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Topic: Associative algebra

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	PlanetMath: non-associative algebra
		A non-associative algebra is an algebra in which the assumption of multiplicative associativity is dropped.
		In much of the literature concerning non-associative algebras, where the meaning of a “non-associative algebra” is clear, the word “non-associative” is dropped for simplicity and clarity.
		Lie algebras and Jordan algebras are two famous examples of non-associative algebras that are not associative.
	planetmath.org /encyclopedia/NonAssociativeRing.html (176 words)

	PlanetMath: non-associative algebra
		A non-associative algebra is an algebra in which the assumption of multiplicative associativity is dropped.
		In much of the literature concerning non-associative algebras, where the meaning of a “non-associative algebra” is clear, the word “non-associative” is dropped for simplicity and clarity.
		Lie algebras and Jordan algebras are two famous examples of non-associative algebras that are not associative.
	www.planetmath.org /encyclopedia/NonAssociativeAlgebra.html (174 words)

	Associative algebra Encyclopedia
		In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner.
		An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.
		A representation of an algebra is a linear map ρ: A → gl(V) from A to the general linear algebra of some vector space (or module) V that preserves the multiplicative operation: that is, ρ(xy)=ρ(x)ρ(y).
	www.hallencyclopedia.com /topic/Associative_algebra.html (1256 words)

	NationMaster - Encyclopedia: Associative algebra
		In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring.
		An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.
		A representation of an algebra is a linear map ρ: A → gl(V) from A to the general linear algebra of some vector space (or module) V that preserves the multiplicative operation: that is, ρ(xy)=ρ(x)ρ(y).
	www.nationmaster.com /encyclopedia/Associative_algebra/Definition (2607 words)

	Lie algebra
		A Lie algebra (pronounced as "lee", named in honor of Sophus Lie) is an algebraic structure in mathematics whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds.
		A subalgebra of the Lie algebra g is a subspace[?] h of g such that [x, y] ∈ h for all x, y ∈ h.
		An ideal of the Lie algebra g is a subspace h of g such that [a, y] ∈ h for all a ∈ g and y ∈ h.
	www.ebroadcast.com.au /lookup/encyclopedia/li/Lie_algebra.html (976 words)

	Abstract algebra/Clifford Algebras - Wikibooks, collection of open-content textbooks
		Specifically, a Clifford algebra is a unital associative algebra which contains and is generated by a vector space V equipped with a quadratic form Q.
		In characteristic not 2 the algebra Cℓ(V,Q) inherits a Z-grading from the canonical isomorphism with the exterior algebra Λ(V).
		One of the principal applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold.
	en.wikibooks.org /wiki/Abstract_algebra/Clifford_Algebras (3723 words)

	PlanetMath: power-associative algebra
		, since the associator is trilinear (linear in each of the three coordinates).
		A theorem, due to A. Albert, states that any finite power-associative division algebra over the integers of characteristic not equal to 2, 3, or 5 is a field.
		This is version 10 of power-associative algebra, born on 2004-10-10, modified 2006-01-30.
	www.planetmath.org /encyclopedia/PowerAssociativeAlgebra.html (133 words)

	Lucian Ionescu's Stuff
		Interpreting the elements of a non-associative algebra as "vector fields" and the multiplication as a connection, we investigate a natural candidate for the algebra of functions, with derivations the former algebra.
		The associator of a non-associative algebra is the curvature of the Hochschild quasi-complex.
		Conditions for the existance of an ``algebra of functions'' having as algebra of derivations the original non-associative algebra, are investigated.
	www.ilstu.edu /~lmiones/research.htm (1098 words)

	Operations on Associative Algebras and their Elements
		Given an associative algebra A and a subalgebra B of A, compute the idealizer of B in A, that is, the largest subalgebra of A in which B is an ideal.
		Let A and B be subalgebras of an associative algebra with underlying module M. This function returns the submodule of M which is spanned by the elements [a, b] = a * b - b * a, a in A, b in B. CommutatorIdeal(A, B) : AlgAss, AlgAss -> AlgAss
		For two subalgebras A and B of an associative algebra, return the left annihilator of B in A; that is, the subalgebra of A consisting of all elements a such that a * b = 0 for all b in B. RightAnnihilator(A, B) : AlgAss, AlgAss -> AlgAss, AlgAss
	www.umich.edu /~gpcc/scs/magma/text894.htm (718 words)

	Associative algebra - Definition, explanation
		An associative algebra A over a field K is defined to be a vector space over K together with a K-bilinear multiplication A x A → A (where the image of (x,y) is written as xy) such that the associativity law holds:
		The preceding definition generalizes without any change to an algebra over a commutative ring K (except that a K-linear space is then called a module and not a vector space).
		One may consider associative algebras over a commutative ring R: these are modules over R together with a R-bilinear map which yields an associative multiplication.
	www.calsky.com /lexikon/en/txt/a/as/associative_algebra.php (1160 words)

	Springer Online Reference Works
		The first examples of associative rings and associative algebras were number rings and fields (the field of complex numbers and its subrings), polynomial algebras, matrix algebras over fields, and function fields.
		The classical part of the theory of associative rings and algebras is formed by the theory of finite-dimensional associative algebras [2].
		A free associative algebra is a ring with free ideals, i.e.
	eom.springer.de /a/a013510.htm (1120 words)

	Algebra Street's Associative, Commutative, and Distributive Math Properties Tutorial
		Associative property problems deal with the grouping of numbers.
		A "hint" that might be helpful in identifying the associative property is to read the problem aloud and see if both sides of the equal mark sound the same outloud.
		Many times associative property problems contain parentheses that are at a different place on one side of the problem than on the other side.
	www.thatmathsite.com /algebra_street/math_properties.shtml (356 words)

	Associative algebra - Education - Information - Educational Resources - Encyclopedia - Music
		An associative algebra A over a field K is defined to be a vector space over K together with a K-bilinear multiplication A x A
		Such an algebra is a ring and contains a copy of the ground field K in the form.
		X form a unitary associative algebra (using composition of operators as multiplication); this is in fact a Banach algebra.
	www.music.us /education/A/Associative-algebra.htm (875 words)

	Olive Hazlett Abstract
		On the Classification and invariantive Characterization of Nilpotent Algebras
		Linear associative algebras in a small number of units, with coordinates ranging over the field C of ordinary complex numbers, have been completely tabulated; that is, their multiplication tables have been reduced to very simple forms.
		But if we have before us a linear associative algebra, the chances are that its multiplication table would not be in any of the tabulated forms, nor even in such a form that we could readily ascertain to which standard form it was equivalent.
	www.agnesscott.edu /lriddle/WOMEN/abstracts/hazlett_abstract.htm (330 words)

	Misha Roitman (Site not responding. Last check: )
		However, a certain subcategory of vertex algebras, obtained by restricting the order of locality of generators, has a universal object, which we call the free vertex algebra corresponding to the given locality bound.
		Algebra, 96(3):279-297, 1994], who proved this for the case when the vertex algebra is non-negatively graded and has finite dimensional homogeneous components.
		We prove that a Lie conformal algebra L with bounded locality function is embeddable into an associative conformal algebra A with the same bound on the locality function.
	www.math.uiuc.edu /~roitman/Research/index.html (1198 words)

	Research in Mathematics, Uppsala University
		Division algebras over arithmetic fields are related to the theory of quadratic forms over these fields, a classical subject in algebraic number theory.
		The notion of a representation of an associative algebra with identity element generalizes and unifies various classical algebraic objects such as, for example, matrix representations of a group, several linear operators acting simultaneously on a vector space, or configurations in a vector space given by linear subspaces satisfying a prescribed inclusion diagram.
		The philosophy is to study a given associative algebra by studying its category of representations which in turn is expected to reveal internal properties of the algebra.
	www.math.uu.se /research/groups/algebraochgeometri.php (831 words)

	Universal enveloping algebra - Definition, explanation
		If L is the Lie algebra corresponding to the Lie group G, U(L) can be identified with the algebra of left-invariant differential operators (of all orders) on G; with L lying inside it as the left-invariant vector fields as first-order differential operators.
		L acts on itself by the Lie algebra adjoint representation, and this action can be extended to a representation of L on U(L): L acts as an algebra of derivations on T(L), and this action respects the imposed relations, so it actually acts on U(L).
		The construction of the group algebra for a given group is in many ways analogous to constructing the universal enveloping algebra for a given Lie algebra.
	www.calsky.com /lexikon/en/txt/u/un/universal_enveloping_algebra.php (991 words)

	Presentations for matrix algebras
		where P is a free algebra in noncommuting variable and I is a two-sided ideal in P. The presentation is obtained by computing a set of primitive idempotents for the algebra and extracting generators for the radical of the algebra.
		This algebra is Morita equivalent to A, and hence shares many of the same homological properties of the algebra A. In the course of obtaining the presentation, several aspects of the algebra are computed.
		The algebra is block upper triangular, where the upper left block is a field extension of degree three and the lower block is a filed extension of degree 2.
	www.math.lsu.edu /magma/text851.htm (2081 words)

	Math Forum - Ask Dr. Math
		That very day, her regular math teacher taught a lesson on the associative property, in which operations in expressions only consisting of addition or multiplication can basically be performed in any order, with the same result.
		It doesn't mean that these expressions don't mean different things (in terms of the operations they say to do), only that their values will always be the same when we carry out those operations.
		Simply put, you can think of the associative property of multiplication and addition to be an exception to the order of operations, if it makes arithmetic easier.
	www.mathforum.org /library/drmath/view/61447.html (1129 words)

	Search Results for "associative"
		A fantastic sequence of haphazardly associative imagery, as seen in dreams or fever.
		Mathematics A group with a binary associative operation such that the operation admits an identity element...
		...of addition and multiplication, in which the set is an abelian group under addition and associative under multiplication and in which the two operations are related...
	www.bartleby.com /cgi-bin/texis/webinator/sitesearch?FILTER=col61&x=10&y=12&query=associative (284 words)

	Citations: New perspectives on the BRST-algebraic structure of string theory - Lian, Zuckerman (ResearchIndex)
		Definition 3.1 A Gerstenhaber algebra is a graded commutative and associative algebra A together with a bracket [ Delta; Delta] A Omega A A of degree Gamma1, such that for all homogeneous elements x, y, and z in A, x; y]....
		A Gerstenhaber algebra is a graded commutative and associative algebra A together with a bracket [ Delta; Delta] A Omega A A of degree Gamma1, such that for all homogeneous elements x, y, and z in A, x; y]....
		A G algebra is a generalization of a Poisson algebra, which incorporates simultaneously....
	citeseer.ist.psu.edu /context/493863/0 (1018 words)

	[ref] 57 Algebras
		An algebra is a vector space equipped with a bilinear map (multiplication).
		If all involved algebras are matrix algebras, and either both are Lie algebras or both are associative then the result is again a matrix algebra of the appropriate type.
		The difference between an algebra homomorphism and an algebra-with-one homomorphism is that in the latter case, it is assumed that the identity of
	www.math.colostate.edu /manuals/gap/CHAP057.htm (3260 words)

	[ref] 60 Algebras
		So a module over an algebra is constructed by giving generators of a vector space, and a function for calculating the action of algebra elements on elements of the vector space.
		This must be a function of two arguments; the first argument is the algebra element, and the second argument is a vector; it outputs the result of applying the algebra element to the vector.
		must be a function of two arguments; the first argument is the algebra element, and the second argument is a vector; it outputs the result of applying the algebra element on the left to the vector.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/doc/htm/ref/CHAP060.htm (5050 words)

	[No title]
		Now a Lie algebra is (the best known) an algebra which is non-associative, but of which the multiplication law satisfies the antisymmetry condition and the Jacobi associativity.
		Abstractly a Lie algebra is associated with derivations (or derivation operators).
		furthermore it's often worthwhile to extract the lie algebra from the group in this way because the lie algebra is often easier to work with than the group is, mainly because the lie algebra lives in the computationally tractable world of linear algebra.
	www.math.niu.edu /~rusin/known-math/99/lie_algebras (1527 words)

	Algebras over Vector Spaces
		An associative algebra is merely an algebra over a vector space with an associative multiplication.
		For particular algebras additional definitional equations are given in the form of an equation to hold for all values of the field and vector variables.
		Such algebras are normally described in the style often used to describe complex numbers where i is merely posited as another previously overlooked number that was there all along.
	www.cap-lore.com /MathPhys/Algebras/index.html (1063 words)

	Why not SEDENIONS?
		The 28 new associative triple cycles of the sedenions are related to the 28-dimensional Lie algebra Spin(0,8), and to the 28 different differentiable structures on the 7-sphere S7 that are used to construct exotic structures on differentiable manifolds.
		Geometrically, the k-grade elements of the Clifford algebra Cl(0,N) of the N-dimensional vector space are identified with the k-dimensional subspaces of the N-dimensional vector space.
		This subalgebra is the LIE ALGEBRA called Spin(0,N), and it is the simply connected covering algebra of the algebra that generates the rotation group of the N-dimensional vector space of Cl(0,N).
	www.valdostamuseum.org /hamsmith/sedenion.html (5107 words)

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