Algorithms for Computations in Local Symmetric Spaces

Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.

Lie groups and algebraic groups are important in many major areas of mathematics and mathematical physics. We find them in diverse roles, notably as groups of automorphisms of geometric structures, as symmetries of differential systems, or as basic tools in the theory of automorphic forms. The author looks at their development, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. Starting from Lie's theory of local analytic transformation groups and early work on Lie algebras, he follows the process of globalization in its two main frameworks: differential geometry and topology on one hand, algebraic geometry on the other. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter. The essays in the first part of the book survey various proofs of the full reducibility of linear representations of \\mathbf{SL}_2{(\\mathbb{C})}, the contributions of H. Weyl to representations and invariant theory for semisimple Lie groups, and conclude with a chapter on E. Cartan's theory of symmetric spaces and Lie groups in the large. The second part of the book first outlines various contributions to linear algebraic groups in the 19th century, due mainly to E. Study, E. Picard, and above all, L. Maurer. After being abandoned for nearly fifty years, the theory was revived by C. Chevalley and E. Kolchin, and then further developed by many others. This is the focus of Chapter VI. The book concludes with two chapters on the work of Chevalley on Lie groups and Lie algebras and of Kolchin on algebraic groups and the Galois theory of differential fields, which put their contributions to algebraic groups in a broader context. Professor Borel brings a unique perspective to this study. As an important developer of some of the modern elements of both the differential geometric and the algebraic geometric sides of the theory, he has a particularly deep understanding of the underlying mathematics. His lifelong involvement and his historical research in the subject area give him a special appreciation of the story of its development.

We prove that the restriction to (2, R) of the underlying HarishChandra module of a Discrete Series representation of a noncompact simple Lie group has no sl(2)-irreducible subrepresentations unless it is holomorphic. We also give a characterization of the holomorphic Discrete Series. Let G be a connected real simple Lie group. Fix a maximal compact subgroup K of G. Let g (t) denote the complex Lie algebra of G (respectively, K). Hence we have the Cartan decomposition g = t + p. Thus, p is Kinvariant under the restriction of the adjoint representation of G to K and p . Let T C K be a maximal torus. Henceforth we assume that T is a Cartan subgroup of G. Let t be the complex Lie algebra of T. Under these hypotheses Harish-Chandra in [HC2] has given a classification of the square-integrable representations of G in terms of the regular characters of T. Let (D(g, t) denote the root system of the pair (g, t) . A root of the pair (g, t) is called compact (noncompact) if its root space is contained in t (respectively, p). We now fix a noncompact root y . Then the root vectors of y span a copy by of sl(2, C) . Moreover by is invariant under the conjugation of g with respect to the Lie algebra of G. Let s,, be the real form of by associated to the conjugation of g with respect the Lie algebra of G restricted to by . Let Hy be the connected subgroup of G corresponding to sy I Let (7r, V) be the Harish-Chandra module underlying a square-integrable, irreducible representation of G. We then have Proposition 1. If the restriction of (7r, V) to by contains an irreducible bysubmodule, then G/K is a Hermitian symmetric space and (7r, V) corresponds to a holomorphic square-integrable representation. A corollary to Proposition 1 is Corollary. If G is as above, G/K is not Hermitian, and H is a semisimple subgroup of G such that its symmetric space is Hermitian, then (7r, V) restricted to [ has no [-holomorphic irreducible subrepresentations. Received by the editors April 29, 1991 and, in revised form, July 11, 1991. 1991 Mathematics Subject Classification. Primary 22E47.

We prove the following theorem: Let K be a connected symmetric subgroup of the group of isometries of a compact connected globally Riemannian symmetric space M.Then, K acts in a variationally complete manner on M. (We refer to the work of R. Bott and H. Samelson [l;2] for a definition of this concept and applications to topology.)Let G be a compact, connected Lie group, L a connected symmetric subgroup of G. Bott and Samelson have proved the following: (1) L acts in a variationally complete manner on G/L, the right coset space, (2) L acts via the linear isotropy representation in a variationally complete way on the tangent space at a point of G/L.Our theorem then generalizes (1), but does not imply (2).The method is essentially the same as Bott's [l], but uses a Lie algebra approach in a stronger way.The proof then provides an exposition of Bott's very important result from a Lie-algebra point of view.Notice also that it suggests the general program of studying the Jacobi-fields on a symmetric space as a problem of Lie algebra theory, even in the nonvariational complete case.1.

Finite and infinite dimensional Lie group structures on Riordan groups

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure ω which is invariant under the geodesic symmetries. When the transvection group G0 of such a symmetric space M is semisimple, its action on (M,ω) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G0-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra $$\mathcal{G}^0 $$ of G0. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is $$\mathcal{G}^0 $$ . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.

The Poincar-Birkhoff-Witt Theorem deals with the structure and universal property of the universal enveloping algebra U (L) of a Lie algebra L , e.g., over R or C .K. H. Hofmann and L. Kramer (HK) [On weakly complete group algebras of Compact Groups, J. Lie Theory 30 (2020) 407-426] recently introduced the weakly complete universal enveloping algebra U(g) of a profinite-dimensional topological Lie algebra g.Here it is shown that the classical universal enveloping algebra U (|g|) of the abstract Lie algebra underlying g is a dense subalgebra of U(g) , algebraically generated by g U(g) .It is further shown that, inspite of U being a left adjoint functor, it nevertheless preserves projective limits in the form U(lim i g/i) = lim i U(g/i) , for profinite-dimensional Lie algebras g represented as projective limits of their finite-dimensional quotients.The required theory is presented in an appendix which is of independent interest.In a natural way, a weakly complete enveloping algebra U(g) is a weakly complete symmetric Hopf algebra with a Lie subalgebra P(U(g)) of primitive elements containing g (indeed properly if g = {0}), and with a nontrivial multiplicative pro-Lie group G(U(g)) of grouplike units, having P(U(g)) as its Lie algebra -in contrast with the classical Poincar-Birhoff-Witt environment of U (L) , thus providing a new aspect of Lie's Third Fundamental Theorem: Indeed a canonical pro-Lie subgroup * (g) of G(U(g)) is identified whose Lie algebra is naturally isomorphic to g.The structure of U(g) is described in detail for dim g = 1 .The primitive and grouplike components and their mutual relationship are evaluated precisely.In (HK), cited above, and in the work of R. Dahmen and K. H. Hofmann [The pro-Lie group aspect of weakly complete algebras and weakly complete group Hopf algebras, J. Lie Theory 29 (2019) 413-455] the real weakly complete group Hopf algebra R[G] of a compact group G was described.In particular, the set P(R[G])) of primitive elements of R[G] was identified as the Lie algebra g of G .It is now shown that for any compact group G with Lie algebra g there is a natural morphism of weakly complete symmetric Hopf algebras g : U(g) R[G] , implementing the identity on g and inducing a morphism of pro-Lie groups * (G) G(R[G]) = G : yet another aspect of Sophus Lie's Third Fundamental Theorem !

Deep compositing is an important practical tool in creating digital imagery, but there has been little theoretical analysis of the underlying mathematical operators. Motivated by finding a simple formulation of the merging operation on OpenEXR -style deep images, we show that the Porter-Duff over function is the operator of a Lie group. In its corresponding Lie algebra, the splitting and mixing functions that OpenEXR deep merging requires have a particularly simple form. Working in the Lie algebra, we present a novel, simple proof of the uniqueness of the mixing function. The Lie group structure has many more applications, including new, correct resampling algorithms for volumetric images with alpha channels, and a deep image compression technique that outperforms that of OpenEXR .

A survey is given of various methods of integrating Hamiltonian systems on symmetric spaces and Lie algebras, and an analysis is also presented of applications of the technique developed to some mechanical problems. Some questions of nonintegrability of Hamiltonian systems on symplectic manifolds are considered, and, in particular, the Morse theory of completely integrable Hamiltonian systems is expounded.

The survey deals with the investigations reviewed inReferativnyi Zhurnal “Matematika” between 1977–1981. In the survey there are reflected the investigations on the structure of Lie groups and Lie algebras, on their finite-dimensional linear representations and universal enveloping algebras, on the theory of invariants and Lie groups of transformations, and also on continuous and discrete subgroups of Lie groups.

Using characters, one can set up the theory of representations for both finite and continuous groups. For continuous groups, another approach—the Lie theory—is also possible. It is shown that a similar theory, based on commutators, can be developed also in the case of finite groups. Essentially, the group algebra of a finite group is converted into a Lie algebra by replacing the usual associative product by the product x∘y=xy−yx. Then the resulting Lie algebra is a direct sum of special unitary Lie algebras.

CHAPTER IV - The Lie Theory

Preliminary Remarks The operator or operational calculus, the method of treating differential operators as algebraic objects, was once thought to have originated with the English physicist and electrical engineer Oliver Heaviside (1850–1925). Indeed, Heaviside revived and brilliantly applied this method to problems in mathematical physics. But the basic ideas can actually be traced back to Leibniz and Lagrange who must be given credit as the founders of the operational method. With his notation for the differential and integral, Leibniz was able to regard some results on derivatives and integrals as analogs of algebraic results. The later insight of Lagrange was to extend this analogy to infinite series of differentials so that, in particular, he could write the Taylor expansion as an exponential function of a differential operator. In fact, this formal approach to infinite series appeared in the work of Newton himself. For Newton, infinite series in algebra served a purpose analogous to infinite decimals in arithmetic: They were necessary to carry out the algebraic operations to their completion. Newton's insightful algorithms using formal power series were of very wide applicability in analysis, algebra, and algebraic geometry; their power lay precisely in their formal nature. Thus, the algebraic analysis of the eighteenth century can trace its origins to Newton's genius. A branch of algebraic analysis focusing on the combinatorial aspects of power series was developed by the eighteenth-century German combinatorial school.

Lie algebras and quantum groups are not usually studied by an undergraduate. However, in the study of these structures, there are interesting questions that are easily accessible to an upper-level undergraduate. Here we look at the expansion of a nested set of brackets that appears in relations presented in a paper of Lum on toroidal algebras. We illuminate certain terms that must be in the expansion, providing a partial answer for the closed form. Lie algebras and quantum groups are not topics that you are apt to hear undergraduates math majors discussing in their spare time. However, there are a surprising number of nontrivial questions in this area that are undergraduate appropriate. In this paper, we will give a brief overview of the broad mathematical setting, and then discuss an accessible problem that involves expanding a nested set of brackets. Lie algebras, their universal enveloping algebras and quantum groups are a fundamental part of representation theory that have many applications within mathematics and mathematical physics. Lie algebras and Lie groups were originally discovered by Sophus Lie in the late nineteenth century [Borel 2001]. Given a Lie algebra, we associate a unique associative algebra called the universal enveloping algebra. In 1985, Jimbo and Drinfeld discovered q-analogues of these universal enveloping algebras called “quantum groups”, which have been a recent area of study (see [Lusztig 1993]). In order to find the quantum analogue of a Lie algebra it is often desirable to understand the defining relationships of the Lie algebra inside of its universal enveloping algebra. The motivation for this project came from a paper by Lum in which he gives a nice presentation of a toroidal Lie algebra that could be useful in understanding this Lie algebra’s quantum group [Lum 1998]. All of these relations utilize a nested set of brackets called t.k/. For simplicity, we have modified t.k/ by a scalar. In this paper we seek to understand the expansion of this object.

This paper addresses the problem of the study of controllability and stability analysis of an elementary but useful example of unicycle. We have considered a three dimensional control system of a unicycle. The control system on unicycle is related to a group, which has a Lie group structure. Here we show how concepts of differential geometry and Lie algebra can be elegantly applied to explain the behavior of such a system. Controllability and a minimum effort problem for the system are studied using well known theorems and necessary optimality conditions. The two unconventional integrators, namely Kahan and Lie–Trotter integrators have been implemented for numerical integration and the results are compared with the conventional Runge-Kutta integrator.