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	Encyclopedia :: encyclopedia : Witt algebra (Site not responding. Last check: 2007-10-21)
		In mathematics, a Witt algebra is the Lie algebra of Killing vector fields defined on the Riemann sphere.
		Witt algebras occur in the study of conformal field theory.
		Consider the commutative algebra of functions with two arguments which are holomorphic in z in the first argument and antiholomorphic in z in the second argument.
	www.hallencyclopedia.com /topic/Witt_algebra.html (253 words)

	PlanetMath: Witt vectors (Site not responding. Last check: 2007-10-21)
		In this entry we define a commutative ring, the Witt vectors, which is particularly useful in number theory, algebraic geometry and other areas of commutative algebra.
		is the projective limit of the Witt vectors of finite length.
		This is version 2 of Witt vectors, born on 2005-05-06, modified 2005-05-06.
	planetmath.org /encyclopedia/WittVectors.html (165 words)

	Springer Online Reference Works
		Witt [1] in 1936 in the context of the description of unramified extensions of
		Witt vectors were subsequently utilized in the study of algebraic varieties over a field of positive characteristic [3], in the theory of commutative algebraic groups [4], [5], and in the theory of formal groups [6].
		) is known as a Witt scheme (or a truncated Witt scheme) and is a ring scheme [3].
	eom.springer.de /w/w098100.htm (658 words)

	New Simple Lie algebras: Melikian Algebras
		Later on several generalizations of the Witt algebra were constructed (by Jacobson, Zassenhaus, Frank, Kaplansky).
		Later on A. Premet [P1] proved that any non-classical simple Lie p-algebra over an algebraically closed field of characteristic p=5 with a torus of maximal dimension 2 is either of Cartan type or is isomorphic to the Melikian algebra L(1,1), which I constructed in [M1], [M2].
		The algebra L(1,1) does not have analogs in characteristics p>5 or characteristic zero.
	www.cs.uwm.edu /~melikian/res/node2.html (341 words)

	Re: What's wrong with string theory?
		We have seen that the Cuntz algebra is a noncommutative generalization of the algebra of functions on a circle.
		Thus our centrix Lie algebra should be thought of as the corresponding noncommutative generalization of the Lie algebra of vector fields on the circle.
		The algebras that I described are the natural generalizations of Virasoro to higher dimensions: abelian extensions of the diffeomorphism algebra (= generalized Witt algebra = algebra of vector fields) by the module of closed dual one-forms.
	www.lns.cornell.edu /spr/2001-01/msg0030595.html (449 words)

	Pharyngula::Witt in the Seattle Times
		I'm afraid all of Witt's claims of scientific legitimacy for the book are false, and I suspect that he knows it, or he wouldn't have been reluctant to mention the title.
		Witt's next complaint is to bring up the "martyrdom" of Richard Sternberg, who apparently was under attack by the PZ Myers Playbook.
		Witt recites the complaints received by the US Office of the Special Counsel from Sternberg, but doesn't bother to mention that the OSC dismissed his complaints, and that he's still working at the Smithsonian…some martyr!
	pharyngula.org /index/weblog/comments/witt_in_the_seattle_times (1977 words)

	13: Commutative rings and algebras
		Of particular interest are several classes of rings of interest in number theory, field theory, algebraic geometry, and related areas; however, other classes of rings arise, and a rich structure theory arises to analyze commutative rings in general, using the concepts of ideals, localizations, and homological algebra.
		Typically one classifies problems as Algebraic Geometry when stated in terms of points, hypersurfaces, divisors, and other geometric objects, and as Commutative Algebra when stated in terms of ideals and coordinate rings, although in practice techniques from both areas are used in tandem.
		The algebraic study of general collections of polynomials is appropriate for this field; the study of individual polynomials or specific collections usually belongs elsewhere.
	www.math.niu.edu /~rusin/known-math/index/13-XX.html (2760 words)

	Dear Stefan:
		This Hopf algebra is one of particular importance among several ``combinatorial Hopf algebras'' that have arisen recently in different contexts, like Gessel's Hopf algebra of quasi-symmetric functions and Loday-Ronco's Hopf algebra of planar binary trees.
		These Hopf algebras are also known as the restricted specializations of quantized enveloping algebras associated to the semisimple Lie algebra algebra g.
		In fact this Hopf algebra is the degree zero part of a larger one which is based on all the planar trees.
	condor.depaul.edu /~scatoiu/seminar/february2002/abstracts.html (1659 words)

	Springer Online Reference Works
		This Hopf algebra is important, e.g., in the theory of curves of non-commutative formal groups (see Formal group) [a1], [a2], [a5].
		(see Witt vector) and it plays an important role in the classification theory of unipotent commutative algebraic groups and in the theory of commutative formal groups (amongst other things) [a3].
		is the free commutative polynomial algebra with as generators the elementary concatenation powers of elementary Lyndon words (see Lyndon word).
	eom.springer.de /L/l110100.htm (256 words)

	Schur duality for the Cartan type Lie algebra (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		As a result, we get a kind of Schur duality between W (n) and a finite dimensional non-semisimple algebra, which is the semi-group ring of the transformation semigroup Tm.
		Introduction Cartan type Lie algebras are Lie subalgebras of algebraic vector fields on a flat affine space F n, where F is a field of...
		1 Commutant algebra and harmonic polynomials of a Lie algebra..
	citeseer.ist.psu.edu /104254.html (337 words)

	Manchester Geometry Seminar (Site not responding. Last check: 2007-10-21)
		Connection between right-symmetric and Lie algebra cohomologies was established.
		It was used in calculation of "hidden" deformations and non-split extensions of four Cartan series of Lie algebras of vector fields.
		Lie algebras of divergence-free vector fields and Hamiltonian vector fields have new central extensions,
	www.maths.man.ac.uk /~tv/Seminar/1999-2000/askar.html (111 words)

	What/why these several algebras?
		Simple finite-dimensional Lie algebras were classified by Cartan and Dynkin in the first half of the previous century.
		W_2 is Virasoro and W_1 is an infinite-dimensional Heisenberg algebra (Note: W_2 (as a W algebra) = W_1 (as a Witt algebra), apart from the central extension!).
		A Lie superalgebra is a Lie algebra g with a Z_2-grading.
	www.lns.cornell.edu /spr/2000-11/msg0029874.html (1011 words)

	Previous Rutgers Algebra Seminars (Since 1995)
		The latter is an algebraic abstraction of the topological equivariant cohomology theory for G-spaces, where G is a compact Lie group.
		The standard way to build flag algebras from a set of flags is to use the determinant to coordinatize the latter (then the former is just the polynomial algebra in the coordinate functions for these coordinates).
		Given an open-string vertex algebra, we show that there exists a vertex algebra, which we call the "meromorphic center" inside the original algebra such that the original algebra yields a module and also an intertwining operator for the meromorphic center.
	www.math.rutgers.edu /~weibel/oldalgebra.sem.html (8477 words)

	Michiel Hazewinkel : Publications
		On Lie algebras of vector fields, Lie algebras of differential operators and nonlinear filtering.
		Operations in the K-theory of endomorphisms, J. of Algebra 84 (1983), 285--304.
		Witt vectors and Leibniz-Hopf algebra, symmetric functions and quasi-symmetric functions.
	homepages.cwi.nl /~mich/publications.html (3419 words)

	Papers and Preprints
		A class of finite commutative rings constructed from Witt rings, (with M. Vo), Bull.
		A sufficent condition for strict total positivity of a matrix, (with G. Csordas), Linear and Multilinear Algebra 45 (1998), 19-34.
		The map of the Witt ring of a domain into the Witt ring of its field of fractions, (with A. Rosenberg and R. Ware), Proc.
	www.math.hawaii.edu /~tom/papers.html (630 words)

	Construction of Lie Algebras
		The Witt algebra W(m, n) is simple unless p=2 and m=1 ([SF88, 4.2.4(1)]) and is restrictable if and only if n=[1,..., 1] ([SF88, 4.2.4(2)]).
		The algebra H(m, n) is the Hamiltonianand CH(m, n) is the conformal Hamiltonian Lie algebra.
		Construct the Witt algebra W(m, n) over the finite field F, where m must be a positive integer and n a sequence of positive integers of length m.
	www.math.lsu.edu /magma/text1062.htm (2092 words)

	Saskatchewan Algebra and Number Theory Mini-meeting (Site not responding. Last check: 2007-10-21)
		Abstract: The derivation algebra of the ring of complex Laurent polynomials is one of the most important examples of a simple infinite dimensional Lie algebra.
		It is called the Witt algebra, and is isomorphic to the Lie algebra of complex vector fields on the circle.
		Abstract: An algebraic variety is usually locally described in terms of equations, parametrization or in terms of its ring of regular functions.
	www.math.uregina.ca /~aherman/skant02.html (1548 words)

	(my) math reviews
		MR 94k:17038 K.V. Kozerenko, Main theorems of invariant theory for the Lie algebra sl(2) in the case of a field of finite characteristic, Unconventional Lie Algebras, 75-102, AMS, 1993.
		Poncin, On the cohomology of the Nijenhuis-Richardson graded Lie algebra of the space of functions of a manifold, J. Algebra 243 (2001), 16-40.
		MR 2005i:16013 A.I. Generalov, Hochschild cohomology of dihedral-type algebras.
	www.justpasha.org /math/rev (1154 words)

	Chari, Pressley: Unitary representations of the Virasoro algebra and a conjecture of Kac
		Chari, Pressley: Unitary representations of the Virasoro algebra and a conjecture of Kac
		Unitary representations of the Virasoro algebra and a conjecture of Kac.
		Proceedings of the Montreal workshop on infinite-dimensional Lie algebras and their applications, S. Kass, ed., to appear.
	www.numdam.org /item?id=CM_1988__67_3_315_0 (195 words)

	[No title]
		This algebra A(1) is built from a homogeneous pre-Koszul algebra A that has a basis of admissibles and is also endowed with a special type of vector space endomorphism F which we call a shift map.
		In particular, the construction produces the (Lie) Steenrod algebra AL when it is applied to the al- gebraic or geometric Dyer-Lashof algebras, and produces the algebraic ____________ 1991 Mathematics Subject Classification.
		If A is a P-B-W algebra with a shift map F satisfying the requirements for constructing A(1), then A(1) is a P-B-W algebra, with a basis consisting of the admissible monomials.
	www.math.purdue.edu /research/atopology/Pengelley-Williams/limalg.txt (1366 words)

	Split Octonions
		The only division algebras over the reals are R, C, H, O and the only normed algebras over the reals are R, C, L, H, M2(R), O, and O".
		That means that all normed algebras over the reals are subalgebras of either the octonions O or the split octonions O".
		Therefore, the Witt Decomposition decomposes the vector space R^(p,q) = R^(p,p+k) into p copies of the Hyperbolic plane plus a k-dimensional definite space (in the example, positive definite).
	www.valdostamuseum.org /hamsmith/so43.html (1585 words)

	MA5P8 - CONFORMAL FIELD THEORY - Section 1 (Site not responding. Last check: 2007-10-21)
		This first part of the course is devoted to the introduction of the Virasoro algebra, one of the key players in conformal field theory (CFT).
		Next, I introduce the concept of an infinitesimal symmetry, to explain how the so-called Witt algebra is generated by the algebra of infinitesimal orientation preserving conformal transformations on the punctured complex plane.
		I prove that up to equivalence it allows a unique one-parameter family of such central extensions, the Virasoro algebra at central charge c with c an arbitrary complex number.
	www.maths.warwick.ac.uk /~wendland/cft/s1.html (133 words)

	David W. Lewis
		His research interests are in the areas of quadratic and hermitian forms and algebras with involutions.
		The level of a Witt ring, Linear and Multilinear Algebra 27 (1990), 163-165.
		Quaternion algebras and the algebraic legacy of Hamilton, Irish Math Soc.Bull.
	mathsa.ucd.ie /staff/dwl (831 words)

	Witt-Birn high school mathematics
		Algebra IB is taken after Algebra IA. (Please read info about Algebra IA above.) Students interested in the technical field should then enroll in Geometry followed by Algebra IIB.
		This is a course designed to enable the student from Algebra IA and IB to satisfactorily complete a Geometry course, which is often required for those continuing their education beyond high school.
		A passing grade in Algebra IA and IB is required for this course, or a low grade in Algebra I with the consent of the instructor.
	www.wittbirn.k12.wi.us /hsmath.htm (775 words)

	Recent Papers and Preprints (Site not responding. Last check: 2007-10-21)
		Witt kernels of triquadratic extensions, (with D. Leep), in "Algebraic and Arithmetic Theory of Quadratic Forms" Contemp.
		Decomposition of Witt rings and Galois groups, (with J. Minác), Canad.
		Witt rings and realizability of small 2-Galois groups, Proc.
	math.uc.edu /~tsmith/papers.html (338 words)

	Conformal field theory - Wikipedia, the free encyclopedia
		However, the infinitesimal conformal transformations form an infinite-dimensional algebra, called the Witt algebra.
		This results in the appearance of a nontrivial central charge, and the Witt algebra is modified to become the Virasoro algebra.
		In particular, it is possible to relate the spectrum of primary operators in a theory to the value of the central charge, c.
	en.wikipedia.org /wiki/Conformal_field_theory (485 words)

	Passman's Abstracts
		Let K[G] be the group algebra of a torsion group G over an infinite field K, and let U=U(G) denote its group of units.
		Here we begin the analysis in the case where the abelian group A is the additive group of a finite-dimensional vector space V over a locally finite field F of prime characteristic p, and the automorphism group G is a simple infinite absolutely irreducible subgroup of GL(V).
		If K^t[G] is a twisted group algebra satisfying a nondegenerate multilinear generalized polynomial identity f=0, then we show that G has certain normal subgroups of finite index which can be viewed as being almost central.
	www.math.wisc.edu /~passman/abstracts.html (3706 words)

	Centre for Particle Theory : Conformal Field Theory and Strings (16 lectures) - Durham University
		They are relevant in (at least) three different areas of modern theoretical physics: they provide toy models for interacting quantum field theories, they describe two-dimensional critical phenomena and they play a central role in string theory, at present the most promising candidate for a unifying theory of all forces.
		This course will introduce the basic concepts and methods of conformal field theory in a pedestrian fashion, with emphasis on the very special case of two dimensions, with aview to using these methods to derive some of the fundamental early results in bosonic string theory.
		Conformal transformations: definition in d dimensions; conformal Killing equation; generators and their algebra when d is not 2; solutions to the 2d conformal Killing equation; Mobius transformations; algebra of local 2d conformal transformations (Witt algebra).
	www.dur.ac.uk /cpt/graduate/lectures/cfts (340 words)

	PlanetMath: Poincaré-Birkhoff-Witt theorem
		Clearly this rewriting proces must end, and this concludes the induction step.
		The proof of linear independence of the PBW-monomials is slightly more difficult, but can be found in most introductory texts on Lie algebras, such as the classic below.
		Cross-references: linear independence, contains, argument, term, linear combination, basis, index, induction, map, span, easy to see, universal enveloping algebra, monomials, linear order, field, Lie algebra
	planetmath.org /encyclopedia/PoincareBirkhoffWittTheorem.html (160 words)

	A q-Schrödinger equation based on a Hopf q-deformation of the Witt algebra
		In an earlier paper a q-Schrödinger equation was obtained based on a particular quantization procedure, called Borel quantization, and a related q-deformation of the Witt algebra.
		This q-deformation is a deformation in the category of Lie algebras and hence the corresponding q-Witt algebra has a trivial Hopf algebra structure.
		In this paper, we extend the above algebra by the addition of a set of shift-type generators, which appear in the expression for the quantum mechanical position operator and hence lead to a new type of quantum kinematics.
	stacks.iop.org /0305-4470/32/4971 (394 words)

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