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Topic: Maximal compact subgroup

	Maximal compact subgroup - Wikipedia, the free encyclopedia
		In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups.
		It is therefore a maximal subgroup amongst compact subgroups, rather than being (alternate possible reading) a maximal subgroup that happens to be compact; which would probably be called a compact maximal subgroup, but in any case is not the intended meaning.
		In that case a maximal compact subgroup K must be a compact Lie group, for which the theory is easier.
	en.wikipedia.org /wiki/Maximal_compact_subgroup (270 words)

	Compact group - Wikipedia, the free encyclopedia
		But it is the compact Hausdorff groups, natural generalisations of finite groups with their discrete topology, that have the properties that carry over in significant fashion.
		Compact groups all carry a Haar measure, which will be invariant by both left and right translation (since the modulus function must be a continuous homomorphism to the positive multiplicative reals, and so 1).
		Inside a general semisimple Lie group there is a maximal compact subgroup, and the representation theory of such groups, developed largely by Harish-Chandra, uses intensively restriction to such a subgroup, and also the model of Weyl's character theory.
	en.wikipedia.org /wiki/Compact_group (461 words)

	Torus - Wikipedia, the free encyclopedia
		This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication).
		This is due in part to the fact that in any compact Lie group one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension.
		The classification theorem for surfaces states that every compact connected surface is either a sphere, an n-torus, or the connected sum of n projective planes.
	en.wikipedia.org /wiki/Torus (908 words)

	PlanetMath: Borel subgroup
		Any Borel group is connected and equal to its own normalizer, and contains a unique Cartan subgroup.
		Cross-references: upper triangular matrices, maximal torus, compact, intersection, contains, normalizer, connected, conjugate, group, subgroup, solvable, maximal, Lie group, semi-simple, complex
		This is version 2 of Borel subgroup, born on 2003-02-13, modified 2003-02-24.
	planetmath.org /encyclopedia/BorelSubgroup.html (84 words)

	Springer Online Reference Works
		A consequence of this is the fact that any locally compact subgroup of a Hausdorff topological group is closed.
		Integration with respect to a Haar measure allows one to transfer to compact groups a significant part of the theory of representations of finite groups (for example, the orthogonality relation for characters, or for matrix entries), and also the Peter–Weyl theorem, which was first obtained for Lie groups.
		The result that every locally compact subgroup has an open subgroup which is a projective limit of subgroups is due to H.
	eom.springer.de /T/t093070.htm (1330 words)

	Cornell Math - Thesis Abstracts (Lie Groups)
		Moreover, we prove that the unrefined minimal K-types of the paired representations are related by the theta correspondence for finite reductive dual pairs when the depth is zero, and are related by the orbit correspondence when the depth is positive.
		We describe on K an analog of the Bargmann-Segal "coherent state" transform, and we prove that this generalized coherent state transform maps L^2(K) isometrically onto the space of holomorphic functions in L^2(G, \mu), where G is the complexification of K and where \mu is an appropriate heat kernel measure on G.
		The generalized coherent state transform is defined in terms of the heat kernel on the compact group K, and its analytic continuation to the complex group G.
	www.math.cornell.edu /Research/Abstracts/lie_groups.html (1159 words)

	Loeks Symmetric Spaces Page
		The most extreme of these are, respectively, the most discrete series (in the group case called fundamental series) corresponding to the conjugacy class of Cartan subspaces with maximal compact part and the most continuous series, corresponding to the conjugacy class with maximal non-compact part.
		He used this to show that all maximal compact subgroups of a real simple Lie group are conjugate.
		On the other hand a maximal compact subgroup determines a Cartan involution of the real Lie group and this involution determines the real Lie group as a real form of its complexification.
	www4.ncsu.edu /~loek/research/symm.html (1193 words)

	[No title]
		The loop space (G^p, BG^p, e), corresponding to a pair (G, p) (whe* re p is a prime, G a compact* Lie group with component group a finite p-group, and (.)^pde* *notes Fp-completion [17][48, x11]) is a p-compact group.
		To state it, we define the maximal torus normali* zer NX (T) to be the loop space such that BNX (T) is the Borel construction of the canoni *cal action of WX (T) on BT.
		Q 6 Then the maximal torus consists of the elements h(t1, t2, t3, t4, t5, t6) = i* =1hffi(ti) and the normalizer N(H) of the maximal* torus is generated by H and the elements ni=* * nffi(1), 1 i 6.
	hopf.math.purdue.edu /Andersen-Grodal-Moller-Viruel/classificationpodd.txt (18041 words)

	PlanetMath:
		A finitely generated group has only finitely many subgroups of a given index owned by avf
		proof that all subgroups of a cyclic group are cyclic owned by Wkbj79
		alternating group is a normal subgroup of the symmetric group owned by mathcam
	planetmath.org /encyclopedia/A (2338 words)

	week178
		Now, a maximal flag is a rather fancy type of figure, built from a bunch of simpler ones satisfying a bunch of incidence relations.
		The smallest parabolic subgroup is B itself, and G/B is the space of "maximal flags".
		In fact, all compact simply-connected Kaehler manifolds with a transitive action of G are of this form.
	math.ucr.edu /home/baez/week178.html (3470 words)

	UCLA Distinguished Lecturers
		Let G be a semisimple Lie group, K a maximal compact subgroup and Y=G/K the associated symmetric space.
		His work with Sarnak and Phillips on the explicit construction of Ramujan graphs via modular forms and the problem of distributing points on the sphere were the subjects of Bourbaki reports and attracted a great deal of attention in computer science and engineering.
		He gave the Woodward lectures in Yale in 1998, the Ritt lectures in Columbia in 1999, the Eilenberg lectures in 2000, and the Porter lectures in Rice in 2001.
	www.math.ucla.edu /dls/2004/lubotzky.html (353 words)

	Category.org - The Online Shopping Center: Books - Group Theory (Site not responding. Last check: 2007-11-01)
		Subgroup growth studies the distribution of subgroups of finite index in a group as a function of the index.
		In the last two decades this topic has developed into one of the most active areas of research in infinite group theory; this book is a systematic and comprehensive account of the substantial theory which has emerged.
		As well as determining the range of possible 'growth types', for finitely generated groups in general and for groups in particular classes such as linear groups, a main focus of the book is on the tight connection between the subgroup growth of a group and its algebraic structure.
	www.category.org /browse/books/13940 (5868 words)

	[No title]
		Mimura 1995 \cite{mimurapaper}] Any connected Lie Group $G$ is homeomorphic to the Cartesian product of a maximal compact subgroup $K$ and a subset which is homeomorphic with a Euclidean space $\mathbb{R}^m$: $$ G\approx K\times\mathbb{R}^m.
		$$ Moreover all maximal compact subgroups are conjugate.
		\end{prop} Notice that the symplectic group $Sp_n$ is a maximal compact subgroup of $\gl_n(\mathbb{H})$ and a Euclidean space is contractible, that is, homotopic to a constant mapping, hence $\pi_k(\mathbb R^m)=\mathbf{0}$ \cite[\S 1.14]{top&geo}.
	www.lehigh.edu /~dmd1/lokshun (488 words)

	[No title]
		TSP_2 Maximal Kolmogorov Subspaces of a Topological Space as Stone Retracts of the Ambient Space by Zbigniew Karno Received July 26, 1994 359.
		LATSUBGR On the Lattice of Subgroups of a Group by Janusz Ganczarski Received May 23, 1995 387.
		WAYBEL33 Compactness of Lim-inf Topology by Grzegorz Bancerek and Noboru Endou Received July 29, 2001 694.
	merak.pb.bialystok.pl /mizardoc/mml.txt (16294 words)

	Yale Math Calendar
		Two compact Riemanian manifolds are said to be isospectral if the (multi-set of) the eigenvalues of their Laplacians are equal.
		The question of finding isospectral non-isometric manifolds has a long history starting with Marc Kac's seminal paper (1966): " Can you hear the shape of a drum?" Various methods are known to construct such manifolds- the most powerful is due to Sunada.
		For locally symetric manifolds M (i.e., M= D\G/K, G semisimple Lie groups, K maximal compact subgroup and D a lattice in G) all the know examples are commensurable to each other.
	www.math.yale.edu /calendar/day.php?LocationID=&Date=2005-03-23 (184 words)

	E6
		In fact it is a maximal compact subgroup, since if there were a larger one, we could average a Riemannian metric group on
		is negative definite on its 52-dimensional maximal compact Lie algebra,
		where the groups on the top are maximal compact subgroups of those on the bottom.
	math.ucr.edu /home/baez/octonions/node17.html (431 words)

	Math Forum Discussions
		doesn't the theorem about the inclusion of a maximal compact subgroup
		compact subgroup here, and isn't su(2) a 3-sphere?
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?threadID=1256112&messageID=4004063 (198 words)

	Professor Pat Eberlein's Home Page
		Symmetric spaces can be represented as coset spaces G/K, where G is a connected semisimple group with no compact factors and K is a maximal compact subgroup of G. Since 1990 I have studied the geometry of 2-step nilpotent, simply connected Lie groups with a Riemannian metric that is preserved by left translations.
		Rational approximation in compact Lie groups and their Lie algebras, preprint,2000 PDF version
		Rational approximation in compact Lie groups and their Lie algebras, II, preprint, 2000 PDF version
	www.math.unc.edu /Faculty/pbe (510 words)

	Math Forum Discussions
		First, there is a fundamental correspondence between compact connected
		algebraic, reductive, and contains G as a maximal compact subgroup.
		Any maximal solvable subgroup is called a Borel subgroup of G^c.
	mathforum.org /kb/thread.jspa?threadID=566577&messageID=1690767 (1889 words)

	Diamond Theory References
		Let G be a simple primitive subgroup of Sn, specified in terms of a set of generating permutations.
		If jGj n 5, efficient algorithms are presented that find "the most natural permutation representation" of G. For example, if G is a classical group then we find a suitable projective space underlying G. A number of related questions are considered.
		http://www.geometrie.tuwien.ac.at/havlicek/anothersimple.pdf Pfeiffer, Gotz The subgroups of M24, or how to compute the table of marks of a finite group National University of Ireland, Galway (websites), 1996 Abstract: Let G be a finite group.
	www.log24.com /notes/refs.html (4893 words)

	Welcome to Adobe GoLive 5 (Site not responding. Last check: 2007-11-01)
		Let G be a connected reductive linear algebraic group over the field
		Q of rational numbers, K a maximal compact subgroup and Gamma an
		compact tori in G. These points play a key role in many arithmetic
	www.math.rutgers.edu /seminars/abstracts/paulacohen.html (156 words)

	Proceedings of the American Mathematical Society
		------, Cohomology on complex homogeneous manifolds with compact subvarieties, Contemp.
		W. Schmid and J. Wolf, A vanishing theorem for open orbits on complex flag manifolds, Proc.
		R. Wells, Parametrizing the compact submanifolds of a period matrix domain by a Stein manifold, Symposium on Several Complex Variables, Park City, Utah, 1970 (R. Brooks, ed), Lecture Notes in Math., vol.
	www.mathaware.org /proc/1996-124-03/S0002-9939-96-03153-X/home.html (227 words)

	PROC - Volume 129, Number 5
		Strictly positive definite functions on a compact group
		Gluing copies of a 3-dimensional polyhedron to obtain a closed nonpositively curved pseudomanifold
		The action of a semisimple Lie group on its maximal compact subgroup
	www.ams.org /proc/2001-129-05 (221 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Maple programs for computing discrete series multiplicities More specifically, this is for branching from a connected semisimple real Lie group G to a maximal compact subgroup K. Blattner's formula (in principle) describes the multiplicities.
		Read the (scant) documentation in the dseries file.
		For an explanation of the other columns, see the dseries file.
	www.math.lsa.umich.edu /~jrs/data/blattner/READ_ME (115 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		TECHNION FACULTY OF MATHEMATICS ALGEBRA SEMINAR SPEAKER: Uri Onn, Technion TOPIC: On the Grassmann representation of GL(n) over a local field restricted to the maximal compact subgroup DATE: Monday, May 19, 2003 TIME: 12:30 PLACE: AMADO 719 ABSTRACT: Let F be a local field, G=GL(n,F) and K be the maximal compact subgroup of G.
		We will discuss the representation of K arising from its action on the lattice of subspaces of F^n.
		If time allows we will also discuss partial results on interpolation for arbitrary n (joint with Shai Haran).
	www.math.technion.ac.il /~techm/20030519123020030519onn (132 words)

	Technical Document Style Guide (Site not responding. Last check: 2007-11-01)
		The equation may be either centered or left-justified, but all equations should be positioned in a consistent way.
		To make your equations more compact, you may use the solidus (/), the exp function, or appropriate exponents.
		The Frattini subgroup of a group G contains G' if, and only if, all maximal subgroups of G are normal [2].
	www.msoe.edu /library/technical_style_guide.shtml (12465 words)

	CiteULike: From p-adic to real Grassmannians via the quantum (Site not responding. Last check: 2007-11-01)
		You can view related articles as a TouchGraph (Java required).
		The action of GL(n,F) on the Grassmann variety Gr(m,n,F) induces a continuous representation of the maximal compact subgroup of GL(n,F) on the space of L^2-functions on Gr(m,n,F).
		The irreducible constituents of this representation are parameterized by the same underlying set both for Archimedean and non-Archimedean fields.
	www.citeulike.org /article/92179 (300 words)

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