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# Topic: Algebraic group

###### In the News (Fri 19 Jul 19)

 Algebraic group - Wikipedia, the free encyclopedia In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. Two important classes of algebraic groups arise, that for the most part are studied separately: abelian varieties (the 'projective' theory) and linear algebraic groups (the 'affine' theory). Note that this means that algebraic group is narrower than Lie group, when working over the field of real numbers: there are examples such as the universal cover of the 2×2 special linear group that are Lie groups, but have no faithful linear representation. en.wikipedia.org /wiki/Algebraic_group   (380 words)

 Linear algebraic group - Wikipedia, the free encyclopedia In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. Such groups were known for a long time before their abstract algebraic theory was developed according to the needs of major applications. The first basic theorem of the subject is that any affine algebraic group is a linear algebraic group: that is, any affine variety V that has an algebraic group law has a faithful linear representation, over the same field. en.wikipedia.org /wiki/Linear_algebraic_group   (471 words)

 Research group in algebra and combinatorics   (Site not responding. Last check: 2007-10-07) Groups are studied in several contexts: as symmetry groups of discrete structures, as linear groups (groups of matrices) and as algebraic groups. Linear and Algebraic Groups: The emphasis lies on recognition problems for finite linear groups and representations, the eigenvalue behaviour of group elements in a representation and on asymptotic problems in representations and locally finite groups. Algebraic group methods are used extensively for the study of representations of Lie type groups. www.mth.uea.ac.uk /admissions/graduate/algrep.html   (455 words)

 [No title] Quillen's theorem for unipotent algebraic groups It is by now a wellknown fact that the dual of the (reduced) Steenrod alge- bra A* is isomorphic to the coordinate algebra of the (infinite dimensional) group scheme G = Aut s(Fa(x; y)) of the strict automorphism of the additive formal group law defined over Fp. The algebra of (reduced) mod p unstable cohomology cooperations is a polynomial algebra P pi B* = Fp[0; 1; : :]:with comultiplication 4n = n-i i. Wilkerson, The cohomology algebras of finite dimensional Hopf algebras, Trans. hopf.math.purdue.edu /PetersonC/Ext_An.txt   (2570 words)

 PlanetMath: affine algebraic group   (Site not responding. Last check: 2007-10-07) is an affine algebraic group over itself with the group law being addition, and as is Cross-references: algebraic torus, matrices, general linear group, algebraic, inverse, map, group, affine space, subset, closed, variety, field This is version 4 of affine algebraic group, born on 2003-08-21, modified 2003-08-22. planetmath.org /encyclopedia/AffineAlgebraicGroup.html   (91 words)

 UWM Math: Lie Theory/Algebraic Groups   (Site not responding. Last check: 2007-10-07) The tangent space at the identity of a Lie group is a Lie algebra. An algebraic group is a group that is simultaneously (and compatibly) an algebraic set (i.e., a set with group operations defined by polynomial equations). The study of quantum groups is also closely linked to the study of Lie algebras and algebraic groups. www.uwm.edu /Dept/Math/Research/Algebra/Lie/Lie.html   (230 words)

 Glossary An algebraic map or regular map or morphism of quasiprojective varieties is a map of whose graph is closed. An algebraic group is a group G in the category of quasiprojective varieties i.e. G is simulateneously a group and variety and the group multiplication G x G → G and inversion G → G are morphisms. www.math.purdue.edu /~dvb/algeom2.html   (786 words)

 Glossary of terms for Fermat's Last Theorem The conjecture that the rank of the group of rational points of an elliptic curve E is equal to the order of the zero of the L-function L(E,s) of the curve at s=1. the kernel of a group homomorphism is a subgroup. A complete algebraic variety which is an algebraic curve that is essentially the quotient space of the upper half of the complex plane by the action of a subgroup of finite index of the modular group. gyral.blackshell.com /flt/flt10.htm   (2633 words)

 AlgebraicGroups - Cmat Wiki   (Site not responding. Last check: 2007-10-07) \section{Algebraic Lie algebras} %\begin{defi} %\label{defi:basicalglie} %\index{Lie algebra!algebraic} %Let $k$ be a field of arbitrary characteristic, $G$ be an affine algebraic group and %$\mathfrak h \subset {\operatorname {\mathcal L}(G)}$ be a $k$--Lie subalgebra. Compute the action of the corresponding Lie algebra on $k^{2}$ and conclude that there are examples of subspaces of $k^{2}$ stable with respect to the action of the Lie algebra but not with respect to the action of $G_{a}$. Prove that this Lie algebra is not associated to an affine algebraic group. www.cmat.edu.uy /moin/AlgebraicGroups   (3010 words)

 Elliptic Curves and Elliptic Functions For every algebraic function, it is possible to construct a specific surface such that the function is "single-valued" on the surface as a domain of definition. The first is the group of all points on the curve E which have an order that divides m for some particular integer m. One of the principal facts of elementary group theory is that any finitely generated abelian group is the direct sum of a finite group and a finite number of infinite cyclic groups (isomorphic to the integers Z). cgd.best.vwh.net /home/flt/flt03.htm   (3513 words)

 UCLA Math: Algebra and Algebraic Geometry   (Site not responding. Last check: 2007-10-07) Algebra, one of the three major branches of pure mathematics, interacts significantly with many other fields. The Department's particular strengths lie in group theory, algebraic geometry, algebraic K-theory, and areas of algebra related to number theory, modular forms, and the theory of algebras. Students specializing in algebra should take most of these during their graduate careers. www.math.ucla.edu /grad_programs/faculty/research_areas/algebra.html   (253 words)

 14: Algebraic geometry Algebraic geometry combines the algebraic with the geometric for the benefit of both. Note that many computations in algebraic geometry are really computations in polynomials rings, hence computational commutative algebra applies. This is essentially the study of formal groups. www.math.niu.edu /~rusin/known-math/index/14-XX.html   (523 words)

 [No title]   (Site not responding. Last check: 2007-10-07) For an algebraic group G, defined over an algebraically closed field of characteristic zero, there is a natural partial order on the set of G-actions on algebraic varieties: X >= Y if there exists a dominant G-equivariant rational map (i.e., a compression) from X to Y. Alternatively, one can consider regular, rather than rational, compressions. Abstract: Let G be an algebraic group and X be an irreducible algebraic variety with a generically free G-action, all defined over an algebraically closed field of characteristic zero. The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type. www.math.ubc.ca /~reichst/abstract.html   (2805 words)

 [No title] Given a recursive sequence v_n of elements in a free monoid, we investigate the quotient of the free associative algebra by the ideal generated by all non-subwords in v_n. Affine space in the given \Theta is represented in the form Hom(W,G), where W=W(X) is the free in \Theta algebra with the finite X and G is an algebra from \Theta. Thus, algebraic sets or algebraic varieties in the space Hom(W,G) are defined by a set of formulas, which are not necessarily equalities. www.math.technion.ac.il /~techm/19990623000019990624asp   (1448 words)

 Knot Table: (Algebraic) Concordance Order   (Site not responding. Last check: 2007-10-07) Levine defined a homomorphism of the concordance group onto an algebraically defined group, isomorphic to the countably infinite direct sum of (an infinite number of) copies of Z_2, Z_4, and Z. The algebraic order of algebraic concordance order of a knot is the order of the image in Levine's alebraic concordance group. Livingston and Naik have shown that many knots of algebraic order 4 are infinite order in the concordance group. Andrius Tamulis proved that may knots of algebraic order 2 are of higher order in the concordance group, and proved that others are either negative amphicheiral, or concordant to negative amphicheiral knots, and thus are of order 2. www.indiana.edu /~knotinfo/descriptions/concordance_order.html   (216 words)

 [No title] The Coxeter group is the symmetry group of an n-dimensional cube. This group is the semidirect product of the permutations of the n axes and the group (Z/2)^n generated by the reflections along these axes. Instead, it's half as big as the Weyl group of B_n: it's the subgroup of the symmetries of the n-dimensional cube generated by permutations of the coordinate axes and reflections along *pairs* of coordinate axes. math.ucr.edu /home/baez/twf_ascii/week187   (2283 words)

 Algebraic Geometry Algebraic geometry is the study of the "shape" of the set of solutions to polynomial equations. The study of this type of question is called "arithmetic algebraic geometry" and is closely related to number theory, group theory, and representation theory. The study of the geometric properties of this continuum is known as "complex algebraic geometry" and is closely related to topology, differential geometry, complex analysis and even theoretical physics. www.math.utah.edu /research/ag   (382 words)

 Algebraic Number Theory Archive   (Site not responding. Last check: 2007-10-07) ANT-0295: 8 Jun 2001, On the structure theory of the Iwasawa algebra of a p-adic Lie group, by Otmar Venjakob. ANT-0185: 7 Jun 1999, An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod-p cohomology of GL(n,Z), by Avner Ash and Warren Sinnott. ANT-0066: 31 Aug 1998, Torsion subgroups of Mordell-Weil groups of Fermat Jacobians, by Pavlos Tzermias. front.math.ucdavis.edu /ANT   (12251 words)

 London postgraduate study group in algebraic Number theory   (Site not responding. Last check: 2007-10-07) This study group is designed for postgrads in algebraic number theory at London colleges, and they present the talks in this study group. The study group is organised by staff members, who attend to give help and guidance on technical questions. This term, the study group will be held on Wednesdays, between 3:00 and 4:00 pm in room 423 of KCL, with a short break for tea and biscuits in room 429 before the London Number Theory Seminar. www.mth.kcl.ac.uk /events/psgant.html   (207 words)

 Short CV: L.R. Renner   (Site not responding. Last check: 2007-10-07) Algebraic groups and monoids, algebraic transformation groups, related combinatorics and geometry. The theory of algebraic monoids is a natural synthesis of algebraic group theory (Chevalley, Borel, Tits) and torus embeddings (Mumford, Kempf, et al). The intention of this monograph is to convince the reader that reductive monoids are among the darlings of algebra. www.math.uwo.ca /~lex/cv/Renner.html   (327 words)

 Department of Mathematics - University of Georgia The geometry group includes algebraic geometry, differential geometry, mathematical physics, and representation theory. A diverse group of mathematicians in the department has a number of overlapping research interests in a broad range of geometric problems. Members of the group are the postdoc Nancy Wrinkle, the graduate students Xander Faber, Chad Mullikin, and Heunggi Park, and the freshman Darren Wolford. www.math.uga.edu /math/research/geometry.html   (504 words)

 algebra help-algebra software-algebra math tutor Both subjects are clearly motivated by their use in resolving singularities of algebraic varieties, for which one of the main tools consists in blowing up the variety along an equimultiple subvariety. Main topic is the bilinear complexity of finite dimensional associative algebras with unity: Upper bounds for the complexity of matrix multiplication and a general lower bound for the complexity and algebraic structure in the case of algebras of minimal rank is shown. Final chapter is on the study of isotropy groups of bilinear mappings and the structure of the variety of optimal algorithms for bilinear mapping. www.softmath.com /algebra11.htm   (1861 words)

 Publisher description for Library of Congress control number 97004011   (Site not responding. Last check: 2007-10-07) It analyses groups which possess a certain very general dependence relation (Shelah's notion of 'forking'), and tries to derive structural properties from this. These may be group-theoretic (nilpotency or solubility of a given group), algebro-geometric (identification of a group as an algebraic group), or model-theoretic (description of the definable sets). In this book, the general theory of stable groups is developed from the beginning (including a chapter on preliminaries in group theory and model theory), concentrating on the model- and group-theoretic aspects. www.loc.gov /catdir/description/cam028/97004011.html   (177 words)

 List of publications of a researcher In Geometric and Algebraic Combinatorics (GAC3, Oisterwijk, The Netherlands, August 14-19, 2005) (pp....-...). Oberwolfach, Arbeitstagung 'Groups and Geometries' (Aschbacher, Kantor, Timmesfeld). Integral representations of finite groups in algebraic groups. oashos01.hosting.kun.nl:8015 /metue/pk_apa_n.medewerker?p_url_id=1161   (1641 words)

 Definition of Radical in chemistry, either an atom or molecule with at least one unpaired electron, or a group of atoms, charged or uncharged, that act as a single entity in reaction. the radical of an algebraic group is a concept in algebraic group theory. the radical of an ideal is an important concept in abstract algebra. www.wordiq.com /definition/Radical   (285 words)

 Galois Cohomology Now the twisted forms of G are in one-to-one correspondence to the 1-cocycles of Gamma on Aut(G) and the forms are conjugate if and only if the cocycles are cohomologous. Returns Aut_K(G) as a Gamma-group with Gamma=Gal(K:k), where A is the automorphism group of G and K is the base field of G. The field k must be a subfield of K. ActingGroup(G) : GrpLie -> Grp, Map Returns Gamma=Gal(K:k) together with the map m from the abstract Galois group Gamma into the set of field automorphisms, such that m(gamma) is the actual field automorphism for every gamma in Gamma. www.math.lsu.edu /magma/text1054.htm   (410 words)

 Department of Mathematics - University of Georgia Diophantine geometry and arithmetic, local and global heights on algebraic varieties, uniform distribution on locally compact groups, applications of harmonic analysis to number theory, the Mahler measure of polynomials. Our number theory group is complemented by a large group in algebraic geometry, including Valery Alexeev, William Graham, Elham Izadi, Roy Smith, and Robert Varley. Members of the group are: Robert Brice, Sungkon Chang, Jerry Hower, Jacob Keenum, Nausheen Lotia, Daeshik Park, Clay Petsche, Charles Pooh, Dong Hoon Shin, and Juhyung Yi. www.math.uga.edu /research/number_theory.html   (607 words)

 [No title]   (Site not responding. Last check: 2007-10-07) These notes provide an introductory overview of the theory of algebraic groups, Lie algebras, Lie groups, and arithmetic groups. The Lie algebra of an algebraic group (continued) Algebraic groups over R and C; relation to Lie groups www.jmilne.org /math/CourseNotes/aag.html   (103 words)

 DIMACS/DIMATIA/Renyi Working Group on Algebraic and Geometric Methods in Combinatorics This working group will concentrate on two broad areas of research: algebraic methods involving the study of homomorphisms of graphs, with special emphasis on problems arising from statistical physics, and problems of combinatorial geometry. This working group will also examine various topics that lie at the interface between the two disciplines of computational geometry and real algebraic geometry, topics such as construction and analysis of arrangements of algebraic surfaces, lower bound proofs, robust computations, and more. The contributions of real algebraic geometry to computational geometry are quite well known, but perhaps less well known are some interactions in the other direction, e.g., separating the combinatorial' from the algebraic' complexity of semialgebraic sets [24]. dimacs.rutgers.edu /Workshops/Algebraic/main.html   (2719 words)

 [No title] This is an expository article on the theory of algebraic stacks. We study the problem of understanding the uniformizing Fuchsian groups for a family of plane algebraic curves by determining explicit first variational formulae for the generators. It is also shown that any infinitesimally divisible measure on a connected nilpotent real algebraic group is embeddable. www.ias.ac.in /mathsci/vol111/feb2001/absfeb2001.html   (678 words)

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