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Topic: Semisimple group

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	Lie group decompositions - Wikipedia, the free encyclopedia
		They are essential technical tools in the representation theory of Lie groups and Lie algebras; they can also be used to study the algebraic topology of such groups and associated homogeneous spaces.
		The same ideas are often applied to Lie groups, Lie algebras, algebraic groups and p-adic number analogues, making it harder to summarise the facts into a unified theory.
		The Iwasawa decomposition KAN of a semisimple group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (a consequence of orthogonalization).
	en.wikipedia.org /wiki/Lie_group_decompositions (276 words)

	PlanetMath: semisimple group
		is called semisimple if it has no proper normal solvable subgroups.
		Every group is an extension of a semisimple group by a solvable one.
		This is version 1 of semisimple group, born on 2002-12-17.
	planetmath.org /encyclopedia/SemisimpleGroup.html (57 words)

	Maximal torus
		In the theory of Lie groups in mathematics, especially those that are compact, a special role is played by the torus groups.
		In a compact Lie group G there is to be found a maximal torus T; that is, a closed subgroup that is a torus, and of the largest possible dimension.
		For example, the Lie group SO(3) of rotations in three dimensions has as maximal torus T a circle group (a 1-torus, that is).
	www.teachtime.com /en/wikipedia/m/ma/maximal_torus.html (246 words)

	PlanetMath: classification of semisimple groups
		"classification of semisimple groups" is owned by bwebste.
		Cross-references: semisimple, group, inner automorphisms, subgroup, injection, conjugation, simple groups, direct product, isomorphic, radical, reducible, normal subgroup, semisimple group
		This is version 1 of classification of semisimple groups, born on 2002-12-17.
	planetmath.org /encyclopedia/ClassificationOfSemisimpleGroups.html (105 words)

	Ergodic Theory, Groups, and Geometry Conference Description
		The study of a general action of a Lie group G is greatly facilitated by the study of actions on vector bundles and principal bundles.
		This leads to the construction of the Gromov representation of the fundamental group of a rigid geometric manifold which admits a simple noncompact automorphism group, which is a key ingredient in the analysis of this situation.
		Arithmeticity theorems for fundamental groups and holonomy groups.
	www.math.umn.edu /~adams/CBMS/descr.html (1702 words)

	APPENDIX B
		In the case of a semisimple Lie group, the group manifold can be endowed with with an affine connection derived from the structure constants, with which the manifold becomes an Einstein space, that is a Riemannian manifold with a Riemann-Christoffel curvature tensor that satisfies the vacuum n-dimensional Einstein equations.
		In the adjoint representation of a semisimple Lie algebra, the algebra basis is represented by matrices whose elements are the structure constants.
		The adjoint action of a Lie group on its algebra defined in equation (B.10) and in the preceding, exponentiates to the adjoint action of the group on itself.
	graham.main.nc.us /~bhammel/FCCR/apdxB.html (7008 words)

	Papers of Rebecca A. Herb
		Fourier inversion and the Plancherel theorem, Proceedings of Conference on Non-Commutative Harmonic Analysis on Lie Groups, 1980.
		The Schwartz space of a general semisimple group II: Wave packets associated to Schwartz functions, Trans.
		The Schwartz space of a general semisimple Lie group V: Schwartz class wave packets, Pacific J. of Math.,174 (1996), 43-139.
	www.math.umd.edu /~rah/rahpub.html (674 words)

	Dave Witte Morris' Lie Theory papers
		Suppose L is a semisimple Levi subgroup of a connected Lie group G, and c is a GL(n,R)-valued Borel cocycle for the action of G on a Borel space X. Assume L has finite center, and that the real rank of every simple factor of L is at least two.
		Let G be a finitely generated group having the property that any action of any finite-index subgroup of G by homeomorphisms of the circle must have a finite orbit.
		The general philosophy is to reduce the study of the cocycle to the study of its restriction to each ergodic component of the action, while being careful to show that all objects arising in the analysis depend measurably on the ergodic component.
	people.uleth.ca /~dave.morris/LieTheory.shtml (2508 words)

	Álgebras de Hopf (Site not responding. Last check: 2007-10-09)
		This representation is a deformation of the quantum Weyl group of Soibelman and Lusztig, which may be called a "dynamical" Weyl group, since it is intimately related with solutions of the dynamical Yang-Baxter equation of Gervais, Neveu, and Felder.
		The quantum coordinate algebra Fq (G) is a deformation of the algebra F(G) of ``polynomial functions'' on a complex semisimple group G. More precisely, the dual of the quantum enveloping algebra Uq(g) (the ``simply connected'' version) corresponding to the complex semisimple Lie algebra g, has a natural algebra structure.
		The irreducible representations of the symmetric group $S_n$ are parameterized by partitions of $n$.
	www.famaf.unc.edu.ar /vaq2001/pf/abstracts_alg-hopf.html (1197 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		In this case the group itself as not a direct product, although a finite covering is. > The semisimple real algebraic groups are all semisimple real Lie groups > but with some restrictions on the fundamental group.
		Since its fundamental group is Z, there are finite coverings of SL(2,R) corresponding to the subgroups nZ of Z. These coverings are real semisimple Lie groups which are not real algebraic, but nevertheless have finite centers.
		In other words, saying that a real Lie group is algebraic means that you can complexify it, and you can only do that when its center is one occurs for a complex Lie group with the complexified Lie algebra.
	www.math.niu.edu /~rusin/known-math/98/classif_lie (561 words)

	Automorphisms (Site not responding. Last check: 2007-10-09)
		The diagonal automorphism of the semisimple group of Lie type G given by the vector v.
		The graph automorphism of the group of Lie type G given by the permutation p.
		Given a group of Lie type automorphism h, this returns an inner automorphism i, a graph automorphism g and a field automorphism f such that h=igf.
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text1102.htm (311 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		The subject of this work is a class of Chern-Simons field theories with non-semisimple gauge group, which may well be considered as the most straightforward generalization of an Abelian Chern-Simons field theory.
		As a matter of fact, these theories, which are characterized by a non-semisimple group of gauge symmetry, have cubic interactions like those of non-Abelian Chern-Simons field theories, but are free from radiative corrections.
		Moreover, at the tree level in the perturbative expansion, there are only two connected tree diagrams, corresponding to the propagator and to the three vertex originating from the cubic interaction terms.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=20420182 (288 words)

	philosophy notes
		The group G is algebraic if it is isomorphic to a closed (in the Zariski topology) subgroup of a matrix group.
		An algebraic group is then an abstract algebraic variety with a binary and a unary operation which satisfy the axioms of groups and are morphisms of algebraic varieties.
		An algebraic group $G$ is a structure whose universe is some subset of $k^n/\iso$ and whose relations are all those definable in the field $k$.
	www.math.uic.edu /~jbaldwin/pub/phil.html (2052 words)

	The Rationality Problem for Semisimple Group Varieties - Chernousov, Platonov (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		11 Classification of algebraic semisimple groups (context) - Tits - 1966
		1 The group of multipliers of the canonical quadratic form and..
		On The Modality Of Parabolic Subgroups Of Linear Algebraic Groups - Röhrle (1999)
	citeseer.ist.psu.edu /chernousov98rationality.html (645 words)

	Random generation in semisimple algebraic groups over local fields (Site not responding. Last check: 2007-10-09)
		Algebra 206 (1998) 438-504.) The idea is that such groups should not be too far from being open subgroups of the group of points of a (possibly smaller) algebraic group over a (possibly smaller) local field.
		To prove that two random elements generate the whole group, the idea is to show first that (with probability 1) they generate a Zariski-dense subgroup and then (again with probability 1) that they do not lie in the "same" group over a proper subfield of K.
		This theorem can be regarded as a topological analogue of the Dixon conjecture for finite groups, which asserts that the probability that two randomly chosen elements of a finite simple group will generate that group tends to 1 as the order of the group grows without bound.
	mlarsen.math.indiana.edu /~larsen/random.html (258 words)

	Rutgers: Lie Group Seminar Spring 1995
		The group W_X is also a reflection group and is analogous to the little Weyl group of a symmetric space.
		The dual correspondence we propose is a bijection between non-trivial conjugacy classes of the finite group and vertices of the diagram.The ends of the diagram give 3 distinguished conjugacy classes, and edges correspond to taking a product with an element in one of these classes.
		The loop Graßmannian is a homogeneous space for a loop group LG with the property that its equivariant D-modules correspond to finite dimensional representations of the Langlands dual of G.
	www.math.rutgers.edu /~knop/seminar/Seminar_Spring95.html (1696 words)

	[No title]
		A compact semisimple Lie group is a compact Riemannian manifold with Riemannian structure defined via the Killing form, which is invariant under the action of the group).
		This element belongs to the center of the enveloping algebra of the Lie algebra of the group.
		On the other hand, I will show using the Lie group structure that the set of critical points of a smooth class function is closed under conjugation.
	www.cs.uaf.edu /dms/Colloquium0102.html (1680 words)

	OUP: Representation Theory of Finite Groups: Algebra and Arithmetic: Weintraub
		And when a group (finite or otherwise) acts on something else (as a set of symmetries, for example), one ends up with a natural representation of the group.
		This book is an introduction to the representation theory of finite groups from an algebraic point of view, regarding representations as modules over the group algebra.
		The book has an extensive development of the semisimple case, where the characteristic of the field is zero or is prime to the order of the group, and builds the foundations of the modular case, where the characteristic of the field divides the order of the group.
	www.oup.co.uk /isbn/0-8218-3222-0 (527 words)

	[No title]
		The image Ad(G) \subseteq GL(Lie(G)) is called the adjoint group of G. The connected component of the centre of G acts trivially in Ad.
		A semisimple ccLs is a finite product of simple ccLg's, which are of the type SU(n), SO(n), Sp(n), and the exceptional groups E_6, E_7, E_8, F_4, G_2.
		Conveivably, a lot of noncompact semisimple and reductive Lie groups appear here, the topology of which should, IMO, reduce to that of ccLg's, since they should deformation retract onto their maximal compact subgroups.
	www.math.niu.edu /~rusin/known-math/00_incoming/lie_g (1053 words)

	Brian C. Hall - Department of Mathematics - University of Notre Dame
		As the name suggests, every matrix Lie group is a Lie group, and while not every Lie group is a matrix Lie group, most of the interesting ones are.
		To define the Lie algebra of a matrix Lie group, I consider the exponential of an n x n matrix, which is defined by the usual power series (and which is described in many standard textbooks on differential equations).
		Meanwhile, on the issue of the semisimple theory, my approach is to consider in detail the representation theory of SU(2) and SU(3) (or, at the level of complex Lie algebras, sl(2,C) and sl(3,C)) before going on to the general case.
	www.nd.edu /~bhall/book (1466 words)

	Diagrams re projection theorem for semisimple subgroups (Site not responding. Last check: 2007-10-09)
		Diagrams illustrating the Projection Theorem in Weight Space and the Dynkin indices of Semisimple Subgroups of Semisimple Groups.
		These diagrams illustrate the paper, A. Stone, "Semisimple subgroups of semisimple groups", Journal of Mathematical Physics, 11, 29-38 (1970).
		This extends the concept of Dynkin index from simple groups to semisimple groups.
	homepage.ntlworld.com /stone-catend/subgr.htm (112 words)

	Abstract: The Universal Cover of SL(2,R) (Site not responding. Last check: 2007-10-09)
		This expository article describes a representation theory for the universal covering group G of SL(2,R) which is motivated by a localization theory due to A. Beilinson and J.
		We explicitly describe the irreducible representations of G using objects familiar to the average calculus student -- the complex plane, vector spaces, the algebra of two-by-two matrices, the derivative of a rational polynomial function -- and some elementary facts about Lie group representations and algebraic D-modules.
		The computations for the universal cover of SL(2,R) in this article have proven important in some recent developments on representations of an arbitray semisimple Lie group with infinite center [10].
	www.panix.com /~shalla/rds02.html (405 words)

	CV for Mogens Flensted Jensen
		An explicit construction of the K-finite vectors in the discrete series for an isotropic semisimple symmetric space, Mém.
		Towards a Paley-Wiener theorem for semisimple symmetric spaces.
		Proceedings of the Nato advanced workshop 'Noncompact Lie groups and their physical applications', San Antonio, Texas 1993.
	www.dina.kvl.dk /~mfj/publications.html (552 words)

	Amazon.com: Representation Theory of Semisimple Groups: An Overview Based on Examples. (PMS-36).: Books: Anthony W. ... (Site not responding. Last check: 2007-10-09)
		A linear connected reductive group is a closed connected group of real or complex matrices that is stable under conjugate transpose.
		The theory of representations of semisimple Lie groups is very complete from a mathematical perspective and is of enormous importance in high energy physics.
		The Iwasawa and Bruhat decompositions and the Weyl group construction are shown to hold for non-compact groups in chapter 5.
	www.amazon.com /exec/obidos/tg/detail/-/0691090890?v=glance (1356 words)

	Atlas: Examples of affine maximal torus fibrations of a compact Lie group by Marcos Salvai
		By a generalization of the method developed by Gluck and Warner to characterize the oriented great circle fibrations of the three-sphere, we obtain, for any compact connected semisimple Lie group G, infinite dimensional spaces of examples of smooth and continuous nonsmooth fibrations of G by Weyl-oriented affine maximal tori.
		Let G be a compact connected semisimple Lie group.
		Using the canonical immersion of G/T into the Lie algebra of G, we also provide, for any G, concrete infinite dimensional spaces of examples of smooth and continuous nonsmooth fibrations of G as in the theorem.
	atlas-conferences.com /c/a/d/q/81.htm (730 words)

	Some remarks on the group of R-equivalence classes of semisimple groups over arithmetic fields (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		Abstract: We compute the group of R-equivalence classes of some simple simply connected algebraic groups of types 3;6 D 4 and E 6 defined over an algebraic number field.
		3 equivalence sur les groupes alg'ebriques r'eductifs d'efinis..
		1 Normal subgroups in the multiplicative group of a division a..
	citeseer.ist.psu.edu /239672.html (393 words)

	Finite modules over non-semisimple group rings., by Cristian D. Gonzalez-Aviles (Site not responding. Last check: 2007-10-09)
		Let $G$ be an abelian group of order $n$ and let $R$ be a commutative ring which admits a homomorphism ${\Bbb Z}[\zeta_{n}]\ra R$, where $\zeta_{n}$ is a(complex) primitive $n$-th root of unity.
		To derive these conditions, we build on work of E.Aljadeff and obtain, as a by-product of our considerations, a new criterion for cohomological triviality which improves the well-known criterion of T.Nakayama.
		We also give applications to abelian varieties and to class groups of abelian fields, obtaining in particular some new class number formulas.
	www.math.uiuc.edu /Algebraic-Number-Theory/0345 (220 words)

	- SHOP.COM (Site not responding. Last check: 2007-10-09)
		In this book the author develops the hyperalgebra associated with certain algebraic groups, with an emphasis on the algebraic rather than geometric approach.
		This avoids the need to cover some of the more sophisticated techniques, while the representation theory of the hyperalgebra is developed from scratch without reference to the associated group.
		The author assumes some familiarity with complex semisimple Lie algebras, but includes a chapter on Hopf algebras and duality to keep the book self-contained.
	www.shop.com /op/aprod-p29659045 (305 words)

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