
	Where results make sense

Topic: Symplectic group

In the News (Mon 13 Feb 12)

EXECUTIVE POWER

Iran

ANALYSIS

Pakistan

Family compact

	What IS a Lie Group?
		F4 is the automorphism group of 3x3 matrices of octonions o11 o12 o13 o21 o22 o23 o31 o32 o33 such that o11, o22, and o33 are real (have no imaginary part), and o12, o13, o23 are the octonion conjugates of o21, o31, o32 respectively.
		Are we happy with G2 as the automorphism group of the octonions, F4 as the isometry of the [octonion] projective plane, E6 (in a noncompact form) as the collineations of the same, and E7 resp.
		A Lie algebra is a logarithm of a Lie group, and a Lie group is an exponential of a Lie algebra.
	www.valdostamuseum.org /hamsmith/Lie.html (3638 words)

	Symplectic group: Just the facts... (Site not responding. Last check: 2007-10-19)
		In mathematics (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement), the name symplectic group can refer to two different, but closely related, types of mathematical groups ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule).
		The latter is sometimes called the compact symplectic group to distinguish it from the former.
		Since all symplectic matrices have unit determinant (A determining or causal element or factor), the symplectic group is a subgroup ((mathematics) a subset (that is not empty) of a mathematical group) of the special linear group (additional info and facts about special linear group) SL(2n, F).
	www.absoluteastronomy.com /encyclopedia/s/sy/symplectic_group.htm (810 words)

	Symplectic matrix - Wikipedia, the free encyclopedia
		The abstract analog of a symplectic matrix is a symplectic transformation of a symplectic vector space.
		Briefly, a symplectic vector space is a 2n-dimensional vector space V equipped with a nondegenerate, skew-symmetric bilinear form ω.
		The condition that L be a symplectic transformation is precisely the condition that M be a symplectic matrix:
	en.wikipedia.org /wiki/Symplectic_matrix (465 words)

	20: Group Theory and Generalizations
		Group theory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way.
		Groups acting on topological spaces are the basis of equivariant topology and homotopy theory in Algebraic Topology.
		Nielsen's theorem: subgroups of free groups are free.
	www.math.niu.edu /~rusin/known-math/index/20-XX.html (2774 words)

	ipedia.com: Orthogonal group Article (Site not responding. Last check: 2007-10-19)
		In mathematics, the orthogonal group of degree n over a field F) is the group of n -by- n orthogonal matrices with entries from F, with the group operation that of matrix multiplication.
		In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication.
		For n > 2 the fundamental group of SO(n,C) is Z/2Z whereas of the fundamental group of SO(2,C) is Z.
	www.ipedia.com /orthogonal_group.html (610 words)

	Lie group - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-19)
		In mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps.
		Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of analytical structures.
		If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra g over F there is a unique (up to isomorphism) simply connected Lie group G with g as Lie algebra.
	www.xahlee.org /_p/wiki/Lie_group.html (1378 words)

	[ref] 48 Group Libraries
		is the analogous one for the corresponding unitary groups.
		The groups are sorted by their orders and they are listed up to isomorphism; that is, for each of the available orders a complete and irredundant list of isomorphism type representatives of groups is given.
		All groups in the library are primitive permutation groups of the indicated degree.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/doc/htm/ref/CHAP048.htm (7861 words)

	School of Mathematics - Lie Groups, Representations and Discrete Mathematics
		In recent years new and important connections have emerged between discrete subgroups of Lie groups, automorphic forms and arithmetic on the one hand, and questions in discrete mathematics, combinatorics, and graph theory on the other.
		A new combinatorial construction of expander graphs was used recently to resolve a group theoretic question on expansion in Cayley graphs.
		Information about a workshop in a related topic, "Automorphic Forms, Group Theory and Graph Expansion" to be held in February 2004 at IPAM can be found at: http://www.ipam.ucla.edu/programs/agg2004/.
	math.ias.edu /pages/activities/special-programs/lie-groups-representations-and-discrete-mathematics.php (346 words)

	Document sans titre
		Suppose that P --> M is a principal bundle with strcture group the loop group LG (LG is the space of smooth maps from the circle to G a Lie group).
		The string class of P is the class in H^3(M;Z) obstructing the existence of a lift of the structure group of P to LG^, the Kac-Moody group.
		A symplectic Lie group is a Lie group G together with a left invariant symplectic 2-form $Omega$.
	www.math.psu.edu /cgmp/seminars/seminarsSpring2004.htm (1345 words)

	[No title]
		By the definition, a symplectically aspherical manifold is a symplectic manifold whose symplectic form is symplectically aspherical.
		The group Zm cannot be realized as the fundamental group of a symplectically aspherical manifold of dimension 2k with 2k > m.
		One more sourse of symplectically aspherical groups comes from the observation of Gompf [G2, Lemma 1] who proved that a branched cov- ering over a 4-dimensional symplectically aspherical manifold is sym- plectically aspherical.
	hopf.math.purdue.edu /Ibanez-Rudyak-Tralle/aspherical.txt (3988 words)

	[No title]
		The group $Sp(\infty)$ we deal with here is the set of invertible operators $g=1+A$ where $g$ preserves a symplectic form on an infinite dimensional vector space and $A$ is of finite rank.
		This group is essentially same as the inductive limit of the classical symplectic group in the sense that the latter group is dense in operator norm topology.
		The symplectic group $Sp(N)$ is non compact on one hand, and the CCR algebra is an unbounded operator algebra.
	www.ma.utexas.edu /mp_arc/papers/02-178 (4586 words)

	Matrix Groups of Large Degree
		If the absolutely irreducible group G preserves a symplectic form (non-degenerate, alternating bilinear) modulo scalars, this function returns the scalars corresponding to the generators of the group of the form.
		If the absolutely irreducible group G preserves a unitary form modulo scalars, this function returns the scalars corresponding to the generators of the group of the form.
		Given a group G of d x d matrices over a finite field E having degree e and a subfield F of E having degree f, write the matrices of G as de/f by de/f matrices over F and return the group and the isomorphism.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text301.htm (3911 words)

	Symplectic Group Sp(2n), properties and analysis -- from Mathematica Information Center
		An abstract of the main properties of the symplectic group, important in Hamilton Mechanics, is given as comments in a first part.
		Various programming of symplectic matrices and of their Symplectic Jordan Form (a new item I believe) are given in a second part.
		In a third part, the analysis of a given symplectic matrix is performed, using a "smoothing" of eigenvalues to identify degeneracies that Eigenvalues[] does not reveals by lack of precision when a matrix has less eigenvectors than eigenvalues.
	library.wolfram.com /infocenter/MathSource/4779 (142 words)

	Symplectic and Contact Geometry - Summer Tutorial 2003
		Symplectic manifolds are an intermediate case between real and complex (Kahler) manifolds.
		Symplectic manifolds still play an important role in recent topics in physics, such as string theory.
		Contact manifolds are the odd-dimensional analogues of symplectic manifolds, and they appear naturally when one is interested in symplectic structures on manifolds with boundary.
	www.math.columbia.edu /~cm/symcon.html (529 words)

	Introduction (Site not responding. Last check: 2007-10-19)
		In this paper we give a classification of the conjugacy classes of the odd symplectic groups together with a representative of each class, called a normal form.
		The odd symplectic groups are nonreductive and are designed to interpolate the classical even ones.
		1], used to determine the conjugacy classes in the even symplectic groups, to find the conjugacy classes and normal forms for the odd symplectic groups.
	www.maths.warwick.ac.uk /~bww/symplectic/node1.html (580 words)

	Standard Groups (Site not responding. Last check: 2007-10-19)
		Construct the orthogonal group Omega(n, K) (which is the kernel of the spinor norm map on SO(n, K)) corresponding to the n-dimensional vector space V over the field K = GF(q), where n is odd, n >= 3 and q is a prime power.
		The Suzuki groups are specified slightly differently, as the degree of the group is always four.
		Given a matrix group G and a permutation group H, construct action of the wreath product on the tensor power of G by H, which is the (image of) the wreath product in its action on the tensor power (of the space that G acts on).
	www.math.uga.edu /~matthews/DOCS/MAGMA/text306.html (1367 words)

	Contents (Site not responding. Last check: 2007-10-19)
		By definition, a symplectic form on a finite-dimensional vector space E is a non-degenerate anti-symmetric bilinear form $\sigma$ on E.
		A symplectic form on a smooth manifold M is a closed smooth differential form $\sigma$ of degree two on M such that, for each m in M, the bilinear form $\sigma_m$ is non-degenerate.
		The first case is the setting of classical mechanics with a symmetry group; quantization of such a situation leads to questions of representation theory of K and spectral theory of differental operators commuting with (spectral degeneration).
	www.math.uu.nl /people/kolk/SpringSchool2004/contents.html (1791 words)

	Re: Weyl group, Symplectic Group, and John Baez
		In article <543dypfo0f.fsf@intech19.enhanced.com>, Camm Maguire wrote: >One bit of group theory which has always fascinated me is Weyl's >comment in his book that the definitions of the operators p and x in >quantum mechanics can basically be seen as generators of a non-trivial >projective representation of R(2).
		Here's one: when we quantize a symplectic manifold using geometric quantization, we need to pick an additional structure called a "polarization", and the symplectic transformations that preserve the polarization then become easier to quantize than the rest.
		I prefer to think of quantization of bosons and fermions in a somewhat different way, which emphasizes the analogy between the symplectic group (which preserves a nondegenerate skew-symmetric form) and the orthogonal group (which preserves a nondegenerate symmetric form).
	www.lns.cornell.edu /spr/1999-07/msg0017253.html (787 words)

	PlanetMath: existence and uniqueness of compact real form (Site not responding. Last check: 2007-10-19)
		, the complex symplectic group, is less well-known.
		Cross-references: conjugate transpose, matrices, quaternion, orthogonal group, special orthogonal group, unitary, special linear group, involution, fixed points, group, real form, compact, real, isomorphism, Lie group, complex, semisimple
		This is version 2 of existence and uniqueness of compact real form, born on 2003-01-27, modified 2003-08-21.
	planetmath.org /encyclopedia/ExistenceAndUniquenessOfCompactRealForm.html (145 words)

	RMK articles : Orthogonal, Complex, Symplectic to Unitary (Site not responding. Last check: 2007-10-19)
		Arnold in his book on "Mathematical Methods of Classical Mechanics" brings to attention the fact that on the space R^2N, the linear transformations that preserve the euclidean structure form the Orthogonal group O(2n), that preserve the symplectic structure form the Symplectic group Sp(2n), and that preserve the complex structure form the Complex linear group GL(n,C).
		These three groups have a common intersection (equal to the intersection of any pair) which is the Unitary group U(n).
		There seems to be some evidence (due to the quaternion representations) that the Symplectic group has the germ of Fermion Spin.
	www22.pair.com /csdc/pd2/pd2fre30.htm (221 words)

	CCR AND THE m DIMENSIONAL HEISENBERG ALGEBRAS
		The matter of the group being toroidal or translational is a matter of the group's global topology; the algebra, however, speaks only to a local topology.
		F is then a symplectic, and formally unitary transformation of H which preserves CCR; preservation of the CCR is preservation of the symplectic structure of commutators.
		But this inhomogeneous group acts naturally on the noncommutative phase space (q, p); it is not a commutative geometry because the "coordinates" q and p to not commute.
	graham.main.nc.us /~bhammel/PHYS/heisalg.html (4859 words)

	ATLAS: Symplectic group S4(7)
		FACT: This is the smallest simple group whose order is a proper power.
		a and b as permutations on 400 points - action on isotropic lines of the symplectic space.
		(7)), the point stabiliser (symplectic); the isotropic line stabiliser (orthogonal).
	web.mat.bham.ac.uk /atlas/html/S47.html (202 words)

	Puzzle reply
		real-valued metric and symplectic structure on the quaternions.
		The reason is that the group of automorphisms of the quaternions
		group of n x n quaternionic matrices preserving lengths.
	world.std.com /~sweetser/quaternions/spr/puzzlereply.html (515 words)

	Hermann Weyl (Site not responding. Last check: 2007-10-19)
		From 1923 to 1938, Weyl developed the theory of compact groups, in terms of matrix representations.
		Non-compact groups and their representations, particularly the Heisenberg group, were also deeply involved.
		It covered symmetric groups, full linear groups, orthogonal groups, and symplectic groups and results on their invariants and representations.
	www.worldhistory.com /wiki/H/Hermann-Weyl.htm (1325 words)

	Réunion d'hiver 2002 de la SMC
		For G=U(n,H), the compact symplectic group (also described as the quaternionic unitary group), however, the decompositions are not multiplicity-free.
		On the level of loop groups, however, there does not seem to be a suitable generalisation.
		It is well-known that an orbifold M, all of whose stabilizer group actions is effective (an ``effective'' or ``reduced'' orbifold) can be presented as M = P/K, where P is a smooth manifold and K is a compact Lie group.
	camel.math.ca /CMS/Events/winter02/abs/sg.f (1848 words)

	vsevcosmos: The reduced spaces of a symplectic Lie group action. [math.SG/0501098] (Site not responding. Last check: 2007-10-19)
		The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, and Molino.
		Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space.
		We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action.
	www.livejournal.com /users/vsevcosmos/7433590.html (257 words)

Try your search on: Qwika (all wikis)