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Topic: Algebra homomorphism

Related Topics

Algebra over a field

Homomorphism

Algebraic structure

Ring homomorphism

Module homomorphism

Isomorphism

Exponential function

	:::► Dictionary of Meaning www.dictionary-of-meaning.com ◄:::
		In abstract algebra, a homomorphism is a structure-preserving map (mathematics) between two map (mathematics) between two algebraic structures (such as group (mathematics)s, ring (mathematics)s, or group (mathematics)s, ring (mathematics)s, or vector spaces).
		A homomorphism is a map (mathematics) from one map (mathematics) from one algebraic structure to another of the same type that preserves all the relevant structure; i.e.
		The notion of a homomorphism can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures.
	www.dictionary-of-meaning.com /homomorphism.html (848 words)

	NationMaster - Encyclopedia: Algebra homomorphism (Site not responding. Last check: 2007-10-26)
		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.
		The ideals are precisely the kernels of homomorphisms, and the fundamental theorem on homomorphisms is valid for Lie algebras.
		Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as addition and multiplication) and relationships (such as equality) connecting the elements.
	www.nationmaster.com /encyclopedia/Algebra-homomorphism (278 words)

	Lie algebra homomorphism
		An ideal of the Lie algebra g is a subspace h of g such that [a, y] â h for all a â g and y â h.
		Given a Lie group, a Lie algebra can be associated to it either by endowing the tangent space to the identity with the differential of the adjoint map, or by considering the left-invariant vector fields as mentioned in the examples.
		This association is functorial, meaning that homomorphisms of Lie groups lift to homomorphisms of Lie algebras, and various properties are satisfied by this lifting: it commutes with composition, it maps Lie subgroups, kernels, quotients and cokernels of Lie groups to subalgebras, kernels, quotients and cokernels of Lie algebras, respectively.
	www.algebra.com /algebra/about/history/Lie-algebra-homomorphism.wikipedia (1642 words)

	Algebra homomorphism - Encyclopedia, History, Geography and Biography
		A homomorphism between two algebras over a field K, A and B, is a map $F:A\rightarrow B$ such that for all k in K and x,y in A, kx) = kF(x)
		Both A and B are algebras over K given by the standard multiplication and addition of polynomials and functions, respectively.
		Hence the mapping $f \rightarrow \hat{f}\,$ is injective and an algebra isomorphism of A and B.
	www.arikah.com /encyclopedia/Algebra_homomorphism (283 words)

	PlanetMath: universal enveloping algebra
		) is isomorphic to the skew polynomial algebra
		Cross-references: polynomial algebra, Lie bracket, left ideal, irreducible, theory, representation, definition, generates, clear, injective, map, Poincaré-Birkhoff-Witt theorem, isomorphic, universal property, two-sided ideal, vector space, generated by, tensor algebra, algebras, algebra, commutator, structure, homomorphism, unity, associative, field, Lie algebra
		This is version 4 of universal enveloping algebra, born on 2002-09-18, modified 2006-03-22.
	planetmath.org /encyclopedia/UniversalEnvelopingAlgebra.html (195 words)

	Lie algebra
		A subalgebra of the Lie algebra g is a subspace[?] h of g such that [x, y] ∈ h for all x, y ∈ h.
		Real and complex Lie algebras can be classified to some extent, and this classification helps in understanding Lie groups, which are the truly interesting objects in geometry, mathematical analysis and physics since they capture symmetries of analytical structures.
		Lie algebras were originally introduced and studied by Sophus Lie and independently by Wilhelm Killing[?] starting in the 1870s for this reason.
	www.ebroadcast.com.au /lookup/encyclopedia/li/Lie_algebra.html (976 words)

	Is String Theory in Knots?
		There is a homomorphism from the braid group onto the symmetric group generated by the second relation.
		The associative algebra is graded by parity of the length of strings.
		The algebra formed by applying the ladder operator an infinite number of times will have the property that it is isomorphic to the algebra formed by applying the ladder operator to itself.
	www.weburbia.com /pg/knots.htm (2891 words)

	Boolean algebra
		The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can always be checked by a trivial brute force algorithm).
		Every Boolean algebra (A,,) gives rise to a ring (A, +, ) by defining a + b = (a ¬b) (b ¬a) (this operation is called "symmetric difference" in the case of sets and XOR in the case of logic) and a b = a b.
		An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x y in I and for all a in A we have a x in I.
	www.brainyencyclopedia.com /encyclopedia/b/bo/boolean_algebra.html (2143 words)

	Homomorphism
		In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure.
		The notion of a homomorphism can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures.
		In that case the image of X in Y under the homomorphism f is necessarily isomorphic to X/~; this fact is one of the isomorphism theorems.
	www.xasa.com /wiki/en/wikipedia/h/ho/homomorphism_1.html (478 words)

	Chern-Weil homomorphism (Site not responding. Last check: 2007-10-26)
		In mathematics, the Chern-Weil homomorphism is a basic construction in the Chern-Weil theory, relating for a smooth manifold M curvature to its de Rham cohomology groups, i.e.
		The Chern-Weil homomorphism is a homomorphism of -algebras from to the cohomology algebra.
		Such a homomorphism exists and is uniquely defined for every principal G-bundle P on M. One can usually think of the bundle P as living inside the K-theory of M,, so that the class of Chern-Weil homomorphisms is parametrized by.
	www.wikimoz.org /wiki/en/wikipedia/c/ch/chern_weil_homomorphism.html (263 words)

	Universal enveloping algebra
		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.xasa.com /wiki/en/wikipedia/u/un/universal_enveloping_algebra.html (984 words)

	HOMOMORPHISM
		Homomorphism A map f between groups A and B is a homomorphism of A into B if f(a1 * a2) = f(a1) * f(a2) for all a1,a2 in A. where the *s are the respective group operations.
		A homomorphism, (or sometimes simply morphism) from one mathematical object to another of the same kind, is a mapping that is compatible with all relevant structure.
		A homomorphism which is also a bijection such that its inverse is also a homomorphism is called an isomorphism; two isomorphic objects are completely indistinguishable as far as the structure in question is concerned.
	www.websters-online-dictionary.org /definition/english/Ho/Homomorphism.html (731 words)

	PlanetMath: Jordan algebra
		is an algebra homomorphism that respects the above two laws.
		However, unlike Lie algebras, not every Jordan algebra is embeddable in an associative algebra.
		This is version 7 of Jordan algebra, born on 2004-12-07, modified 2004-12-17.
	www.planetmath.org /encyclopedia/JordanAlgebraHomomorphism.html (193 words)

	Springer Online Reference Works
		Algebras in genetics originate from the work of I.M.H. Etherington [a2], who put the Mendelian laws into an algebraic form.
		Schafer [a5] showed that every special train algebra is a genetic algebra and that every genetic algebra is a train algebra.
		In [a6] necessary and sufficient conditions have been given for a Bernstein algebra to be a Jordan algebra.
	eom.springer.de /G/g043970.htm (378 words)

	Examensarbete vid Matematiska Institutionen
		Lie algebras are algebras with an anti-commutative binary operation, which satisfies the Jacoby identity.
		A representation of a Lie algebra is a homomorphism from this Lie algebra to the Lie algebra of linear operators on some vector space.
		A Lie algebra usually has many representations and one of the problems is to classify the easiest (simple) reprsentations for a given Lie algebra.
	www.math.uu.se /studie/grundutb/exjobb (882 words)

	[ref] 57 Algebras
		Algebra homomorphisms are vector space homomorphisms that preserve the multiplication.
		So the default methods for vector space homomorphisms work, and in fact there is not much use of the fact that source and range are algebras, except that preimages and images are algebras (or even ideals) in certain cases.
		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)

	APPENDIX J
		Ideals are of interest because in algebraic geometry, they are abstract descriptions of algebraic varieties in rings of polynomials and because they are also the kernels of ring homomorphisms.
		A Ring Homomorphism is a mapping from a ring R to a ring R' which preserves the ring operations.
		The characteristic of a field is its characteristic as a division algebra.
	graham.main.nc.us /~bhammel/FCCR/apdxJ.html (6145 words)

	Homomorphism
		A stronger form of relationship (than equivalence) between algebras which does assert such a structure preserving property is homomorphism.
		To be a homomorphism, this result must hold for all operations of every arity.
		We can get a feel for the nature of homomorphisms and the meaning of equation (10.1) by looking at one particular homomorphism that has been used over the years to ease the effort involved in multiplying real numbers.
	scom.hud.ac.uk /scomtlm/book/node271.html (270 words)

	PlanetMath: tensor algebra
		From the point of view of category theory, one can describe the tensor algebra construction as a functor
		Cross-references: algebra, homomorphism, algebras, forgetful functor, category, functor, category theory, point, right, bimodule, module, non-commutative, ground ring, cover, definition, tensor product, multiplication, component, commutative ring
		This is version 10 of tensor algebra, born on 2002-12-18, modified 2007-03-22.
	planetmath.org /encyclopedia/TensorAlgebra.html (112 words)

	The Initial Algebra (Site not responding. Last check: 2007-10-26)
		The uniqueness property of the homomorphism amounts to the `no junk, no confusion' condition described above; the presence of either of these in the model would mean that there were other possible homomorphisms.
		When abstract data types are specified algebraically, the initial model is often taken as the meaning of the specification.
		It is also the case that the initial algebra always exists and can be constructed from the term algebra, partitioned or quotiented by the equations.
	homepages.feis.herts.ac.uk /~comqejb/algspec/node10.html (349 words)

	Algebras, Symmetries, Spaces
		After discussing several aspects of noncommutative geometry from a rather subjective point of view, algebraic techniques are shown to offer a powerful tool for studying specific manifolds in the realm of commutative geometry, with possible generalization to infinite dimensions.
		A possible conclusion stemming from the (till now unsuccessful) experience with relativistic quantum field theory is that the classical space-time model breaks down at very small distances and it has to be replaced by some kind of a 'quantum space'.
		It is at least interesting to renounce, for a while, the "algebra paradigm", including its current season's overcoat, the 'noncommutative geometry paradigm'.
	quantumfuture.net /arkadiusz-jadczyk/papers/algebras/index.html (1287 words)

	ABSTRACT ALGEBRA ON LINE: Contents
		It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course.
		It is based on the books Abstract Algebra, by John A. Beachy and William D. Blair, and Abstract Algebra II, by John A. Beachy.
		An algebraic extension of an algebraic extension is algebraic(6.2.10)
	www.math.niu.edu /~beachy/aaol/contents.html (401 words)

	Identity element is preserved in homomorphism
		abstract algebra group homomorphisms - I know the definition of a group homomorphism, but I'm really stuck with showing there is no homomorphism and don't know how to show it.
		Modern Algebra - #8 - Let Φ be a homomorphism of group G into a group G'.
		Prove that the map given by, where is the residue of a modulo n, is a ring homomorphism.
	www.brainmass.com /homeworkhelp/math/algebra/38310 (214 words)

	algebra homomorphism - Information from Reference.com
		On continuity of algebra homomorphisms and uniqueness of metric...
		Fredholm Theory Relative to a Banach Algebra Homomorphism.
		R algebra homomorphism (R is real) by C (January 7, 2007).
	www.reference.com /search?q=algebra%20homomorphism&db=web (328 words)

	Lie group Summary
		To tie the algebraic structure together with the geometric structure, the operations of multiplication and inversion are required to be differentiable.
		The global structure of a Lie group is in general not completely determined by its Lie algebra; for example, if Z is any discrete subgroup of the center of G then G and G/Z have the same Lie algebra (see the table of Lie groups for examples).
		Its Lie algebra is (more or less) the Witt algebra, which has a central extension called the Virasoro algebra, used in string theory and conformal field theory.
	www.bookrags.com /Lie_group (4005 words)

	Springer Online Reference Works
		An algebraic analogue of the concept of a local Lie group (cf.
		The theory of formal groups has numerous applications in algebraic geometry, class field theory and cobordism theory.
		Formal groups also are an important tool in algebraic geometry, especially in the theory of Abelian varieties.
	eom.springer.de /F/f040820.htm (889 words)

	Abstract of: Universal coalgebra: a theory of systems
		Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras (Aczel and Mendler, 1989).
		Thus the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to: coalgebra, homomorphism of coalgebras, and bisimulation, respectively.
		Some standard results from universal algebra are reformulated (using the afore mentioned correspondence) and proved for a large class of coalgebras, leading to a series of results on, e.g., the lattices of subcoalgebras and bisimulations, simple coalgebras and coinduction, and a covariety theorem for coalgebras similar to Birkhoff's variety theorem.
	db.cwi.nl /rapporten/abstract.php?abstractnr=604 (276 words)

	Algebra homomorphism software downloads (Site not responding. Last check: 2007-10-26)
		Algebra - One on One is an educational game for those wanting a fun way to learn and practice algebra.
		This free collection of tests in arithmetic, pre-algebra, algebra, trigonometry, hyperbolic trig and calculus offers step-by-step solutions together with extensive reference material and interactive training technique.
		The calculation is done by the algebra system Yacas or by the Java interpreter BeanShell.
	www.free-download-soft.com /Algebra+homomorphism (477 words)

	Tensor algebra - Definition, explanation
		The tensor algebra T(V) is also called the free algebra on the vector space V.
		Because of the generality of the tensor algebra, many other algebras of interest are constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e.
		Examples of this are the exterior algebra, the symmetric algebra, Clifford algebras and universal enveloping algebras.
	www.calsky.com /lexikon/en/txt/t/te/tensor_algebra.php (616 words)

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