The term algebra is defined below after first defining a ring . ring In mathematics , a ring is an associative ring with a map A A which is an antiautomorphism , and an Semigroup with involution involution . More precisely, is required to satisfy the following properties math x y x y math math x y y ... over any ring. algebra A algebra A is a ring that is an associative algebra over another ring R , with the agreeing ... x mu y x lambda y mu lambda x mu y math A homomorphism math f colon A to B math is algebra homomorphism ... on the complex numbers. A operation on a algebra is an operation on an algebra over a ring that behaves ... familiar example of a algebra is the field of complex numbers C where is just complex conjugation . More generally, the conjugation involution in any Cayley Dickson algebra such as the complex numbers ... example is the Matrix ring matrix algebra of n × n matrix mathematics matrices over C with given ... space is also a star algebra. In Hecke algebra , an involution is important to the Kazhdan ... ring of an elliptic curve becomes a algebra over the integers, where the involution is given by taking ... see Milne s lecture notes on abelian varieties . Hopf algebra Examples Involutive Hopf algebras are important ... familiar example being The group Hopf algebra a group ring , with involution given by math g mapsto ... elements form a Jordan algebra The skew Hermitian elements form a Lie algebra If 2 is invertible ... symmetrizing and anti symmetrizing , so the algebra decomposes as a direct sum of symmetric and anti symmetric Hermitian and skew Hermitian elements. This decomposition is as a vector space, not as an algebra, because the idempotents are operators, not elements of the algebra. Skew structures ... algebra characteristic is 2, in which case it s identical to the original , as math ... B algebra C algebra von Neumann algebra Baer ring operator algebra This article is no longer a stub, but there is more to be said about algebras which are not B or C algebras. DEFAULTSORT Algebra Category ... more details
A algebra or, more explicitly, a closed algebra is the name occasionally used in physics ref John A. Holbrook, David W. Kribs, and Raymond Laflamme. Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction. Quantum Information Processing . Volume 2, Number 5, p. 381&ndash 419. Oct 2003. ref for a finite dimensional C algebra . The dagger, , is used in the name because physicists typically use that symbol to denote a hermitian adjoint and are often not worried about the subtleties associated with an infinite number of dimensions. Mathematicians usually use the asterisk, , to denote the hermitian adjoint. algebra feature prominently in quantum mechanics , and especially quantum information science . References references math stub Category C algebras ... more details
about the branch of mathematics pp move indef sprotect small yes Algebra is the branch of mathematics ... , topology , combinatorics , and number theory , algebra is one of the main branches of pure mathematics . Elementary algebra is often part of the curriculum in secondary education and introduces ... be done for a variety of reasons, including equation solving . Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations ... algebra . History Main History of algebra See also Timeline of algebra File Image Al Kit b al mu ta ar ... Greece Greeks created a geometric algebra where terms were represented by sides of geometric ... Khwarizmi s Algebra made use of lettered diagrams but all coefficients in the equations used in the Algebra ... , sometimes called the father of algebra , was an Alexandria n Greek mathematics Greek mathematician ... 1 4460 2221 8 ref While the word algebra comes from the Arabic language al jabr , wikt ... later wrote The Compendious Book on Calculation by Completion and Balancing , which established algebra ... Al Khwarizmi The Beginnings of Algebra author Roshdi Rashed publisher Saqi Books date November 2009 isbn 0 86356 430 5 ref The roots of algebra can be traced to the ancient Babylonian mathematics Babylonians ... ref http library.thinkquest.org 25672 diiophan.htm Diophantus, Father of Algebra ref as well ... level. ref http www.algebra.com algebra about history History of Algebra ref For example, the first ... been known as the father of algebra but in more recent times there is much debate over whether al Khwarizmi ... point to the fact that the algebra found in Al Jabr is slightly more elementary than the algebra found .... ref supported by geometric proofs, while treating algebra as an independent discipline in its own right. ref Gandz and Saloman 1936 , The sources of al Khwarizmi s algebra , Osiris i, p. 263 277 In a sense, Khwarizmi is more entitled to be called the father of algebra than Diophantus because Khwarizmi ... more details
Enveloping algebra in mathematics may refer to The universal enveloping algebra of a Lie algebra The enveloping algebra of a general Algebra over a field Non associative algebras non associative algebra disambig ... more details
Affine algebra may refer to affine Lie algebra , a type of Kac Moody algebras the Lie algebra of the affine group finitely generated algebra disambig ... more details
In abstract algebra , a derivative algebra is an algebraic structure of the signature A , , , , 0, 1, sup D sup where A , , , , 0, 1 is a Boolean algebra structure Boolean algebra and sup D sup is a unary operator , the derivative operator , satisfying the identities 0 sup D sup 0 x sup DD sup x x sup D sup x y sup D sup x sup D sup y sup D sup . x sup D sup is called the derivative of x. Derivative algebras provide an algebraic abstraction of the derived set mathematics derived set operator in topological space topology . They also Lindenbaum Tarski algebra play the same role for the modal logic wK4 K p p p that Boolean algebra structure Boolean algebra s play for ordinary propositional logic . References Esakia, L., Intuitionistic logic and modality via topology , Annals of Pure and Applied Logic, 127 2004 155 170 McKinsey, J.C.C. and A. Tarski Tarski, A. , The Algebra of Topology , Annals of Mathematics, 45 1944 141 191 Category Abstract algebra Category Boolean algebra Category Topology zh algebra stub ... more details
Wiktionarypar algebraAlgebra is one of the main branches of mathematics. The term is also used in other ways. As a specialized branch of mathematics The term algebra may also refer to a more specialized branch of mathematics within the general field of Algebra Elementary algebra , i.e. high school algebra. Abstract algebra Linear algebra Relational algebra Universal algebra The term is also traditionally used for the field of Computer algebra , dealing with software systems for symbolic mathematical computation, which often offer capabilities beyond what is normally understood to be algebra . As a mathematical structure In ring theory Algebra ring theory Algebra over a commutative ring a module equipped with a bilinear product In logic Boolean algebra structure Heyting algebra In set theory and measure theory Algebra over a set a collection of sets closed under finite unions and complementation Sigma algebra a collection of sets closed under countable unions and complementation In linear algebra and the study of vector space s Algebra over a field a vector space equipped with a bilinear vector product Associative algebra a module mathematics module equipped with an associative bilinear vector product Superalgebra a math mathbb Z 2 math graded algebra Lie algebra In functional analysis Banach algebra an associative algebra A over the real number real or complex number complex numbers which at the same time is also a Banach space . Operator algebra continuous function topology .... C algebra a Banach algebra equipped with a unary Involution mathematics involution operation. Von Neumann algebra or W algebra In category theory F algebra math F math algebra F coalgebra math F math coalgebra Other Algebra singer Algebra Blessett , singer from the U.S, goes by the stage name Algebra . Algebra song Algebra , a song by Jason Der lo See also Algebraic disambiguation mathdab bg cs Algebra rozcestn k de Algebra Begriffskl rung fr Alg bre homonymie it Algebra disambigua ... more details
Matrix algebra may refer to Matrix theory , is the branch of mathematics that studies matrix mathematics matrices Matrix ring , thought of as an algebra over a field or a commutative ring disambig pl Algebra macierzy ... more details
unreferenced date September 2009 In mathematics , more specifically in algebra , an tale or separable algebra ring theory algebra is a special type of algebra. Definition Let math K math be a field mathematics field and math mathfrak R math be a math K math algebra. Then math mathfrak R math is called tale or separable algebra separable if math mathfrak R otimes K bar K cong bar K times ... times bar K math or equivalently if math mathrm Spec , mathfrak R to mathrm Spec ,K math is an tale morphism . See also tale group scheme tale DEFAULTSORT Etale Algebra Category Algebra ... more details
Difference algebra is analogous to differential algebra but concerned with difference equation s rather than differential equation s. References Alexander Levin 2008 , http books.google.co.uk books?id 15pgjT5PeY0C Difference algebra , Springer, ISBN 9781402069468 Richard M. Cohn 1979 , http books.google.co.uk books?id Fs8oAAAACAAJ& Difference algebra , R.E. Krieger Pub. Co., ISBN 9780882756516 algebra stub Category Abstract algebra ... more details
Noref date November 2009 In mathematics , a topological algebra A over a topological field K is a topological vector space together with a continuous multiplication math cdot A times A longrightarrow A math math a,b longmapsto a cdot b math that makes it an algebra over a field algebra over K . A unital associative algebra associative topological algebra is a topological ring . An example of a topological algebra is the algebra C 0,1 of continuous real valued functions on the closed unit interval 0,1 , or more generally any Banach algebra . The term was coined by David van Dantzig it appears in the title of his Thesis doctoral dissertation 1931 . The natural notion of subspace in a topological algebra is that of a topologically closed subalgebra . A topological algebra A is said to be generated by a subset S if A itself is the smallest closed subalgebra of A that contains S . For example by the Stone Weierstrass theorem , the set id sub 0,1 sub consisting only of the identity function id sub 0,1 sub is a generating set of the Banach algebra C 0,1 . Category Topological vector spaces Category Topological algebra Category Algebras topology stub pl Algebra topologiczna uk ... more details
See also List of abstract algebra topics Algebra is one of the main branches of mathematics , and concerns ... root s. In addition to working directly with numbers, algebra also covers symbols , variables, and Set ... as an overview of and topical guide to algebra Essence of algebra Main article Algebra Arithmetic Equation s Polynomials Variable mathematics Variables Branches or classifications of algebra Pre algebra Elementary algebra Abstract algebra Linear algebra Universal algebra History of algebra Main article History of algebra General algebra concepts Algebra Cubic equation Fundamental theorem of algebra Linear equation Quadratic equation Quartic equation Quintic equation Polynomial Boolean algebraAlgebra of sets Talk Algebra of sets Algebraic normal form Talk Algebraic normal form Ampheck Talk ... algebra structure Talk Boolean algebra structure Boolean algebras canonically defined Talk Boolean ... function Talk Boolean function Boolean algebra logic Talk Boolean algebra logic Implicant Boolean implicant ... form Boolean algebra Talk normal form Boolean algebra Characteristic function Talk Characterisitic function Compactness theorem Talk compactness theorem Complete Boolean algebra Talk Complete Boolean algebra Consensus theorem Talk Consensus theorem Augustus De Morgan De Morgan, Augustus Talk ... Free Boolean algebra Talk free Boolean algebra Heyting algebra Talk Heyting algebra Indicator function Talk Indicator function Interior algebra Talk interior algebra William Stanley Jevons Jevons, William ... Karnaugh map Laws of Form Talk Laws of Form Lindenbaum Tarski algebra Talk Lindenbaum Tarski algebra ... Minimal negation operator Monadic Boolean algebra Talk monadic Boolean algebra Charles Peirce Peirce ... theorem for Boolean algebras Stone space Topological Boolean algebra Talk topological Boolean algebra Truth table Talk truth table Two element Boolean algebra Talk Two element Boolean algebra ... order logic Algebra lists Main List of abstract algebra topics See also Portal Algebra Table of mathematical ... more details
Journal of Algebra ISSN 0021 8693 is a leading international mathematical research journal in abstract algebraalgebra . An imprint of Academic Press , it is presently published by Elsevier . Journal of Algebra was founded by Graham Higman , who was its editor from 1964 to 1984. From 1985 until 2000, Walter Feit served as its editor in chief. In 2004, Journal of Algebra announced vol. 276, no. 1 and 2 the creation of a new section on Computational Algebra, with a separate editorial board. The first issue completely devoted to Computational Algebra was vol. 292, no. 1 October 2005 . External links http www.sciencedirect.com science journal 00218693 Journal of Algebra at ScienceDirect sci journal stub Category Mathematics journals Category Publications established in 1964 nl Journal of Algebra ... more details
In functional analysis , the Calkin algebra , named after J. W. Calkin, is the quotient space linear algebra quotient of B H , the ring algebra ring of bounded linear operator s on a separable space separable infinite dimensional Hilbert space H , by the ideal ring theory ideal K H of Compact operator on Hilbert space compact operator s. Since the compact operators is a in fact, the only maximal norm closed ideal in B H , the Calkin algebra is simple algebra simple . As a quotient of two C algebra s, the Calkin algebra is a C algebra itself. There is a short exact sequence math 0 rightarrow K H rightarrow B H rightarrow B H K H rightarrow 0 math which induces a six term cyclic exact sequence in K theory . Those operators in B H which are mapped to an invertible element of the Calkin algebra are called Fredholm operator s, and their index can be described both using K theory and directly. One can conclude, for instance, that the collection of unitary operators in the Calkin algebra are homotopy classes indexed by the integers Z . This is in contrast to B H , where the unitary operators are path connected. As a C algebra, the Calkin algebra is remarkable because it is not isomorphic to an algebra of operators on a separable Hilbert space instead, a larger Hilbert space has to be chosen the GNS theorem says that every C algebra is isomorphic to an algebra of operators on a Hilbert space for many other simple C algebras, there are explicit descriptions of such Hilbert spaces, but for the Calkin algebra, this is not the case . The same name is now used for the analogous construction for a Banach space . The Calkin algebra is the Corona algebra of the algebra of compact operators on a Hilbert space. References Calkin, J.W. 1941 . Two sided ideals and congruences in the ring of bounded operators in Hilbert space . Annals of Mathematics , 42 , 839 873. Category Operator theory Category C algebras Category K theory de Calkin Algebra ... more details
In mathematics In abstract algebra and mathematical logic a derivative algebra abstract algebra derivative algebra is an algebraic structure that provides an abstraction of the derivative operator in topological space topology and which provides algebraic semantics for the modal logic wK3 . In differential geometry a derivative algebra is a vector space with a product operation that has similar behaviour to the standard cross product of 3 vector geometric vector s. Citation needed date July 2009 disambig ... more details
In mathematics , a projectionless C algebra is a C algebra in which 0 and 1 are the only projection mathematics projection s. Examples C , the complex number s Function spaces C sub 0 sub 0, 1 and C 0, 1 References unreferenced date September 2008 Category C algebras algebra stub ... more details
A uniform algebra A on a compact space compact Hausdorff space Hausdorff topological space X is a closed with respect to the uniform norm algebra over a field subalgebra of the C algebra C X the continuous complex valued functions on X with the following properties the constant functions are contained in A for every x , y math in math X there is f math in math A with f x math ne math f y . This is called separating the points of X . As a closed subalgebra of the commutative Banach algebra C X a uniform algebra is itself a unital commutative Banach algebra when equipped with the uniform norm . Hence, it is, by definition a Banach function algebra . A uniform algebra A on X is said to be natural if the maximal ideal s of A precisely are the ideals math M x math of functions vanishing at a point x in X . Abstract characterization If A is a unital algebra unital commutative Banach algebra such that math a 2 a 2 math for all a in A , then there is a compact space compact Hausdorff space Hausdorff X such that A is isomorphic as a Banach algebra to a uniform algebra on X . This result follows from the spectral radius formula and the Gelfand representation. mathanalysis stub Category Functional analysis Category Banach algebras ... more details
In abstract algebraalgebra and logic , a modal algebra is a structure math langle A, land, lor, ,0,1, Box rangle math such that math langle A, land, lor, ,0,1 rangle math is a Boolean algebra structure Boolean algebra , math Box math is a unary operation on A satisfying math Box1 1 math and math Box x land y Box x land Box y math for all x , y in A . Modal algebras provide models of propositional logic propositional modal logic s in the same way as Boolean algebras are models of classical logic . In particular, the variety universal algebra variety of all modal algebras is the equivalent algebraic semantics of the modal logic K in the sense of abstract algebraic logic , and the lattice order lattice of its subvarieties is dually isomorphic to the lattice of normal modal logic s. Stone s representation theorem can be generalized to the J nsson Tarski duality , which ensures that each modal algebra can be representation theorem represented as the algebra of admissible sets in a modal general frame . See also interior algebra Heyting algebra References A. Chagrov and M. Zakharyaschev, Modal Logic , Oxford Logic Guides vol. 35, Oxford University Press, 1997. ISBN 0 19 853779 4 algebra stub Category Modal logic Category Boolean algebra zh ... more details
In mathematics , the Griess algebra is a commutative Algebra over a field Non associative algebras non associative algebra on a real number real vector space of dimension 196884 that has the Monster group M as its automorphism group . It is named after mathematician R. L. Griess , who constructed it in 1980 and subsequently used it in 1982 to construct M . The Monster fixes vectorwise a 1 space in this algebra and acts absolutely irreducibly on the 196883 dimensional orthogonal complement of this 1 space. The Monster preserves the standard inner product on the 196884 space. Griess s construction was later simplified by Jacques Tits and John H. Conway . The Griess algebra is the same as the degree 2 piece of the monster vertex algebra , and the Griess product is one of the vertex algebra products. References R. L. Griess, Jr, The Friendly Giant , Inventiones Mathematicae 69 1982 , 1 102 algebra stub Category Nonassociative algebras ... more details
In theoretical physics , a supersymmetry algebra or SUSY algebra is a symmetry algebra incorporating supersymmetry , a relation between boson s and fermion s. In a supersymmetry supersymmetric world, every boson would have a partner fermion of equal rest mass . Bosonic field s Commutative operation commute while fermionic field s anticommute. In order to relate the two kinds of fields in a single algebra, the introduction of a graded algebra Z sub 2 sub grading under which the even elements are bosonic and the odd elements are fermionic is required. Such an algebra is called a Lie superalgebra . On the other hand, the spin statistics theorem shows that bosons have integer spin, while fermions have half integer spin. Consequently, the odd elements in a supersymmetry algebra need to have half integer spin, in contrast to the tensor ial symmetries which are more traditional symmetries in physics. Just as one can have representations of a Lie algebra , one can also have representation of a Lie superalgebra representations of a Lie superalgebra . For each Lie algebra, there exists an associated Lie group which is connected space connected and simply connected . Unique up to isomorphism, this Lie group is canonically associated with the Lie algebra, and the representations of the algebra can be extended to create group representations. In the same way, representations of a Lie superalgebra can sometimes be extended into representations of a Lie supergroup . See also super Poincar algebra superconformal algebra N 1 supersymmetry algebra in 1 1 dimensions N 1 supersymmetry algebra in 1 1 dimensions N 2 superconformal algebra N 2 superconformal algebra physics stub Category Supersymmetry Category Lie algebras ko it Algebra supersimmetrica ... more details
In mathematics , an algebra bundle is a fiber bundle whose fiber s are algebra over a field algebra s and local trivialization s respect the algebra structure. It follows that the transition function s are algebra isomorphism s. Since algebras are also vector space s, every algebra bundle is a vector bundle . Examples include the tensor bundle , exterior bundle , and symmetric bundle associated to a given vector bundle , as well as the Clifford bundle associated to any Riemannian vector bundle. See also Lie algebra bundle References 1. W. Greub, S. Halperin and R. Vanstone, Connections, Curvature and Cohomology, Vo. 2, Academic Press, New Yark, 1973 2. C. Chidambara and B.S. Kiranagi, On Cohomology of Associative algebra bundles, J. Ramanujan Math. Soc., Vol. 9 1 , 1994. pp. 1 12 3. B.S. Kiranagi and R. Rajendra, Revisiting Hochschild Cohomology for Algebra Bundles, Journal of Algebra and Its Applications Vol. 7, No. 6 2008 685 715. DEFAULTSORT Algebra Bundle Category Vector bundles geometry stub ... more details
for the Lie algebras or groups Malcev Lie algebra In mathematics , a Malcev algebra or Maltsev algebra or Ruth Moufang Moufang Sophus Lie Lie algebra over a field is a nonassociative algebra that is antisymmetric, so that math xy yx math and satisfies the Malcev identity math xy xz xy z x yz x x zx x y. math They were first defined by Anatoly Maltsev 1955 . Examples Any Lie algebra is a Malcev algebra. Any alternative algebra may be made into a Malcev algebra by defining the Malcev product to be xy &minus yx . The imaginary octonions form a 7 dimensional Malcev algebra by defining the Malcev product to be xy &minus yx . References Alberto Elduque and Hyo C. Myung Mutations of alternative algebras , Kluwer Academic Publishers, Boston, 1994, ISBN 0 7923 2735 7 springer id M m062170 author V.T. Filippov title Mal tsev algebra Citation last1 Mal cev first1 A. I. title Analytic loops id MathSciNet id 0069190 year 1955 journal Mat. Sb. N.S. volume 36 pages 569 576 language Russian issue 78 Category Nonassociative algebras Category Lie algebras he uk ... more details
In mathematics , the symmetric algebra S V also denoted Sym V on a vector space V over a field mathematics field K is the Free object free commutative unital algebra unital associative algebra over K containing ... tensor s in V . A Frobenius algebra whose bilinear form is symmetric bilinear form symmetric is also called a symmetric algebra , but is not discussed here. Construction It turns out that S V is in effect ... this way has some advantage. It is possible to use the tensor algebra T V to describe the symmetric algebra S V . In fact we pass from the tensor algebra to the symmetric algebra by forcing it to be commutative if elements of V commute, then tensors in them must, so that we construct the symmetric algebra from the tensor algebra by taking the quotient algebra of T V by the ideal ring theory ideal ... algebra , into summands S sup k sup V which consist of the linear span of the monomial s in vectors ... operators defined on V sup k sup . Distinction with symmetric tensors The symmetric algebra and symmetric tensors are easily confused the symmetric algebra is a quotient of the tensor algebra, while the symmetric tensors are a subspace of the tensor algebra. The symmetric algebra must be a quotient to satisfy its universal property since every symmetric algebra is an algebra, the tensor algebra maps to the symmetric algebra . Conversely, symmetric tensors are defined as invariants given the natural action of the symmetric group on the tensor algebra, the symmetric tensors are the subspace ... , as described in quadratic form s. In characteristic 0 symmetric tensors and the symmetric algebra can be identified. In any characteristic, there is a symmetrization map from the symmetric algebra ... algebra and the quotient to the symmetric algebra is multiplication by math k math on the k th graded ... spaces, and one can identify symmetric tensors with elements of the symmetric algebra. One divides by math ... group in characteristic 0, over an algebraically closed field, the group algebra is Semisimple ... more details
In mathematics , specifically in ring theory , an algebra ring theory algebra is simple if it contains no non trivial two sided ideal ring theory ideal s and the set ab a , b are elements of the algebra &ne 0 . The second condition in the definition precludes the following situation consider the algebra math begin bmatrix 0 & alpha 0 & 0 end bmatrix , , alpha in mathbb C math with the usual matrix operations. This is a one dimensional algebra in which the product of any two elements is zero. This condition ensures that the algebra has a minimal nonzero left ideal, which simplifies certain arguments. An immediate example of simple algebras are division algebras, where every element has a multiplicative inverse, for instance, the real algebra of quaternions . Also, one can show that the algebra of n × n matrices with entries in a division ring is simple. In fact, this characterizes all finite dimensional simple algebras up to isomorphism, i.e. any finite dimensional simple algebra is isomorphic to a matrix algebra over some division ring . This result was given in 1907 by Joseph Wedderburn ... Mathematical Society . Wedderburn s thesis classified simple and semisimple algebra s. Simple algebras are building blocks of semi simple algebras any finite dimensional semi simple algebra is a Cartesian ... to semisimple ring s in the Artin Wedderburn theorem . Examples A central simple algebra sometimes called Brauer algebra is a simple finite dimensional algebra over a field mathematics field F whose center of an algebra center is F . Simple universal algebras In universal algebra , an abstract algebra A is called simple if and only if it has no nontrivial congruence relation s, or equivalently ... in the sense of universal algebra. See also simple group simple ring central simple algebra References ... , 2003, ISBN 0 8218 1024 3. P.37. Category Algebras Category Ring theory it Algebra semplice ja nl Enkelvoudige algebra ... more details
A braid algebra can be A Gerstenhaber algebra also called an antibracket algebra . An algebra related to the braid group . disambig Short pages monitor This long comment was added to the page to prevent it being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Longcomment. Please do not remove the monitor template without removing the comment as well. ... more details
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