Lie Groups and Lie Algebras

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

We give a general theory of matrix elements (ME’s) of the unitary irreducible representations (UIR’s) of linear semisimple Lie groups and of reductive Lie groups. This theory connects together the following things, (1) MEUIR’s of all the representation series of a noncompact Lie group, (2) MEUIR’s of compact and noncompact forms of the same complex Lie group. The theory presented is based on the results of the theory of the principal nonunitary series representations and on a theorem which states that ME’s of the principal nonunitary series representations are entire analytic functions of continuous representation parameters. The principle of analytic continuation of Clebsch–Gordan coefficients (CGC’s) of finite dimensional representations to CGC’s of the tensor product of a finite and an infinite dimensional representation and to CGC’s of the tensor product of two infinite dimensional representations is proved. ME’s for any UIR of the group U(n) and of the group U(n,1) are obtained. The explicit expression for all CGC’s summed over the multiplicity of the irreducible representation in the tensor product decomposition is derived.

An estimation problem in compact Lie groups

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field. The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras—the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras. Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size n-by-n may be related to complex matrices of size 2n-by-2n. The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of sl[(2, ℂ). There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones. Section 10 contains a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory as in Chevalley [1946] is summarized, and the correspondence is made with the concrete theory of closed linear groups, where the Lie algebra is obtained as the space of derivatives at t = 0 of smooth curves in the group passing through the identity at t = 0. The section ends with a discussion of the adjoint representation. The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with ℝ1, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.

Weights for symmetric and general linear groups

A finitely generated subgroup${\rm\Gamma}$of a real Lie group$G$is said to be Diophantine if there is${\it\beta}>0$such that non-trivial elements in the word ball$B_{{\rm\Gamma}}(n)$centered at$1\in {\rm\Gamma}$never approach the identity of$G$closer than$|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group$G$is said to be Diophantine if for every$k\geqslant 1$a random$k$-tuple in$G$generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most$5$, or derived length at most$2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.

The concepts of normalized irreducible tensorial matrices (NITM) are extended to all finite and compact unitary groups by a development that clarifies their relationship to group theory and matrix algebra. NITM for a unitary group G are shown to be elements of a basis obtained by symmetry adapting to G the matrix basis of a matrix space M(α1 × α2). Elements [X] ∈ M(α1 α2) transform under Ga ∈ G according to [Ga] [X][G−1a], where [Ga] and [G−1a] belong to irreducible representations of G. The usual properties of NITM and the Wigner–Eckart theorem follow from these results, which are valid for both finite and compact unitary groups. The NITM span M(α1 × α2) are orthonormal under the trace and transform irreducibly with respect to G. This NITM basis of M(α1 × α2) is said to be simple. A compound NITM basis of a matrix space results when the space is partitioned into two or more subspaces, each spanned by a simple NITM basis. NITM determined from Griffith's V coefficients for the octahedral group are tabulated and used to construct a six‐coordinate superposition Hamiltonian.

Introduction Representations of $\mathbf {SU}(2)$ The general theory of linear representations of compact groups Lie theory of representations of compact connected Lie groups Induced representations of compact groups Spherical functions on compact groups Examples spherical harmonics The general theory of spherical functions Fourier and Plancherel transforms Extension of the Plancherel transform The subtleties of harmonic analysis Differential properties of spherical functions on Lie groups Spherical functions on semisimple Lie groups More on $\mathbf {SL}(2, \mathbf R)$ Automorphic functions Groups of isometries and Bessel functions Other special functions References.

If G G is a connected, simply connected, semisimple Lie group with metric given by the negative of the Killing form and zeta-function Z ( t ) Z(t) , then \[ Z ( t ) = Vol ⁡ G ( 4 π t ) dim ⁡ G / 2 exp ⁡ | δ | 2 t + exponentially small error as t ↓ 0. Z(t) = \frac {{\operatorname {Vol} G}} {{{{(4\pi t)}^{\dim G/2}}}}\exp |\delta {|^2}t + {\text {exponentially small error as }}t \downarrow 0. \]

In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible representations of the current group for $SL(2,R)$, and was used for this the well-known Fock space-structure That construction could be applied to the arbitrary locally compact group,and is based on a so called special representation of the original group $G$, with nontrivial 1-cohomology. A new construction uses the special property of one dimensional extensions (semi-direct product)of the nilpotent groups which allows immediately to produce the special representation of the group and then to apply the quasi-Poisson construction from the previous papers by authors in order to construct the representation of current group. The parabolic subgroup of the semisimple Lie group of rank one has such semidirect product, and special representation of it can be extended onto whole semisimple group. As result we obtain needed new model of the irreducible representation of semi-simple current groups.

In this chapter, we introduce the reader to the homology theory for algebras known as cyclic homology. As we shall see in the next section, cyclic homology may be viewed as the “linearization” of K-theory, in the same sense in which the matrix ring M n (R) is the “linearization” of the general linear group GL(n, R). For motivation, it is useful to think of the case where the ring R is ℝ or ℂ. Then GL(n, R)is a Lie group, and the space BGL(n, R)+ giving rise to the higher K-groups is by its construction an H-space whose homology agrees with the homology of this Lie group (never mind for the moment that we are forgetting the topology!). On the other hand, one of the basic principles of Lie theory is that the best way to study Lie groups (which are “non-linear” objects) is often by way of their Lie algebras (which can be studied using linear algebra). For example, it is a famous result that for any compact Lie group G with Lie algebra g, the topological cohomology H●(G;ℝ) (here we are just thinking of G as a space and ignoring the group structure) is canonically isomorphic to the Lie algebra cohomology H●(g;ℝ), which can at least in principle be computed using only finite-dimensional linear algebra. This suggests that some construction with the Lie algebra M n (R) of GL(n, R)ought to yield a reasonable approximation to the K-theory of R, at least in the limit as n → ∞. While the actual construction of cyclic homology and of the Chern character which relates it to K-theory is a bit roundabout, this philosophy turns out to be basically correct.KeywordsExact SequenceCommutative RingHomology TheoryChern CharacterCyclic HomologyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper focuses on the relevance of a certain class of left-invariant Hamiltonians (affine-quadratic) on a reductive semi-simple Lie algebra g for the theory of integrable systems and the equations of applied mathematics.Any semi-simple Lie group G that contains a closed subgroup K is reductive, in the sense, that the orthogonal complement p in g of the Lie algebra k of K , relative to the Killing form, satisfies [k, p] p.This implies that K acts (by adjoint action) on p and therefore induces the semi-direct product p K .Consequently, g , as a vector space, carries two Lie algebra structures: semi-simple, and semi-direct.Hence, the dual g * carries two Poisson structures as well.Any affine-quadratic function H on g can be simultaneously viewed as a Hamiltonian for either Poisson structure.We will show that certain coadjoint orbits relative to the semi-direct action are the cotangent bundles of SO(n).This implies that the equations of an n -dimensional top can be represented on such coadjoint orbits.In this situation there is a canonical affine-quadratic Hamiltonian whose Hamiltonian equations on these coadjoint orbits coincide with the equations of the top.This implies that the integrable cases of the top correspond to the the integrable cases of the overseeing affine Hamiltonian.More generally, we will identify a subclass of affine-quadratic Hamiltonians, called isospectral, that provides new insights into the theory of integrable systems based on the contributions of S. V. Manakov, A. T. Fomenko, A. S. Mischenko and O. Bogoyavlensky listed in the references.

CheMPS2: A free open-source spin-adapted implementation of the density matrix renormalization group for ab initio quantum chemistry

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.

On a remarkable class of flat pseudo-Riemannian Lie groups