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Topic: Linear algebraic group

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	Glossary of terms for Fermat's Last Theorem
		the kernel of a group homomorphism is a subgroup.
		A group that is isomorphic to a subgroup of a general linear group.
		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.
	cgd.best.vwh.net /home/flt/flt10.htm (2633 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.
		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.
		A linear algebraic group G consists of a finite number of irreducible components, that are in fact also the connected components: the one G
	en.wikipedia.org /wiki/Linear_algebraic_group (481 words)

	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 (390 words)

	PlanetMath: special linear group
		is defined to be the subgroup of the general linear group
		Cross-references: standard basis, matrix, linear transformation, group, field, determinant, invertible linear transformations, general linear group, subgroup, vector space
		This is version 4 of special linear group, born on 2002-02-22, modified 2005-05-04.
	planetmath.org /encyclopedia/SpecialLinearGroup.html (108 words)

	Linear algebra Article, Linearalgebra Information (Site not responding. Last check: 2007-11-02)
		Linear algebra is the branch of mathematics concernedwith the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations.
		Linear algebra had its beginnings in the study of vectors in Cartesian 2-space and 3-space.
		Linear operators take elementsfrom a linear space to another (or to itself), in a manner that is compatible with the addition and scalar multiplication givenon the vector space(s).
	www.anoca.org /space/vector/linear_algebra.html (792 words)

	Student understanding of topics in linear algebra
		The increasing demand for student understanding of linear algebra and the growing concern that the present linear algebra course does not adequately meet the needs of the students it serves, prompts further study and curriculum development.
		The first course in linear algebra is a service course for a wide variety of disciplines such as computer science, electrical engineering, other engineering fields such as aerospace engineering and systems engineering, physics, economics, statistics, and operations research.
		After taking the linear algebra course, students often come away knowing how to perform certain algorithms but they have not acquired the intuition relating knowledge of the mathematics to selection of the method for analysis, design, and control of physical systems (Wang 237).
	www.physics.umd.edu /rgroups/ripe/perg/plinks/linalg.htm (1312 words)

	math lessons - Linear algebraic group
		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.
	www.mathdaily.com /lessons/Linear_algebraic_group (463 words)

	20: Group Theory and Generalizations
		Groups acting on vector spaces are subgroups of the matrix groups studied in Linear Algebra.
		Groups acting on topological spaces are the basis of equivariant topology and homotopy theory in Algebraic Topology.
		Nielsen's theorem: subgroups of free groups are free.
	www.math.niu.edu /~rusin/known-math/index/20-XX.html (2774 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 other kind is a linear algebraic group, which is (isomorphic to) an algebraic subgroup of a general linear group - i.
		One of the pricipal 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).
	www.mbay.net /~cgd/flt/flt03.htm (3513 words)

	Springer Online Reference Works
		It was generalized twenty years later to algebraic groups by M.
		The naturally arising problem as to whether the number of such double classes for an arbitrary algebraic group is finite is connected with the reduction theory for subgroups of principal adèles, i.e.
		is a function field, it was also proved that the number of double classes of this kind for the adèle group of the algebraic group is finite, and an analogue of the reduction theory was developed [6].
	eom.springer.de /a/a010740.htm (409 words)

	[No title]
		Differential algebra} \vskip.1in The notion of the Galois group of a linear ordinary differential equation or a holonomic system of such equations is well known.
		The Galois group $G$ of the system is, by definition, the group of all automorphisms $g$ of the field $L$ such that $g$ fixes all elements of $F$ and $[g,d/dz]=0$.
		This theory allows to prove that certain linear differential equations of second and higher order cannot be reduced to a sequence of first order equations, by showing that their Galois groups are not solvable.
	www.maths.tcd.ie /EMIS/journals/ERA-AMS/1995-01-001/1995-01-001.tex.html (1449 words)

	XI. INVARIANCE GROUP of G(n)
		The largest continuous group of linear transformations U(G, n) on each Hilb(n) that does this is conjugate in the general linear group GL(n, C) to the non-compact Lie group U(n-1, 1).
		This group is adopted as the group of canonical transformations.
		The algebra of a noncompact translation group is isomorphic to the algebra of a compact toroidal group.
	graham.main.nc.us /~bhammel/FCCR/XI.html (2441 words)

	Reductive algebraic groups
		A reductive algebraic group is an almost direct product of a finite number of simple algebraic groups and a finite number of copies of
		itself is the basic example of a reductive linear algebraic group.
		of reductive algebraic groups over local fields are always of type I. This statement provides a large class of examples of type I groups for which the questions raised in the previous section are meaningful.
	www.stieltjes.org /archief/rep9899/node14.html (202 words)

	KSDA - Graduate Center series
		Differential algebra is a natural subject for study by computable model theorists, yet there are precious few results for computable differential fields.
		A Picard-Vessiot extension of a linear ODE is the differential algebraic analogue of the splitting field of a polynomial in ordinary Galois theory.
		We offer a formulation of linear ordinary differential equations midway between what one encounters in a first undergraduate ODE course and what one encounteres in a graduate Differential Geometry course (in the latter instance under the heading of "connections").
	www.sci.ccny.cuny.edu /~ksda/gradcenter.html (1066 words)

	[No title]
		In this paper, we construct homology groups Hi(X, G), where G is an algebraic group and X is a variety, by considering cycles on the simplicial scheme BG x X, an idea first suggested by Andrei Suslin.
		Hi(X, G; A), whose source is the usual group homology of the discrete group G(R) of R-points of the algebraic group G. Moreover, this map is an isomorphism if R = k and k is algebraically closed, so that these groups capture the homology of the discrete group G(k).
		HOMOLOGY OF LINEAR GROUPS 7 Proof.The abelian group H1(Spec (R), Gm ; Z) is generated by classes of non- zero complex numbers, [z] for z 2 Cx, modulo the relations [z] = [__z] and [zw] = [z] + [w], for all z, w 2 Cx.
	hopf.math.purdue.edu /Knudson-Walker/hom11-19.txt (4410 words)

	a-First name :Shadi (Site not responding. Last check: 2007-11-02)
		We define a closed subgroup H of a pro-affine algebraic group G to be observable if every (finite-dimensional) rational H-module is an H sub-module of a rational G-module.
		We extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed field K of characteristic 0.
		Let A be a subLie algebra of a solvable Lie algebra L over an algebraically closed field of characteristic 0.
	www.aub.edu.lb /~webpubof/research/23report_addendum/fas/mathematics.htm (227 words)

	Concrete Resolution of Differential Problems using Tannakian Categories (Site not responding. Last check: 2007-11-02)
		Let C be the subfield of constants of k, n be the order of L, and consider the Picard-Vessiot extensions k' of k associated with L, i.e., the differential field extensions of k that contain an n-dimensional C-vector space of solutions of L and do not enlarge the constant field C.
		Then the differential Galois group of L is defined as the group G of differential field automorphisms (i.e., field automorphisms that respect the differential structure) of any Picard-Vessiot extension k' that additionally respect the action of k on k'.
		While the (algebraic) Galois group of a polynomial is a subgroup of a permutation group
	pauillac.inria.fr /algo/seminars/sem98-99/weil.html (2425 words)

	M390C Linear Algebraic Groups--Daniel Allcock (Site not responding. Last check: 2007-11-02)
		Algebraic groups are the algebraic geometry analogue of Lie groups.
		The topological and Lie-algebra arguments used in the development of Lie groups must be replaced by global algebra-geometric arguments, but amazingly, the resulting classification of simple linear algebraic groups is exactly the same as the classification of simple Lie groups.
		There are three standard texts for this course, each called "Linear Algebraic Groups", one by Borel, one by Humphreys and one by Springer.
	www.ma.utexas.edu /dev/math/Courses/S07Descriptions/M390C_Allcock.html (249 words)

	Gerhard Röhrle
		Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p.
		Let G be a simple algebraic group over an algebraically closed field k; assume that Char k is zero or good for G. Let \cB be the variety of Borel subgroups of G and let e in Lie G be nilpotent.
		We prove analogues for reductive algebraic groups of some results for finite groups due to Knoerr and Robinson which play a central role in their reformulation of Alperin's conjecture for finite groups.
	www.maths.soton.ac.uk /staff/Roehrle (499 words)

	Springer Online Reference Works
		One of the classical groups, defined as the group of automorphisms of a skew-symmetric bilinear form
		This algebraic group, also called a symplectic group, is a simple simply-connected linear algebraic group of type
		As a consequence, its tangent mapping at a fixed point belongs to the symplectic group of the tangent space.
	eom.springer.de /S/s091820.htm (301 words)

	Raghuram's homepage (Site not responding. Last check: 2007-11-02)
		Sep 15: Jordan decomposition in GL(n) and in a linear algebraic group.
		Sep 22: Solvable, Nilpotent and Unipotent algebraic groups.
		Oct 27: The Lie algebra of an algebraic group.
	www.math.uiowa.edu /~araghura/fall2005/22m330 (182 words)

	INI : Abstracts : MAA : Solvability and unsolvability of equations in finite terms (Site not responding. Last check: 2007-11-02)
		With each linear differential equation, Picard associated its Galois group, which is a Lie group (and, moreover, a linear algebraic group).
		Picard and Vessiot showed that this particular group is responsible for the solvability of equations by quadratures.
		In particular, he showed that the general algebraic equation of degree at least 5 is unsolvable by radicals exactly for topological reasons.
	www.newton.cam.ac.uk /programmes/MAA/Askold.html (339 words)

	[No title]
		As an application, we* * compute the rational cohomology of a family of commutative unipotent groups V (n) and d* iscuss the connection of these cohomology rings with that of the Steenrod algebra*.
		The coord* inate ring k[G(n)] is a polynomial algebra* k[a1; : :;:an-1] with the comultiplication Xi j 4ai= aj api-j: j=0 On the otherhand, let P (n) be the finite dimensional subalgebra of the Steenro* d algebra* 0 pn generated by the reduced powers P p; : :;:P.
		Its dual Hopf algebra is n+1 pn p P (n)* ~=k[1; : :;:n+1]=(p1 ; 2 : :;:n+1); 11 P i pj with 4i= j=0j i-j: There is a Hopf algebra epimorphism by (3.3) in [4].
	www.math.purdue.edu /research/atopology/PetersonC-Yagita/Witt.txt (1531 words)

	KSDA - Graduate Center 2005-2006
		We now move to differential algebraic geometry, which Alexandru Buium, in his book, Differential algebra and Diophantine Geometry, calls a "new geometry", and, to differential algebraic groups, the group objects of the new geometry.
		In the affine case, the objects of differential algebraic geometry are the sets of zeros of differential polynomial ideals.
		The group of an equation will be defined, the type of information about solutions provided by the group will be illustrated, and methods of calculation will be indicated.
	www.sci.ccny.cuny.edu /~ksda/gradcenter2005.html (925 words)

	Contents (Site not responding. Last check: 2007-11-02)
		One, a first course in modern algebra, emphasizing applications and once-a-week computer labs, to students who have only had a ``fundmentals in mathematics'' course and a matrix theory course.
		The ``standard treatment'', which emphasizes the classification of groups, is put aside in favor of a treatment which emphsizes examples and computations.
		The automorphism group of a linear error-correcting code and more campanology and the theory of the Rubik's cube are discussed as applications.
	web.usna.navy.mil /~wdj/book/node1.html (878 words)

	A purity theorem for linear algebraic groups, by Ivan Panin (Site not responding. Last check: 2007-11-02)
		A purity theorem for linear algebraic groups, by Ivan Panin
		Given a characteristic zero field and a dominant morphism of linear algebraic groups with a commutative target one can form a functor from commutative algebras to abelian groups.
		The functor takes an algebra to the group of points of the target group modulo the group of points of the sourse.
	www.math.uiuc.edu /K-theory/0729 (96 words)

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