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Topic: Universal enveloping algebra

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	Algebra
		Homological algebra Homological algebra is that branch of algebraic topology.
		Universal algebra Universal algebra is the field of algebraic structures.
		Virasoro algebra In spanned by elements L The factor of 1/12 is merely a matter of convention.
	www.brainyencyclopedia.com /topics/algebra.html (1269 words)

	Casimir invariant - Wikipedia, the free encyclopedia
		In mathematics, a Casimir invariant of a Lie algebra is a member of the center of the universal enveloping algebra of the Lie algebra.
		The number of independent elements of the center of the universal enveloping algebra is also the rank in the case of a semisimple Lie algebra.
		In any irreducible representation of the Lie algebra, by Schur's Lemma, any member of the center of the universal enveloping algebra commutes with everything and thus is proportional to the identity.
	en.wikipedia.org /wiki/Casimir_invariant (166 words)

	PlanetMath: universal enveloping algebra (Site not responding. Last check: 2007-09-06)
		A universal enveloping algebra of a Lie algebra
		) is isomorphic to the skew polynomial algebra
		This is version 3 of universal enveloping algebra, born on 2002-09-18, modified 2004-09-06.
	evil-wire.luvfeed.org /cache/3797 (163 words)

	Graduate Study in Algebra
		Algebra is one of the oldest branches of mathematics, and the study of algebra in the Department of Mathematics has traditionally been rich and strong.
		The research strengths of the faculty are in the theory of rings (commutative and noncommutative), the theory of groups, algebraic number theory, the representation theory of groups and algebras, and algebraic geometry.
		The core graduate courses in algebra are Abstract Algebra I and II (Math 500 and 501).
	www.math.uiuc.edu /GraduateProgram/researchmath/gradalgebra.html (1660 words)

	Station Information - Universal enveloping algebra
		In mathematics, the universal enveloping algebra construction of abstract algebra is applied to a Lie algebra L in order to pass from a non-associative structure to a more familiar and associative algebra over a field U(L) while preserving the representation theory.
		For example if L is a vector space V as abelian Lie algebra, the left-invariant differential operators are the constant coefficient operators, which are indeed a polynomial algebra in the partial derivatives of first order.
		The algebra of differential operators in n variables with polynomial coefficients may be obtained starting with the Lie algebra of the Heisenberg group.
	www.stationinformation.com /encyclopedia/u/un/universal_enveloping_algebra.html (468 words)

	PlanetMath: Hopf algebra
		The category of commutative Hopf algebras is anti-equivalent to the category of affine group schemes.
		The prime spectrum of a commutative Hopf algebra is an affine group scheme of multiplicative units.
		This is version 9 of Hopf algebra, born on 2002-10-18, modified 2005-07-12.
	planetmath.org /encyclopedia/Antipode.html (266 words)

	Universal Enveloping Algebras
		Universal enveloping algebras are the Lie theoretic analogues of group algebras.
		Nowadays enveloping algebras and their deformations are studied in a number of interrelated fields such as quantum groups, Lie superalgebras or restricted Lie algebras.
		One thus obtains an algebraic family of finite dimensional algebras that is parametrized by the maximal spectrum of O(L).
	www.mathematik.uni-bielefeld.de /~rolf/EnvelopingAlg.htm (469 words)

	Knowledge King - Hopf algebra (Site not responding. Last check: 2007-09-06)
		The other Hopf algebra we can construct is the convolution product algebra of distributions over G. This time, the action of this Hopf algebra upon noncommutative spaces is as a left (right) module.
		Its universal enveloping algebra can be turned into a Hopf algebra by εx=0, Δx=x⊗1+1⊗x and Sx=-x for all elements of the Lie algebra.
		There's an injective homomorphism from this Hopf algebra to the Hopf algebra of convolutions over G such that the image of this homomorphism is the subalgebra generated by the Dirac delta distribution and its derivatives over the identity of G. See also superalgebra, anyonic Lie algebra
	www.knowledgeking.net /encyclopedia/h/ho/hopf_algebra.html (224 words)

	[No title]
		The graded abelian Lie algebra on {xj}, denoted Lab(xj), is the free graded module on the basis {xj}, with the trivial Lie bracket.
		Er be the composition of algebra isomorphisms H(mr) H(ffr-1) (sWr) -- - !
		Morphisms of universal enveloping algebras Let R be a commutative ring, L1 and L2 connected, graded Lie algebras over R which are R-free of finite type.
	hopf.math.purdue.edu /ScottJA/ls-bss.txt (3926 words)

	Hopf algebra -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-06)
		Suppose g is a (Click link for more info and facts about Lie algebra) Lie algebra over the field K and U is its (Click link for more info and facts about universal enveloping algebra) universal enveloping algebra.
		The most exciting Hopf algebras however are certain "deformations" or " (The act of dividing into quanta or expressing in terms of quantum theory) quantizations" of those from example 3 and 4 which are neither commutative nor co-commutative.
		The algebra of all (Click link for more info and facts about continuous function) continuous functions on a (Click link for more info and facts about Lie group) Lie group is a locally compact quantum group.
	www.absoluteastronomy.com /encyclopedia/H/Ho/Hopf_algebra.htm (662 words)

	[No title]
		In a first approach, a quantum algebra can be seen as a deformation of the enveloping algebra of a (simple) Lie algebra; this structure can then be endowed with other quantities that turns it into a quasi-triangular Hopf algebra~\cite{majid}.
		Considering the algebra generated by a subset of such triples (and with the corresponding relations), it is thus rather trivial to construct quantum subalgebras of the original quantum algebra.
		The quantum enveloping algebra $u_q(3)$ as defined here can be made into a Hopf algebra, by defining a co-product, a co-unit and an antipode; however, for the purposes of this paper only the $q$-deformed algebra as determined by (\ref{rel}) is needed.
	allserv.rug.ac.be /~jvdjeugt/files/tex/sharp.tex (4612 words)

	Universal enveloping algebra (Site not responding. Last check: 2007-09-06)
		If L is the Lie algebra corresponding to the Lie group G, U(L) can be identified with the algebra of left-invariant differential operators (of all orders) on G; with L lying inside it as the left-invariant vector fields as first-order differential operators.
		L acts on itself by the Lie algebra adjoint representation, and this action can be extended to a representation of L on U(L): L acts as an algebra of derivations on T(L), and this action respects the imposed relations, so it actually acts on U(L).
		The construction of the group algebra for a given group is in many ways analogous to constructing the universal enveloping algebra for a given Lie algebra.
	www.worldhistory.com /wiki/U/Universal-enveloping-algebra.htm (1044 words)

	Tensor algebra -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-06)
		The construction generalizes in straightforward manner to the tensor algebra of any (A self-contained component (unit or item) that is used in combination with other components) module M over a (Click link for more info and facts about commutative ring) commutative ring.
		The tensor algebra T(V) is also called the (Click link for more info and facts about free algebra) free algebra on the vector space V.
		Because of the generality of the tensor algebra, many other algebras of interest are constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e.
	www.absoluteastronomy.com /encyclopedia/t/te/tensor_algebra.htm (708 words)

	Publications of the network -Algebraic Lie Representations
		Capelli elements in the classical universal enveloping algebra, to appear in Combinatorial methods in Representation theory, Advanced Studies in Pure Mathematics.
		The minimal primitive spectrum of the enveloping algebra of the Lie superalgebra osp(1,2l).
		The Zhang transformation and the annhilation theorem for the quantum enveloping algebra of the Lie superalgebra osp(1,2l).
	home.mathematik.uni-freiburg.de /algebra/TMRLiteraturVerzeichnisNeu.html (697 words)

	Enveloping algebra (Site not responding. Last check: 2007-09-06)
		Construction of the universal enveloping algebra is the reverse process; the important constraint is to preserve the representation theory.
		For example if L is a vector space V as abelian Lie algebra, the left-invariant differentialoperators are the constant coefficient operators, which are indeed a polynomial algebra in the partial derivatives of first order.
		The algebra of differential operators in n variables with polynomial coefficients may be obtained starting with theLie algebra of the Heisenberg group.
	www.therfcc.org /enveloping-algebra-223201.html (614 words)

	Background (Site not responding. Last check: 2007-09-06)
		The universal enveloping algebra U(L) of L is the associative algebra with identity, generated by n symbols which are also denoted by x_1,..., x_n.
		If the Lie algebra L happens to be (split) semisimple and of characteristic 0, then the universal enveloping algebra has a nice basis described by [Kos66].
		In the universal enveloping algebra we use the divided powers y_i^((n)) = (y_i^n/n!), x_i^((n)) = (x_i^n/n!), and the binomials (h_i choose k) = (h_i(h_i - 1)...
	magma.maths.usyd.edu.au /magma/htmlhelp/text1103.htm (293 words)

	IRMA Strasbourg - Publication 2001 (Site not responding. Last check: 2007-09-06)
		The structural feature (including its PBW-basis) of the braided universal enveloping algebra $\Cal U(L)$ of a $\theta$-Lie algebra $L$ is investigated as an object in $\Cal G_{\Lambda,\theta}$ and a class of quantum groups arising from $\Cal U(L)$ is constructed.
		The quantum affine space $k[A_q^{n0}]$, as the braided universal enveloping algebra of an abelian $\theta$-Lie algebra, is a braided Hopf algebra.
		Especially, the quantized universal enveloping algebra of any abelian Lie algebra is found, which is a quantum group associated to the quantum affine space $k[A_q^{n0}]$.
	www-irma.u-strasbg.fr /irma/publications/2001/01026.shtml (179 words)

	[No title] (Site not responding. Last check: 2007-09-06)
		Subject: question for the toplist From: "A Lazarev, Mathematics" Date: Wed, 06 Oct 2004 19:35:24 +0100 Let k be a commutative ring of char 0 and L be a nilpotent graded Lie algebra over k, free and of finite rank as a k-module.
		Let UL be the COMPLETED universal enveloping algebra of L by which I mean inverse lim(UL/I^n) where I is the augmentation ideal of UL.
		Note that in his paper 'Rational homotopy theory' Quillen has a corresponding statement for an ungraded Lie algebra and under the assumption that k be a field.
	www.lehigh.edu /~dmd1/al107.txt (128 words)

	Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebra of gl2 - Doty, Giaquinto ...
		Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebra of gl2 (2000)
		Presenting Schur algebras as quotients of the universal..
		Doty and A.Giaquinto, Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebra of gl(2).
	citeseer.ist.psu.edu /467133.html (453 words)

	Hopf Algebra Structure (Site not responding. Last check: 2007-09-06)
		The Hopf algebra structure that is used by default is the one described in Section Representations of U_q(L).
		Given a quantized universal enveloping algebra U and (anti-) automorphisms f and g of U where g is the inverse of f (this is not checked by Magma) set U to use the corresponding twisted Hopf algebra structure.
		Returns the comultiplication of degree d of the quantized enveloping algebra U. This is a map from U into the d-fold tensor power of U.
	www.math.lsu.edu /magma/text1093.htm (358 words)

	Clearing up the market cycle... best Special Lie Algebra (Site not responding. Last check: 2007-09-06)
		Universal Lie algebra extensions via commutative structures Universal Lie algebra extensions via commutative structures We consider some special type extensions of an arbitrary Lie algebra, which we call universal extensions.
		Primeness Of The Enveloping Algebra Of The Special Lie Superalgebras Primeness Of The Enveloping Algebra Of The Special Lie Superalgebras A primeness criterion due to Bell is shown to apply to the universal enveloping algebra of the Cartan type...
		Lie algebra extensions related with linear bundles of Lie brackets Lie algebra extensions related with linear bundles of Lie brackets We consider some special type extensions of an arbitrary Lie algebra ${\cal G}$, arising in the theory of...
	ascot.pl /th/Fourier5/Special-Lie-Algebra.htm (624 words)

	Introduction (Site not responding. Last check: 2007-09-06)
		This chapter describes the functionality for universal enveloping algebras of Lie algebras in Magma.
		If a Lie algebra is semisimple and defined over a field of characteristic 0, then it is possible to write down an integral basis of the universal enveloping algebra that has nice properties.
		To accomodate this possibility, two constructions of a universal enveloping algebra are provided: a general construction, and one in which this integral basis is used.
	www.math.lsu.edu /magma/text1102.htm (93 words)

	Weyl algebra (Site not responding. Last check: 2007-09-06)
		In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients(in one variable),
		The Weyl algebra is an example of a simple ring that is not a matrix ring over a divisionring.
		It is a quotient of the universal enveloping algebra of the Liealgebra of the Heisenberg group.
	www.therfcc.org /weyl-algebra-203018.html (181 words)

	Abstract from Pacific Journal of Mathematics - 203-1-9 - V.K. Kharchenko (Site not responding. Last check: 2007-09-06)
		We propose a notion of a quantum universal enveloping algebra for any Lie algebra defined by generators and relations which is based on the quantum Lie operation concept.
		This enveloping algebra has a PBW basis that admits a monomial crystallization by means of the Kashiwara idea.
		The similar statement is valid for Hall--Shirshov basis of any Lie algebra defined by one relation, but it is not so in the general case.
	nyjm.albany.edu:8000 /PacJ/2002/203-1-9nf.htm (91 words)

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