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Topic: Tensor product of R algebras

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	Tensor product - Wikipedia, the free encyclopedia
		It means that if a pair of tensors are juxtaposed (placed side-by-side) then they combine by mere aggregation to form a new tensor which can be subsequently called the tensor product of the pair of juxtaposed tensors.
		The tensor product inherits all the indices of its factors.
		In J the tensor product is the dyadic form of / (for example a / b or a / b / c).
	en.wikipedia.org /wiki/Tensor_product (1120 words)

	ZeroDivisor Algebras, Charles Muses
		To maintain algebraic consistency with the split-quaternionic structure of 1, i, e, and f, pure 1 and pure i matrices should be in the same subspace and pure e and pure f matrices should be in the same subspace.
		New algebraic phenomena that appear in dimensions 128 and 256, the existence of nonzero a such that a^2 = 0 in 128 dimensions and a^4 = 0 for a^2 =/= 0 and a^3 =/= 0 in 256 dimensions, are related to the appearance of spinors that are not pure.
		The 128-dimensional algebra is related to the 126 = 28+70+28 (128-2=7+21+35+35+21+7 = dim of Cl(7) - scalar and pseudoscalar) root vectors of the 133-dimensional Lie algebra E7.
	www.valdostamuseum.org /hamsmith/NDalg.html (3990 words)

	Tensor product of R algebras (Site not responding. Last check: 2007-10-22)
		In mathematics, there is a construction in abstract algebra of the tensor product of commutativerings; which puts a ring structure on the tensor product as abeliangroups of two commutative rings R and S. This structure on R
		We observe the multilinearnature of the product a
		This construction is of constant use in algebraic geometry :working in the opposite category to that of commutativeR-algebras, it provides pullbacks of affine schemes, otherwise known as fiberproducts.
	www.therfcc.org /tensor-product-of-r-algebras-210864.html (154 words)

	Glossary of tensor theory - Encyclopedia, History, Geography and Biography
		Einstein notation This states that in a product of two indexed arrays, if an index letter in the first is repeated in the second, then the (default) interpretation is that the product is summed over all values of the index.
		The free algebra on a set 'X is for practical purposes the same as the tensor algebra on the vector space with X as basis.
		The wedge product is the anti-symmetric form of the $\otimes$ operation.
	www.arikah.com /encyclopedia/Tensor_notation (679 words)

	Tensor product of R-algebras - Wikipedia, the free encyclopedia
		In mathematics, there is a construction in abstract algebra of the tensor product of commutative rings; which puts a ring structure on the tensor product as abelian groups of two commutative rings R and S.
		We observe the multilinear nature of the product
		This construction is of constant use in algebraic geometry: working in the opposite category to that of commutative R-algebras, it provides pullbacks of affine schemes, otherwise known as fiber products.
	en.wikipedia.org /wiki/Tensor_product_of_R-algebras (171 words)

	Brauer group - Encyclopedia, History, Geography and Biography
		For example when K is the real number field R, the Brauer group Br(R) is a cyclic group of order two: there are just two types of division algebra, R and the quaternion algebra H.
		The product in the Brauer group is based on the tensor product: the statement that H has order two in the group is equivalent to the existence of an isomorphism of R-algebras
		A central simple algebra (CSA) over K is a finite-dimensional (associative) algebra K which is a simple ring for which the center is exactly K.
	www.arikah.net /encyclopedia/Brauer_group (487 words)

	Tensor Product of Algebras
		We're going to prove that the tensor product of two r algebras is another r algebra, but first we need a lemma about associative homomorphisms.
		This satisfies the criteria of an r algebra, as described in the introduction.
		be r algebras, whose tensor product is t.
	www.mathreference.com /ring-alg,prod.html (848 words)

	Non abelian tensor products of groups: bibliography
		G.J. Ellis, `The non-abelian tensor product of finite groups is finite', J. Algebra, 111, 203-205, 1987.
		L.-C. Kappe, `Non abelian tensor products of groups: the commutator connection', Proceedings Groups St Andrews at Bath 1997, Lecture Notes LMS 261 (1999) 447-454.
		N.Inassaridze, E.Khmaladze and M.Ladra, Non-abelian tensor product of Lie algebras and its derived functors, Extracta Mathematicae 17 (2002) 281-288.
	www.bangor.ac.uk /%7Emas010/nonabtens.html (1505 words)

	Group scheme - Wikipedia, the free encyclopedia
		There are numerous examples familiar in algebra, including the general linear group, and elliptic curves.
		The theory of commutative group varieties occupies the place in theory that was investigated in nineteenth century mathematics, in the search for the most general addition theorems.
		Every algebraic torus and abelian variety is part of the theory, as are group extensions formed from both kinds of object (which have been called quasi-abelian varieties), used in the theory of differential forms (of the second kind and third kind, in classical terminology), and the geometric forms of class field theory.
	www.wikipedia.org /wiki/Group_scheme (460 words)

	PlanetMath: tensor product
		The tensor product is a formal bilinear multiplication of two modules or vector spaces.
		In essence, it permits us to replace bilinear maps from two such objects by an equivalent linear map from the tensor product of the two objects.
		This is version 5 of tensor product, born on 2002-02-17, modified 2005-08-11.
	planetmath.org /encyclopedia/TensorProduct.html (253 words)

	Glossary of tensor theory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		A dyadic tensor has rank two, and may be represented as a square (A rectangular array of elements (or entries) set out by rows and columns) matrix.
		This states that in a product of two indexed arrays, if an index letter in the first is repeated in the second, then the (default) interpretation is that the product is summed over all values of the index.
		The (Click link for more info and facts about free algebra) free algebra on a set X is for practical purposes the same as the tensor algebra on the vector space with X as basis.
	www.absoluteastronomy.com /encyclopedia/G/Gl/Glossary_of_tensor_theory.htm (1303 words)

	The Construction of Extensions and their Elements
		Given two matrix algebras R and T, where R and T have the same coefficient ring S, return the direct sum D of R and T (with the action given by the direct sum of the action of R and the action of T).
		Given an element a of the matrix algebra Q and an element b of the matrix algebra R, form the direct sum of matrices a and b.
		The square is returned as an element of the matrix algebra T, which must be the direct sum of the parent of a and the parent of b.
	www.math.ufl.edu /help/magma/text452.html (365 words)

	Glossary of tensor theory (Site not responding. Last check: 2007-10-22)
		Einstein summation convention This states that in a product of two indexed arrays, if an index letter in the first is repeated in the second, then the (default) interpretation is that the product is summed over all values of the index.
		In the tensor algebra T(V) of a vector space V, the operation becomes a normal (internal) binary operation.
		The wedge product is the anti-symmetic form of the operation.
	www.portaljuice.com /glossary_of_tensor_theory.html (504 words)

	What ARE Clifford Algebras and Spinors?
		If the algebra A is also generated as a ring by the copies of R and X or, equivalently, as a real algebra by {1} and X, then A is said to be a (real) Clifford algebra for X (Clifford's term...
		Since odd dimensional Clifford algebras are the sum of two matrix algebras, spinors for odd dimensional Clifford algebras are rows (or columns) of one of the matrix algebras, the matrix subalgebras of the even-grade elements of the graded Clifford algebra.
		The non-isomorphism of the octonions O with the Clifford algebra Cl(3) is due to the nonassociativity of the 7-dimensional vector cross-product.
	www.valdostamuseum.org /hamsmith/clfpq.html (5336 words)

	SET THEORY, QUANTUM SET THEORY & CLIFFORD ALGEBRAS
		A Boolean algebra is not at all what one would call an "algebra" in the modern sense of a vector space over a field with a multiplication defined as a binary operation on the vectors.
		The difference here, however, is that the spin algebra is complexified, so that instead of a three dimensional su(2) algebra, we have an eight real dimensional M(2) algebra which is a gl(2, C) algebra to describe a Qpoint with irreducible quantum extension resulting from the incompressibility of the Planck extensions of space and time.
		The cheat is resolved by taking a direct (tensor) product of n copies of SU(2) instead of C because there are independent yes-no classical (dichotomic) independent choices for the occupation of each container.
	graham.main.nc.us /~bhammel/QSET/qset1.html (10449 words)

	AMCA: Tensor product of modules over non-unital algebras and Lie-Rinehart algebras by Jan Kubarski (Site not responding. Last check: 2007-10-22)
		If we consider the typical situation where the base ring R is unital, then non-unitality of the R-algebra A means that there is no homomorphism of rings l:R->A such that l(r)a=ra=al(r).
		There are some simple anomalies in the theory of A -modules over non-unital R-algebra A which caused that the planned researches on Lie-Rinehart algebras for algebras not necessarily with 1 failed.
		The aim of this paper is to construct the notion of a tensor product of modules over non-unital algebras which does not possesses the anomalies and its applications for the Picard group of non-unital algebras and Lie-Rinehart algebras.
	at.yorku.ca /c/a/j/c/10.htm (277 words)

	Brauer group (Site not responding. Last check: 2007-10-22)
		The product in the Brauer group is based on the tensor product : the statement that H has order two in thegroup is equivalent to the existence of an isomorphism of R-algebras
		In order to define the Brauer group, one calls acentral simple algebra (CSA) over K a finite-dimensional (associative) algebra K which is a simple ring, and for which the center is exactly K.
		A slick way tosee this is to use a characterisation: a central simple algebra over K is a K-algebra that becomes a matrixring when we extend the field of scalars to an algebraic closure of K.
	www.therfcc.org /brauer-group-210865.html (441 words)

	Algebras of electromagnetics
		Discusses tensors, Clifford algebras (spinors are elements of minimal left or right ideals of Clifford algebras, which explains why the word spinor appears so often in Clifford algebra literature) and applications.
		Kot, G. James: "Clifford algebra in electromagnetics", Proceedings of the International Symposium on Electromagnetic Theory, URSI International Union of Radio Science, Aristotle University of Thessaloniki, 25-28 May 1998, Thessaloniki, Greece.
		Algebra of forms with its de Rham operator is a standard example of cohomology in practice.
	users.tkk.fi /~ppuska/elmag_alg.html (2561 words)

	Glossary of tensor theory - InfoSearchPoint.com (Site not responding. Last check: 2007-10-22)
		If v and w are vectors in vector spaces V and W respectively, then $v \otimes w$ is a tensor in $V \otimes W$ .
		In the tensor algebra T(V) of a vector space V, the operation $\otimes$ becomes a normal (internal) binary operation.
		The wedge product is the anti-symmetic form of the $\otimes$ operation.
	www.infosearchpoint.com /display/Glossary_of_tensor_theory (552 words)

	Diagonals In Tensor Products Of Operator Algebras (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		In this paper we give a short, direct proof, using only properties of the Haagerup tensor product, that if an operator algebra A possesses a diagonal in the Haagerup tensor product of A with itself, then A must be isomorphic to a finite dimensional C # -algebra.
		2 The homology of Banach and topological algebras (context) - Helemskii - 1989
		1 The ideal structure of the Haagerup tensor product of C # -..
	citeseer.ist.psu.edu /373692.html (303 words)

	Current Commutation Relations, Continuous Tensor Products and Infinitely Divisible Group Representations, by R. F. ...
		Dubin and I had already shown that in general, continuous tensor products of cyclic group representations do not exist.
		In the trivial case when the group is R, then the generator of the group defines a random variable, and the cyclic vector defines a probability measure on the line; then the theory reduces to the concept of infinite-divisibility of this measure.
		The theory can be easily reformulated in terms of infinitely divisible positive-definite functions on groups, namely, the expectation of the unitary operators representing the group in the cyclic vector (known as the characteristic function of the cyclic representation), which is always a positive semi-definite continuous function on the group.
	www.mth.kcl.ac.uk /~streater/infidivi.html (631 words)

	Seminormal representations of Weyl groups and Iwahori-Hecke algebras - Ram (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		The method is to generalize the Jucys-Murphy elements in the group algebras of the symmetric groups to arbitrary Weyl groups and Iwahori-Hecke algebras.
		In that paper the representations of the algebras of types An, Bn, Dn and G 2 were computed and it is the purpose of this paper to extend these computations to F...
		Affine Hecke algebras, cyclotomic Hecke algebras and Clifford..
	citeseer.ist.psu.edu /ram97seminormal.html (731 words)

	[ref] 61 Lie Algebras (Site not responding. Last check: 2007-10-22)
		A Lie algebra L is an algebra such that xx=0 and x(yz)+y(zx)+z(xy)=0 for all x,y,z ∈ L.
		Representations of Lie algebras are dealt with in the same way as representations of ordinary algebras (see Representations of Algebras).
		The tensor product is returned as an algebra module.
	www.gap-system.org /Manuals/doc/htm/ref/CHAP061.htm (4974 words)

	Tensor product of R-algebras - The Jiggies Reference Guide (Site not responding. Last check: 2007-10-22)
		In mathematics, there is a construction in abstract algebra of the tensor product of commutative rings; which puts a ring structure on the tensor product as abelian groups of two commutative rings R and S. This structure on R $\otimes$
		More generally, if R is a commutative ring and A and B are commutative R-algebras, we can make A $\otimes$
		We observe the multilinear nature of the product a $\otimes$ b.
	www.jiggies.com /reference/Tensor_product_of_R-algebras (197 words)

	Tensor Product Multiplicities, Canonical Bases And Totally Positive Varieties (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		The original goal of this project was to find explicit "polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra g.
		Here "polyhedral " means that the multiplicity in question is to be expressed as the number of lattice points in some convex polytope, or in more down-to-earth terms, the number of...
		2 Tensor product multiplicities and convex polytopes in partit..
	citeseer.ist.psu.edu /303428.html (420 words)

	Atlas: Tensor products of locally $m-$convex $H^-$algebras. Structure theorems by Marina Haralampidou (Site not responding. Last check: 2007-10-22)*
		Murray and J. Von Neumann defined a tensor product of Hilbert spaces, which is a Hilbert space too (1936).
		-algebras is an algebra of the same type, when it is equipped with the projective tensor product topology.
		The existence of an orthogonal basis in a tensor product algebra, as before, is crucial for its structure.
	atlas-conferences.com /cgi-bin/abstract/cado-44 (208 words)

	A stratification of generic representation theory and generalized Schur algebras, by Nicholas J. Kuhn (Site not responding. Last check: 2007-10-22)
		Let F(q) be the category whose objects are functors from finite dimensional Fq-vector spaces to Fq-vector spaces, and with morphisms the natural transformations between such functors.
		In fact, (essentially) all our subcategories are equivalent to categories of modules over various finite dimensional algebras, and our lattice can be interpreted in terms of lattices of idempotent two sided ideals in these generalized Schur algebras.
		Our tensor product theorem is then used to study when various of our generalized Schur algebras are Morita equivalent.
	www.math.uiuc.edu /K-theory/0437 (314 words)

	Finiteness Of A Non-Abelian Tensor Product Of Groups (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		Some sufficient conditions for finiteness of a generalized non-abelian tensor product of groups are established extending Ellis' result for compatible actions.
		The non-abelian tensor product of groups was introduced by Brown and Loday [2,3] following works of A.Lue [4] and R.K.Dennis [7].
		4 The non-abelian tensor product of finite groups is finite (context) - Ellis - 1987
	citeseer.ist.psu.edu /inassaridze96finitenes.html (291 words)

	Current Commutation Relations and Continuous Tensor Products, by R. F. Streater. (Site not responding. Last check: 2007-10-22)
		We define a current Lie algebra to be the set of pairs (X,f) where X is in a Lie algebra and f is a test function on some space say R^n.
		We show that if the continuous tensor product of a representation of the Lie algebra exists, then we get a representation of the current algebra by differentiation.
		It is remarked that continuous tensor products exist for several representations of the oscillator group, and that the corresponding currents are Boson quantum fields.
	www.mth.kcl.ac.uk /~streater/current.html (182 words)

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