic maps with respect to different representations and to make more explicit the relation between previous work by different authors on harmonic two-spheres in classical compact Lie groups and their inner symmetric spaces and the Morse theoretic approach to the study of such harmonic two-spheres introduced by Burstall and Guest. As an application, we will also give some explicit descriptions of harmonic spheres in low-dimensional spin groups making use of spinor representations.

The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

In this paper, we prove the cut-off phenomenon in total variation distance for the Brownian motions traced on the classical symmetric spaces of compact type, that is to say: 1. the classical simple compact Lie groups: special orthogonal groups SO(n), special unitary groups SU(n) and compact symplectic groups USp(n); 2. the real, complex and quaternionic Grassmannian varieties (including the real spheres, and the complex or quaternionic projective spaces when q = 1): SO(p + q)/(SO(p)×SO(q)), SU(p + q)/S(U(p)×U(q)) and USp(p + q)/(USp(p)×USp(q)); 3. the spaces of real, complex and quaternionic structures: SU(n)/SO(n), SO(2n)/ U(n), SU(2n)/USp(n) and USp(n)/UU(n). Denoting μt the law of the Brownian motion at time t, we give explicit lower bounds for dTV(μt,Haar) if \(t t_{\text{cut-of\/f}}\). This provides in particular an answer to some questions raised in recent papers by Chen and Saloff-Coste. Our proofs are inspired by those given by Rosenthal and Porod for products of random rotations in SO(n), and by Diaconis and Shahshahani for products of random transpositions in \(\mathfrak{S}_{n}\).

Idempotent states on compact quantum groups and their classification on $\mathrm{U}_q(2)$, $\mathrm{SU}_q(2)$, and $\mathrm{SO}_q(3)$

Unlike for locally compact groups, idempotent states on locally compact quantum groups do not necessarily arise as Haar states of compact quantum subgroups. We give a simple characterisation of those idempotent states on compact quantum groups that do arise as Haar states on quantum subgroups. We also show that all idempotent states on the quantum groups \mathrm{U}_q(2) , \mathrm{SU}_q(2) , and \mathrm{SO}_q(3) ( q \in (-1,0)\cup (0,1] ) arise in this manner and list the idempotent states on the compact quantum semigroups \mathrm{U}_0(2) , \mathrm{SU}_0(2) , and \mathrm{SO}_0(3) . In the Appendix we provide a short new proof of the coamenability of deformations of classical compact Lie groups based on their representation theory.

A theory of induction and classification of tensor C*-categories

This paper addresses the problem of describing the structure of tensor C*-categories \mathcal{M} with conjugates and irreducible tensor unit. No assumption on the existence of a braided symmetry or on amenability is made. Our assumptions are motivated by the remark that these categories often contain non-full tensor C*-subcategories with conjugates and the same objects admitting an embedding into the Hilbert spaces. Such an embedding defines a compact quantum group by Woronowicz duality. An important example is the Temperley–Lieb category canonically contained in a tensor C*-category generated by a single real or pseudoreal object of dimension ≥ 2 . The associated quantum groups are the universal orthogonal quantum groups of Wang and Van Daele. Our main result asserts that there is a full and faithful tensor functor from \mathcal{M} to a category of Hilbert bimodule representations of the compact quantum group. In the classical case, these bimodule representations reduce to the G -equivariant Hermitian bundles over compact homogeneous G -spaces, with G a compact group. Our structural results shed light on the problem of whether there is an embedding functor of \mathcal{M} into the Hilbert spaces. We show that this is related to the problem of whether a classical compact Lie group can act ergodically on a non-type I von Neumann algebra. In particular, combining this with a result of Wassermann shows that an embedding exists if \mathcal{M} is generated by a pseudoreal object of dimension 2.

Filtrations, Factorizations and Explicit Formulae for Harmonic Maps

We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere, to the unitary group for a general class of factorizations by unitons. We show how these specialize to give explicit formulae for such harmonic maps to each of the classical compact Lie groups and their inner symmetric spaces—the nonlinear σ-model of particle physics. Our methods also give an explicit Iwasawa decomposition of the algebraic loop group.

A Rigidity Result for Extensions of Braided Tensor C*–Categories Derived from Compact Matrix Quantum Groups

Let G be a classical compact Lie group and G_\mu the associated compact matrix quantum group deformed by a positive parameter \mu (or a nonzero and real \mu in the type A case). It is well known that the category Rep(G_\mu) of unitary f.d. representations of G_\mu is a braided tensor C*-category. We show that any braided tensor *-functor from Rep(G_\mu) to another braided tensor C*-category with irreducible tensor unit is full if |\mu|\neq 1. In particular, the functor of restriction to the representation category of a proper compact quantum subgroup, cannot be made into a braided functor. Our result also shows that the Temperley--Lieb category generated by an object of dimension >2 can not be embedded properly into a larger category with the same objects as a braided tensor C*-subcategory.

Non-Abelian Kuramoto models and synchronization

We describe non-Abelian generalizations of the Kuramoto model for any classical compact Lie group and identify their main properties. These models may be defined on any complex network where the variable at each node is an element of the unitary group U(n), or a subgroup of U(n). The nonlinear evolution equations maintain the unitarity of all variables which therefore evolve on the compact manifold of U(n). Synchronization of trajectories with phase locking occurs as for the Kuramoto model, for values of the coupling constant larger than a critical value, and may be measured by various order and disorder parameters. Limit cycles are characterized by a frequency matrix which is independent of the node and is determined by minimizing a function which is quadratic in the variables. We perform numerical computations for n = 2, for which the SU(2) group manifold is S3, for a range of natural frequencies and all-to-all coupling, in order to confirm synchronization properties. We also describe a second generalization of the Kuramoto model which is formulated in terms of real m-vectors confined to the (m − 1)-sphere for any positive integer m, and investigate trajectories numerically for the S2 model. This model displays a variety of synchronization phenomena in which trajectories generally synchronize spatially but are not necessarily phase-locked, even for large values of the coupling constant.

Formula omitted]-singular dichotomy for orbital measures of classical compact Lie groups

We prove that for any classical, compact, simple, connected Lie group G, the G-invariant orbital measures supported on non-trivial conjugacy classes satisfy a surprising L 2 -singular dichotomy: Either μ h k ∈ L 2 ( G ) or μ h k is singular to the Haar measure on G. The minimum exponent k for which μ h k ∈ L 2 is specified; it depends on Lie properties of the element h ∈ G . As a corollary, we complete the solution to a classical problem – to determine the minimum exponent k such that μ k ∈ L 1 ( G ) for all central, continuous measures μ on G. Our approach to the singularity problem is geometric and involves studying the size of tangent spaces to the products of the conjugacy classes.

Harmonic morphisms from the classical compact semisimple Lie groups

In this article, we introduce a new method for manufacturing harmonic morphisms from semi-Riemannian manifolds. This is employed to yield a variety of new examples from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with their standard Riemannian metrics. We develop a duality principle and show how this can be used to construct the first known examples of harmonic morphisms from the non-compact Lie groups $${\bf SL}_n({\mathbb{R}})$$ , SU *(2n), $${\bf Sp}(n, {\mathbb{R}})$$ , SO *(2n), SO(p, q), SU(p, q) and Sp(p, q) equipped with their standard dual semi-Riemannian metrics.

A Characterization of Right Coideals of Quotient Type and its Application to Classification of Poisson Boundaries

Let $G$ be a co-amenable compact quantum group. We show that a right coideal of $G$ is of quotient type if and only if it is the range of a conditional expectation preserving the Haar state and is globally invariant under the left action of the dual discrete quantum group. We apply this result to theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on $SU_q(N)$ for co-amenable compact quantum groups with the commutative fusion rules. More precisely, we prove that the Poisson integral is an isomorphism between the Poisson boundary and the right coideal of quotient type by maximal quantum subgroup of Kac type. In particular, the Poisson boundary and the quantum flag manifold are isomorphic for any q-deformed classical compact Lie group.

Harmonic morphisms from the compact semisimple Lie groups and their non-compact duals

In this paper we prove the local existence of complex-valued harmonic morphisms from any compact semisimple Lie group and their non-compact duals. These include all Riemannian symmetric spaces of types II and IV. We produce a variety of concrete harmonic morphisms from the classical compact simple Lie groups SO ( n ) , SU ( n ) , Sp ( n ) and globally defined solutions on their non-compact duals SO ( n , C ) / SO ( n ) , SL n ( C ) / SU ( n ) and Sp ( n , C ) / Sp ( n ) .

Isospectral metrics and potentials on classical compact simple Lie groups

Given a compact Riemannian manifold (M, g), the eigenvalues of the Laplace operator � form a discrete sequence known as the spectrum of (M, g). (In the case the M has boundary, we stipulate either Dirichlet or Neumann boundary conditions.) We say that two Riemannian manifolds are isospectral if they have the same spectrum. For a fixed manifold M, an isospectral deformation of a metric g0 on M is a continuous family F of metrics on M containing g0 such that each metric g ∈ F is isospectral to g0. We say that the deformation is nontrivial if none of the other metrics in F are isometric to g0 and that the deformation is multidimensional if F can be parameterized by more that one variable. For two functions φ, ψ ∈ C ∞ (M), we say that φ and ψ are isospectral potentials on (M, g) if the eigenvalue spectra of the Schrodinger operators ~ � + φ and ~ � + ψ are equal for any choice of Planck's constant ~ . In this paper, we prove the existence of multiparameter isospectral deformations of metrics on SO(n) (n = 9 or n ≥ 11), SU(n) (n ≥ 8), and Sp(n) (n ≥ 4). For these exam- ples, we follow a metric construction developed by Schueth who had given one-parameter families of isospectral metrics on orthogonal and unitary groups. Our multiparameter families are obtained by a new proof of nontriviality establishing a generic condition for nonisometry of metrics arising from the construction. We also show the existence of non-congruent pairs of isospectral potentials and nonisometric pairs of isospectral conformally equivalent metrics on Sp(n) for n ≥ 6. The industry of producing isospectral manifolds began in 1964 with Milnor's pair of 16-dimensional isospectral, nonisometric tori (Mi). Several years later, in the early 1980's, new examples began to appear sporadically (e.g. (Vi), (Ik), (GWil1)). These isospectral constructions were ad hoc and did not appear to be related until 1985, when Sunada began developing the first unified approach for producing isospectral manifolds. The method described a program for taking quotients of a given manifold so that the resulting manifolds were isospectral. Sunada's original theorem and subsequent generalizations ((Bd1), (Bd2), (DG), (Pes), (Sut)) explained most of the previously known isospectral examples and led to a wide variety of new examples. See, for example, (BGG), (Bus), and (GWW). In 1993 Gordon produced the first examples of closed isospectral manifolds with dif- ferent local geometry (Gor) and then, in a series of papers, generalized the construction to the following principle based on torus actions.

More on quantum groups from the quantization point of view

Star products on the classical double group of a simple Lie group and on corresponding symplectic groupoids are given so that the quantum double and the “quantized tangent bundle” are obtained in the deformation description. “Complex” quantum groups and bicovariant quantum Lie algebras are discussed from this point of view. Further we discuss the quantization of the Poisson structure on the symmetric algebraS(g) leading to the quantized enveloping algebraU h (g) as an example of biquantization in the sense of Turaev. Description ofU h (g) in terms of the generators of the bicovariant differential calculus onF(G q ) is very convenient for this purpose. Finaly we interpret in the deformation framework some well known properties of compact quantum groups as simple consequences of corresponding properties of classical compact Lie groups. An analogue of the classical Kirillov's universal character formula is given for the unitary irreducble representation in the compact case.

Canonical forms of tensor representations and spontaneous symmetry breaking

An algorithm for constructing canonical forms for any tensor representation of the classical compact Lie groups is given. This method is used to find a complete list of the symmetry breaking patterns produced by Higgs fields in the third-rank antisymmetric representations of U(n), SU(n) and SO(n) for n<or=7. A simple canonical form is also given for kth-rank symmetric tensor representations.