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[No title] In case of a Lie algebra, or Lie group, of endomorphisms of a finite dimensional vector space V - which is the case considered in the paper - there appears to be a conflicting terminology going back to the older literature. For arbitrary (connected) Lie groups, this is done in such a way that a Lie group is reductive iff its corresponding Lie algebra is reductive. This means that an algebraic group G is \emph{defined} to be reductive in such a way that this holds iff a) its Lie algebra g is reductive (as an abstract Lie algebra) and,\emph{in addition}, b) the centre c(g) of g consists of semisimple endomorphisms of V, i.e. www.math.niu.edu /~rusin/known-math/99/lie_alg2   (594 words)

Molecular Mechanisms Group The Group was site-visited in June 1996 and a new quinquennial programme formally commenced in April 1997. Reductive elimination of the leaving group occurred from the (indol—3—yl)methyl derivatives (Figure 1, pathway I) but not the 2—substituted regioisomers (pathway II), indicating that only the C—3 position may be utilised in bioreductively—activated drug delivery. Reductive elimination of a model leaving group was shown for 5—nitroindole, but it was too slow to compete with disproportionation of the nitro radical-anion. www.graylab.ac.uk /lab/report98/pw/pw.html   (3623 words)

Springer Online Reference Works that are a semi-simple algebraic group and an algebraic torus, respectively. The generalized Hilbert theorem on invariants is true for reductive groups. Connected reductive groups have a structure theory that is largely similar to the structure theory of reductive Lie algebras (root system; Weyl group, etc., see [2]). eom.springer.de /r/r080440.htm   (379 words)

Reductive group - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab1.cs.wisc.edu)   (Site not responding. Last check: 2007-10-08) In mathematics, a reductive group is an algebraic group G such that the unipotent radical of the identity component of G is trivial. More generally, in the case of Lie groups, a reductive Lie group G is sometimes defined as one such that its Lie algebra g is the Lie algebra of a real algebraic group that is reductive, in other words a Lie algebra that is the sum of an abelian and a semisimple Lie algebra. The concept of reductive is not quite the same as for Lie groups as it is for algebraic groups because a reductive Lie group can be the group of real points of a unipotent algebraic group. en.wikipedia.org.cob-web.org:8888 /wiki/Reductive_group   (383 words)

PlanetMath: reductive For example, a finite group is reductive over a field A complex Lie group is reductive if and only if it is a direct product of a semisimple group and an algebraic torus. This is version 3 of reductive, born on 2003-01-28, modified 2003-08-22. planetmath.org /encyclopedia/Reductive.html   (88 words)

5.2.3 Lie Groups and Representation Theory A representation of a Lie group G is a continuous homomorphism from G into the space of continuous linear operators on a topological vector space which may be infinite dimensional. The former is often called noncommutative harmonic analysis, as it is the analog for a noncommutative group G of Fourier analysis for the group R or the circle group, where the irreducible representations are one-dimensional. Chang: Representation theory for reductive Lie groups (a typical example of a reductive group is the group of all invertible matrices of fixed size). www.math.okstate.edu /grad/long-hbk/5_2_3Lie_Groups_Representat.html   (592 words)

Springer Online Reference Works (via CobWeb/3.1 planetlab1.cs.wisc.edu)   (Site not responding. Last check: 2007-10-08) Characters occurring in the context of the representation theory of finite reductive groups (cf. In the case of a reductive group over the algebraic closure of a finite field, a character sheaf fixed by the Frobenius mapping has a characteristic function which is a class function on the finite group However, since the theory of character sheaves is valid in the wider context of reductive groups over arbitrary algebraically closed fields, the ideas of the Deligne–Lusztig theory have an analogue in arbitrary connected reductive groups. eom.springer.de.cob-web.org:8888 /d/d120100.htm   (805 words)

07w5034 Group Embeddings: Geometry and Representations (via CobWeb/3.1 planetlab1.cs.wisc.edu)   (Site not responding. Last check: 2007-10-08) One can choose a finite dimensional representation V of a reductive group G, then the Zariski closure in End(V) of the group generated by the scalars and the image of G will be an interesting reductive monoid [Sol]. The correponding problem was settled by Donkin for reductive groups, and there is a natural conjecture for the reductive monoid case. [Rit] Rittatore, A. Algebraic monoids and group embeddings. www.pims.math.ca.cob-web.org:8888 /birs/birspages.php?task=displayevent&event_id=07w5034   (2007 words)

Springer Online Reference Works Mumford [1] (in a different, but equivalent form) with the aim of finding a property of semi-simple groups, defined over an algebraically closed field of arbitrary characteristic, which, from the point of view of the geometric theory of invariants (cf. Invariants, theory of), would serve as a substitute for the classical property of complete reducibility of rational linear representations of semi-simple groups defined over fields of characteristic zero (this latter property not holding for ground fields of positive characteristic). Mumford's hypothesis was first proved in [4]; the proof was extended in [5] to the general case of reductive group schemes over a field. eom.springer.de /m/m065570.htm   (405 words)

David C. Murphy: Research Equivariant Embeddings of Algebraic Groups, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2004. Given a variety X with an action of an algebraic group G, the closures of G-orbits in X are parametrized by the Hilbert scheme of X (assuming it exists; e.g., X is projective). I want to explore the connection between the group embedding problem and moduli of principal bundles as well as the representations of the group associated to its embeddings. max.cs.kzoo.edu /~dmurphy/researchindex.html   (1030 words)

research.html However, the group Sl(2,R) is a Lie group and we can effectively exploit the interaction of the group structure and differential structure to analyze various function spaces. Since every Lie group is the semidirect product of a reductive group and a solvable group, there is a natural fork in the road when studying the associated representation theory and harmonic analysis. A representation of a group is a homomorphism into a group of unitary operators on some Hilbert space. www.math.washington.edu /~colling/Research/researchbak.html   (1133 words)

Semisimple - Wikipedia, the free encyclopedia A connected Lie group is called semisimple when its Lie algebra is; and the same for algebraic groups. Every finite dimensional representation of a semisimple Lie algebra, Lie group, or algebraic group in characteristic 0 is semisimple, i.e., completely reducible, but the converse is not true. (See reductive group.) Moreover, in characteristic p>0, semisimple Lie groups and Lie algebras have finite dimensional representations which are not semisimple. en.wikipedia.org /wiki/Reductive   (350 words)

Graduate Program in Mathematics: Areas of Interest (via CobWeb/3.1 planetlab1.cs.wisc.edu)   (Site not responding. Last check: 2007-10-08) Groups and algebras in physics - Lorentzian Kac-Moody groups and algebras, discrete and infinite dimensional groups in string theory and M-theory. The group conducts a weekly seminar which is informal by nature and usually concerns either the speaker's own research or some topic of current interest to the group. The theory of Lie groups and their representations is playing an ever greater role in mathematics (differential equations and classical analysis, algebraic geometry and number theory, topology and differential geometry) and mathematical physics (non-linear dynamical systems, elementary particle theory, quantum field theory). www.math.rutgers.edu.cob-web.org:8888 /grad/interests.html   (4534 words)

CJM - Motivic Haar Measure on Reductive Groups We define a motivic analogue of the Haar measure for groups of the form G(k((t))), where k is an algebraically closed field of characteristic zero, and G is a reductive algebraic group defined over k. A classical Haar measure on such groups does not exist since they are not locally compact. We give an explicit construction of the motivic Haar measure, and then prove that the result is independent of all the choices that are made in the process. journals.cms.math.ca /cgi-bin/vault/view/gordon3396   (166 words)

bibliography for automorphic and modular forms, L-functions, representations, and number theory [Bernstein-Zelevinsky 1977] I.N. Bernstein and A.V. Zelevinsky, `Induced representations of reductive p-adic groups I', Ann. This is done in the context of real Lie groups, but easily generalizes to an adelic version, if one knows enough about the other necessary technical matters in that case. [Mackey 1958] G.W. Mackey, `Multiplicity-free representations of finite groups', Pac. www.math.umn.edu /~garrett/m/b/bib.html   (3638 words)

[No title] This is seen already with mu_3 over Q. p6, in the section on nonconnected groups, the G should be an M. p7, in the table, the group at left has a composition series whose quotients are the groups on the right. In the definition of the Clifford group, the condition should be gamma(t)Vt^{-1}=V. p42, "is the sequence" is too strong --- with a little effort they can be shown to be isomorphic. p121, roughly speaking, the reason the Weyl group W is constant is that W(k) equals the Weyl group of the root datum, which doesn't change when the field is enlarged. www-personal.umich.edu /~jmilne/aag.html   (924 words)

Hanspeter Kraft homepage (via CobWeb/3.1 planetlab1.cs.wisc.edu)   (Site not responding. Last check: 2007-10-08) In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alternating groups. We show that a linear subspace of a reductive Lie algebra g that consists of nilpotent elements has dimension at most equal to the number of positive roots, and that any nilpotent subspace attaining this upper bound is equal to the nilradical of a Borel subalgebra of g. We study the geometry of the nullcone N(V^k) for several copies of a representation V of a reductive group G and its behavior for different k. www.math.unibas.ch.cob-web.org:8888 /~kraft   (2086 words)

A Harish-Chandra Homomorphism for Reductive Group Actions.   (Site not responding. Last check: 2007-10-08) Let G be a connected reductive group and X a smooth G-variety. Then the center Z(X) of the ring of G-invariant differential operators on X is a polynomial ring. More precisely, Z(X) is isomorphic to the ring of invariants of a finite reflection group. www.math.rutgers.edu /~knop/papers/HC.html   (53 words)

[math/0407491] Complete toric varieties with reductive automorphism group   (Site not responding. Last check: 2007-10-08) We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots. www.arxiv.org /math.AG/0407491   (109 words)

Citebase - On quasi-reductive group schemes The paper was motivated by a question of Vilonen, and the main results have been used by Mirkovic and Vilonen to give a geometric interpretation of the dual group (as a Chevalley group over Z) of a reductive group. We also obtain results about group schemes over a Dedekind scheme or a noetherian scheme. We show that in residue characteristic 2 there are indeed non-smooth quasi-reductive groups and they can be classified when R is strictly henselian. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0405381   (130 words)

Journal of the American Mathematical Society   (Site not responding. Last check: 2007-10-08) As a corollary, under the same hypotheses, we see that any supercuspidal representation is compactly induced from a representation of an open subgroup which is compact modulo the center. F. Bruhat and J. Tits, Groupes réductifs sur un corps local I, Publ. M. Tadic, Geometry of dual spaces of reductive groups, J. d'Analyse Math. www.ams.org /jams/0000-000-00/S0894-0347-06-00544-3/home.html   (461 words)

Adrian Vasiu   (Site not responding. Last check: 2007-10-08) Unipotent, normal subgroup schemes of reductive groups C. Acad. Older versions of some of the papers of Work on reductive group schemes, crystalline theories, and Shimura varieties Points of integral canonical models of preabelian type, p-divisible groups, and applications part 2C, 1/31/00, p. math.arizona.edu /~adrian   (451 words)

Thierry Levasseur at MSRI - Invariant differential operators under the action of a reductive group (via CobWeb/3.1 ...   (Site not responding. Last check: 2007-10-08) Thierry Levasseur - Invariant differential operators under the action of a reductive group Abstract: Let G be a reductive group acting on a complex affine variety X and X//G be the categorical quotient. If D(X) is the ring of algebraic differential operators on X, we review old and new results on the natural map from D(X)G to D(X//G). www.msri.org.cob-web.org:8888 /publications/ln/msri/2000/interact/levasseur/1/index.html   (76 words)

Citations: ecicka: Open subsets of projective spaces with a good quotient by an action of a reductive group - ...   (Site not responding. Last check: 2007-10-08) Citations: ecicka: Open subsets of projective spaces with a good quotient by an action of a reductive group - lynicki-Birula, Swi (ResearchIndex) (via CobWeb/3.1 planetlab1.cs.wisc.edu) Bia lynicki-Birula, J.  Swiecicka: Open subsets of projective spaces with a good quotient by an action of a reductive group. When the intersection W of all translates g U, g 2 G, is open in X and admits a good quotient W W= G In this note we continue the study of the case G = SL 2 started in [2] and.... citeseer.ist.psu.edu.cob-web.org:8888 /context/1740632/0   (500 words)

Citations: Galois- xed points in the Bruhat-Tits building of a reductive group - Prasad (ResearchIndex) (via CobWeb/3.1 ...   (Site not responding. Last check: 2007-10-08) Citations: Galois- xed points in the Bruhat-Tits building of a reductive group - Prasad (ResearchIndex) (via CobWeb/3.1 planetlab1.cs.wisc.edu) Prasad, Galois- xed points in the Bruhat-Tits building of a reductive group, Bull. Online articles have much greater impact More about CiteSeer.IST Add search form to your site Submit documents Feedback citeseer.ist.psu.edu.cob-web.org:8888 /context/2255434/0   (240 words)

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