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Topic: Lie supergroup

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	Encyclopedia: Lie supergroup
		The concept of supergroup is a generalization of a that of group.
		A Lie supergroup is a supermanifold G together with a morphism
		This is a generalization of a Lie group.
	www.nationmaster.com /encyclopedia/Lie-supergroup (171 words)

	Lie superalgebra (Site not responding. Last check: 2007-11-05)
		In mathematics, a Lie superalgebra is a kind of generalisation of a Lie algebra.
		Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry.
		Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map from itself to itself which respects the Z
	www.encyclopedia-1.com /l/li/lie_superalgebra.html (264 words)

	Lie
		Lie (album) Lie is an album by 1968, its distribution began during the Manson murder trial.
		Representation of a Lie superalgebra In the theory of Lie superalgebras, a representation of a Lie superalgebra L is the...
		Representation of Lie algebras In Lie groups, and g and h are the Lie algebras of G and H respectively, then the induced...
	www.brainyencyclopedia.com /topics/lie.html (506 words)

	Supersymmetry (Site not responding. Last check: 2007-11-05)
		The even part of the star Lie superalgebra is the direct sum of the Poincaré algebra; and a reductive Lie algebra B (such that its self-adjoint part is the tangent space of a real compact Lie group).
		The Lie bracket for the odd part is given by a symmetric intertwiner {.,.} from the odd part "squared" to the even part.
		And just as reps of a Lie algebra can be extended into reps of the canonical Lie group associated with it (basically, the unique Lie group up to isomorphism which is connected AND simply connected), we can also extend reps of a Lie superalgebra into reps of a Lie supergroup (in some cases).
	www.1-free-software.com /en/wikipedia/s/su/supersymmetry.html (1408 words)

	Lie subgroup Definition / Lie subgroup Research
		Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of analytical structures.
		Then the Lie algebra In mathematics, a Lie algebra (named after Sophus Lie, pronounced "lee") is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds.
		Lie algebras were introduced to study the concept of infinitesimal transformations....
	www.elresearch.com /Lie_subgroup (273 words)

	[No title]
		Lie superalgebras \endheading \smallpagebreak A {\it Lie superalgebra} over $\L$ is a free graded $\L$-module, $\g$, endowed with a graded Lie bracket, which is bi-$\L$-linear, anticommutative and satisfies the graded Jacobi identity.
		The Lie algebra of $\dot I_q^0$ is canonically isomorphic to the underlying Fr\'echet-Lie algebra of $I_q^0$, and the corresponding exponential mapping is identity, ${\Bbb I}d: I_q^0\to\dot I_q^0$.
		Lie supergroups \endheading \smallpagebreak A {\it Banach-Lie supergroup} modeled over a free graded $\L$-module $M$ is a group object in the category of Banach-Lie supermanifolds over $\L$.
	www.ma.utexas.edu /mp_arc/e/92-77.amstex (6301 words)

	Supersymmetry
		For each Lie algebra, there exists an associated Lie group which is connected and simply connected.
		Unique up to isomorphism, this Lie group is canonically associated with the Lie algebra, and the algebra's representations can be extended to create group representations.
		In particular, its reduced intertwiner from to the ideal of the Poincaré algebra generated by translations is given as the product of a nonzero intertwiner from to (1/2,1/2).
	pedia.newsfilter.co.uk /wikipedia/s/su/supersymmetry.html (1446 words)

	Encyclopedia: Group object (Site not responding. Last check: 2007-11-05)
		A topological group is a group object in the category of topological spaces with continuous functions.
		A Lie group is a group object in the category of smooth manifolds with smooth maps.
		A Lie supergroup is a group object in the category of supermanifolds.
	www.nationmaster.com /encyclopedia/Group-object (633 words)

	Symplectic geometry seminar (Site not responding. Last check: 2007-11-05)
		We then extend this definition to construct twisted representation rings for Lie supergroups.
		As an application, given a compact Lie group, we can consider its parity reversed tangent bundle as a Lie supergroup, whose representation ring agrees with that of the original group via a Thom isomorphism.
		If we have a Lie subgroup H of maximal rank inside a compact Lie group G, then we show that the induction map from the representation ring of TH to that of TG is in fact the Dirac induction map from R(H) to R(G).
	www.math.toronto.edu /symplec/fall04/sem110804.html (120 words)

	[No title]
		The Lie superalgebra $q(2)$ has a basis consisting of 4 even elements $e_{ij}^{\bar 0}$ ($i,j=0,1$) and 4 odd elements $e_{ij}^{\bar 1}$ ($i,j=0,1$), satisfying the bracket relation \beq \lb e_{ij}^\si, e_{kl}^\theta \rb = \de_{jk} e_{il}^{\si+\theta} - (-1)^{\si\theta} \de_{il} e_{kj}^{\si+\theta}, \eeq where $\si,\theta\in\Z_2=\{\bar 0, \bar 1\}$, and $i,j,k,l\in\{0,1\}$.
		Here, $\lb\,,\,\rb$ stands for the Lie superalgebra bracket, which could be a commutator or an anti-commutator, depending on the grading of the elements considered.
		Phys.} {\bf 40} 147 %-158 \bibitem{PS2} Penkov I and Serganova V 1997 %Characters of irreducible $G$-modules and cohomology of $G/P$ %for the Lie supergroup $G+Q(N)$ {\em J. Math.
	allserv.rug.ac.be /~jvdjeugt/files/tex/q2realizations3.tex (3981 words)

	[No title]
		The importance of Lie algebra cohomology for physics in general was in the air lately; conferences entitled ``Cohomological Physics'' started to mushroom.
		The Lie superalgebra $\fgl (Par)$ is generated by the generators of its maximal nilpotent subalgebras, the upper and the lower triangular ones.
		Denote the elements of the Lie algebra $\fsp (2m; \Ree) \subset \fh (M)$ by $$ e^{ii} = y_{i}\partial _{i}, \; \; \; e_{ii} = x_{i}\delta _{i}, \; \; \; \; \; \; e^{i}_{j} = x_{j}\partial _{i} + y_{i}\delta _{j}.
	math.la.asu.edu /~sf2000/leites.tex (9320 words)

	Fowell et al. 1994
		The Newark Supergroup is composed predominantly of fluvial and lacustrine strata that fill a series of rift basins formed by the Pangean breakup.
		The regularity and duration of the Newark Supergroup lacustrine cycles are compatible with the hypothesis that orbitally controlled seasonal and latitudinal variations in solar radiation detennined the length and intensity of Mesozoic rainy seasons.
		The abruptness and apparent synchroneity of the tetrapod, ichnofossil, and palynofloral extinctions in the Newark Supergroup are consistent with catastrophic extinction scenarios involving volcanism or bolide impacts.
	bcornet.tripod.com /MassExt/Fowell94.htm (3866 words)

	[No title]
		The first is the Lie superalgebra $gl(n/1)$, for which the representations (even the atypical ones) are now well known~: e.g., a Gelfand-Zetlin basis has been introduced and its transformations have been determined~\cite{pal78,pal82}, and representations of $sl(n/1)$ have been studied~\cite{pal79,pal81}.
		In the present paper, the purpose is to study a new class of irreducible finite-dimensional representations of the Lie superalgebra $q(n)$, which have certain properties that are required in a physical context.
		Recall that $Q(n)$ is a simple Lie superalgebra for $n\geq 2$, and that it is one of the series of classical (but ``strange'') Lie superalgebras in the classification of Kac~\cite{Kac1}.
	allserv.rug.ac.be /~jvdjeugt/files/tex/qn3.tex (5111 words)

	On Enlargability Of Infinite-Dimensional Lie Superalgebras (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		We show that in infinite dimensions a Lie superalgebra coming from a Rogers supergroup may not come from a DeWitt one.
		Thus, we produce an evidence that a whole class of Lie superalgebras can be enlarged only by means of the "naive" approach to supermanifolds, which therefore cannot be merely thrown away.
		Introduction The term "enlargability" in Lie theory means a possibility to associate a Lie group to a Lie algebra; those Lie algebras coming from Lie groups are called "enlargable Lie...
	citeseer.lcs.mit.edu /22083.html (387 words)

	International Journal of Mathematics and Mathematical Sciences (Site not responding. Last check: 2007-11-05)
		We propose and study an appropriate analog of normal Lie subgroups in the supergeometrical context.
		We prove that the ringed space obtained taking the quotient of a Lie supergroup by a normal Lie subsupergroup, is still a Lie supergroup.
		We show how one can construct Lie supergroup structures over topologically nontrivial Lie groups and how the previous property of normal Lie subsupergroups can be used, in order to explicitly obtain the coproduct, counit, and antipode of these structures.
	www.hindawi.com /journals/ijmms/volume-30/S0161171202012395.html (368 words)

	Rutgers Topology/Geometry Seminar
		Using ideas of twisted equivariant K-theory, we construct twisted versions of the representation rings for Lie superalgebras and Lie supergroups, built from projective Z_2-graded representations with a given cocycle.
		As an application, given a compact Lie group G, we construct a Lie supergroup by taking the cotangent bundle T*G and reversing the parity of its fibers.
		An inclusion of Lie groups H --> G induces a pushforward homomorphism between the twisted representation rings of the corresponding Lie supergroups TH and TG, which pulls back via a Thom isomorphism to give an additive homomorphism from R(H) --> R(G).
	www.math.rutgers.edu /~ctw/seminar (780 words)

	TWD -- Creationism and the Grand Canyon: The Colorado Plateau
		These fossils are used to assign the Supergroup rocks to the interval of time called the Proterozoic, the "age of first life." The Cardenas Lava can be dated directly using radiometric and paleomagnetic methods; the results of these tests match the stratigraphic indicators of the Supergroup's age.
		The Grand Canyon Supergroup is made of sediments that were deposited and lithified (turned to stone) late in Creation Week and in the 1600 years between the Creation and the Flood (Austin p.66).
		From there on up, the Grand Canyon Supergroup rocks record a series of transgressions and regressions, indicating the area was low and flat, and that as the sea rose and fell, the environment cycled repeatedly between coastal plain, tidal flats, and shallow offshore shelf.
	www.erinet.com /jwoolf/gc_rocks.html (7502 words)

	Supersymmetry - Wikipedia, the free encyclopedia
		If we relax the condition from symmetry groups to symmetry supergroups, we arrive at SUSY.
		In the same way, representations of a Lie superalgebra can sometimes be extended into representations of a Lie supergroup.
		See supersymmetry algebra for a more detailed discussion, including a description of SUSY in Minkowski spacetime.
	en.wikipedia.org /wiki/Supersymmetry (2242 words)

	Sherpa Guides \| Tennessee \| The Tennessee Mountains \| Upper Unaka Mountains
		Throughout most of Tennessee, ancient rocks formed during the Precambrian Era over 1 billion years ago lie below the surface of sedimentary rock; however, in the Unaka Range the ancient rocks are exposed.
		About 600 million years ago, sediments collected in a sinking trough (geosyncline) on the Unakas' western side and formed the Ocoee Supergroup and Chilhowee Group that make up most of the present-day Unaka Range.
		A period of folding took place about 470 million years ago during the Ordovician Period, when the older rock was forced over the younger rock of the Ocoee and Chilhowee formations.
	www.sherpaguides.com /tennessee/upper_unakas (998 words)

	Ian M. Musson: Papers/Preprints
		It is further shown that the finite dimensional irreducible modules over a (not necessarily classical) finite dimensional complex Lie superalgebra form a complete set if and only if the even part of the Lie superalgebra is reductive and the universal enveloping superalgebra is semiprime.
		The invariant $k(\cal O)$ is in many cases, equal to the odd dimension of the orbit $G\cdot\cal O$ where $G$ is a Lie supergroup with Lie superalgebra ${\mathfrak g}$.
		Abstract: If $\fg$ is a semisimple Lie algebra, we describe the prime factors of $\mcU(\fg)$ that have enough finite dimensional modules.
	www.uwm.edu /%7Emusson/preprints.html (998 words)

	Quantum supergroup structure of (1+1)-dimensional quantum superplane
		Quantum supergroup structure of (1+1)-dimensional quantum superplane, its dual and its differential calculus
		We show that the (1+1)-dimensional quantum superplane introduced by Manin is a quantum supergroup, according to the Faddeev-Reshetikhin-Takhtajan approach, when it is extended by the inverse of the bosonic variable.
		Second, we construct a right-invariant differential calculus on it and then deduce the corresponding quantum Lie superalgebra which as a commutation superalgebra appears classical, and as Hopf structure is a non-cocommutative q -deformed one.
	stacks.iop.org /0305-4470/34/3403 (278 words)

	Twist deformation of the rank-one Lie superalgebra
		It is difficult to overestimate the role of the rank-one Lie algebra sl (2) in the theory of Lie groups and their applications.
		constructed in [ 3 ] - [ 5 ], is the corresponding analogue for the quantum supergroups.
		Although the deformed Lie superalgebra is finite dimensional it can be used for further deformation of infinite-dimensional Hopf superalgebras (e.g.
	ej.iop.org /EJ/article/0305-4470/31/4/001/ja31004l1.html (1135 words)

	[No title]
		We shall represent the Chern class of a complex line superbundle by means of the curvature of a canonical connection which, is a sense, it is determined by the bundle itself.
		Finally, the Lie supergroup $UOSP(1,2)$ is defined to be the exponential map of $uosp(1,2)$, \be UOSP(1,2) =: \{exp(X) ~~ X \in uosp(1,2)\}~.
		The total space is the $(1,2)$-dimensional supergroup $UOSP(1,2)$ while the structure supergroup is $\cu(1)$.
	www.ma.utexas.edu /mp_arc/papers/99-323 (5320 words)

	Dawn's C-M Age Abstract (Site not responding. Last check: 2007-11-05)
		This new age demonstrates that the Campbellrand-Malmani carbonate platform was deposited during late Archean time, and is the oldest extensively preserved carbonate platform known.
		Although a previously reported Pb-Pb age of 2557±49 Ma for dolomites at the base of the Campbellrand Subgroup is broadly consistent with this U-Pb zircon age, new Pb-isotopic data for carbonate cements and massive sulfides lie along the same Pb-Pb array as the dolomites.
		This relationship implies that the "isochron" is the result of post depositional resetting of the U-Pb system in dolomite by fluids responsible for sulfide mineralization, and that the age which corresponds to its slope has no geological significance.
	www.gps.caltech.edu /users/sumner/Ddate.html (172 words)

	Coleman-Mandula theorem - Wikipedia, the free encyclopedia
		Supersymmetry may be considered the only possible "loophole" of the theorem because it contains additional generators ( supercharges) that are not scalars but rather spinors.
		This loophole is possible because supersymmetry is a Lie supergroup, not a Lie group.
		This page was last modified 05:29, 19 February 2005.
	www.wikipedia.org /wiki/Coleman-Mandula_theorem (172 words)

	[No title]
		If this were not the case the weak kernel of $\Omega$ would impose algebraic conditions on $\Gamma$, and the strong kernel of $\Omega$ (defined as the set of supervector fields $Z$ such that $\Omega (Z,X)=0$, $\forall X$) defines an invariant Lie superalgebra, the gauge superalgebra of $\Gamma$ and $L$.
		Under such circumstances, the tangent supermanifold and the dynamical equation can be reduced quotienting out the gauge degrees of freedom defined by the strong kernel of $\Omega_L$.
		In other words, the linear symmetry supergroup for $\Gamma$ is the supergroup of linear transformations $Gl(2n2n, \R)$.
	www.ma.utexas.edu /mp_arc/e/92-189.latex (3075 words)

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