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 ... over any ring. algebra A algebra A is a ring that is an associative algebra over a commutative ring ... , math x,y in A math . A homomorphism math f colon A to B math is algebra homomorphism that is compatible ... numbers. A operation on a algebra is an operation on an algebra over a ring that behaves similarly to taking ... 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, quaternion s and octonion ... ring matrix algebra of n × n matrix mathematics matrices over C with given by the conjugate ... is also a star algebra. In Hecke algebra , an involution is important to the Kazhdan Lusztig polynomial ... curve becomes a algebra over the integers, where the involution is given by taking the dual ... notes on abelian varieties . Hopf algebra Examples Involutive Hopf algebras are important examples ... being The group Hopf algebra a group ring , with involution given by math g mapsto g 1 . math ... form a Jordan algebra The skew Hermitian elements form a Lie algebra If 2 is invertible, then math ... 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 Given a ring, there is also the map math x mapsto x math . This is not a ring structure unless the characteristic algebra ... elements, and the imaginary numbers are the skew Hermitian. See also 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 Algebras fr ... more details
Orphan date January 2012 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 the 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. algebras feature prominently in quantum mechanics , and especially quantum information science . References references Category C algebras physics stub algebra stub ... more details
about the branch of mathematics pp move indef sprotect small yes Algebra from Arabic language Arabic al jebr meaning reunion of broken parts ref cite web title algebra work Online Etymology Dictionary ... , topology , combinatorics , and number theory , algebra is one of the main branches of pure mathematics . Elementary algebra , often part of the curriculum in secondary education , 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 Timeline of algebra File Image Al Kit b al mu ta ar f is b ... Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually ... s Algebra made use of lettered diagrams but all coefficients in the equations used in the Algebra are specific ... called the father of algebra , was an Alexandria n Greek mathematics Greek mathematician and the author ... 8 ref While the word algebra comes from the Arabic language lang ar transl ar al jabr restoration ... algebra as a mathematical discipline that is independent of geometry and arithmetic . ref citation title 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 ... Diophantus, Father of Algebra ref as well as Indian mathematics Indian mathematicians such as Brahmagupta ... s Brahmasphutasiddhanta are on a higher level. ref http www.algebra.com algebra about history History of Algebra ref For example, the first complete arithmetic solution including zero and negative ... of algebra but in more recent times there is much debate over whether al Khwarizmi, who founded the discipline ... Edition Wiley, 1991 , pages 178, 181 ref Those who support Diophantus point to the fact that the algebra found in Al Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica ... 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
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
Wiktionarypar algebra The word Algebra describes one of the main branches of mathematics. It can also ... al Khw rizm . As a 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 ... and linear algebraAlgebra ring theory Algebra over a commutative ring a module equipped with a bilinear product 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 s, Poisson algebra s, and Jordan algebra s are important examples of potentially nonassociative algebras. 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 continuous linear operator s on a topological vector space with multiplication given by the composition. algebra An algebra with a notion of adjoint of an operator adjoints . C algebra a Banach algebra equipped with a unary Involution mathematics involution operation. Von Neumann algebra or W algebra See also coalgebra ... s, set mathematics sets , and lattice order lattices . In logic Boolean algebra structure Heyting algebra In 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 The term algebra can also describe more general structures In category theory and computer science ... 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 topology stub algebra stub ... 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
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 . The current Editor in Chief of the Journal of Algebra is Michel Brou , Universit Paris Diderot , whereby Gerhard Hiss, Rheinische Westf lische Technische Hochschule Aachen RWTS is Editor of the Computational Algebra section. 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 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 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
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 Algebras ... more details
Unreferenced date December 2009 In mathematics , a supercommutative algebra is a superalgebra i.e. a Z sub 2 sub graded algebra such that for any two homogeneous element s x , y we have math yx 1 x y xy. , math Equivalently, it is a superalgebra where the supercommutator math x,y xy 1 x y yx , math always vanishes. Algebraic structures which supercommute in the above sense are sometimes referred to as skew commutative associative algebras to emphasize the anti commutation, or, to emphasize the grading, graded commutative or, if the supercommutativity is understood, simply commutative . Any commutative algebra is a supercommutative algebra if given the trivial gradation i.e. all elements are even . Grassmann algebra s also known as exterior algebra s are the most common examples of nontrivial supercommutative algebras. The supercenter of any superalgebra is the set of elements that supercommute with all elements, and is a supercommutative algebra. The even subalgebra of a supercommutative algebra is always a commutative algebra . That is, even elements always commute. Odd elements, on the other hand, always anticommute. That is, math xy yx 0 , math for odd x and y . In particular, the square of any odd element x vanishes whenever 2 is invertible math x 2 0. , math Thus a commutative superalgebra with 2 invertible and nonzero degree one component always contains nilpotent elements. See also Commutative algebra Lie superalgebra DEFAULTSORT Supercommutative Algebra Category Algebras Category Super linear algebra it Algebra supercommutativa ... 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
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 pl Algebra funkcyjna ... more details
In mathematics, a Hecke algebra can be one of several algebras, similar to the algebra of Hecke operator s studied by Erich Hecke . The algebra of Hecke operators can be interpreted as an algebra of double cosets, and as a result the term Hecke algebra is also used for several similar algebras related to double cosets. In particular it can mean Iwahori Hecke algebra of a Coxeter group. Hecke algebra of a pair g,K where g is the Lie algebra of a Lie group G and K is a compact subgroup of G . H G , K , the Hecke algebra of a locally compact group G with respect to a compact subgroup K The Hecke algebra of a locally profinite group such as an algebraic group over a local field , given by the direct limit of the algebras H G , K for K a compact open subgroup. The algebra generated by Hecke operator s acting on modular forms The algebra spanned by the double coset s HgH of a finite index subgroup H of a group G . The centralizer algebra of an induced representation . Category Representation theory Category Set indices ja zh yue Hecke zh Hecke ... more details
B algebras were mathematics mathematical structures studied in functional analysis . As it is now known that all B algebras are C algebras and vice versa , the term B algebra is no longer widely used. General Banach algebras A Banach algebra A is a Banach algebra over the field of complex number s, together with a map A A called involution which has the following properties x y x y for all x , y in A . math lambda x bar lambda x math for every in C and every x in A here, math bar lambda math denotes the complex conjugate of . xy y x for all x , y in A . x x for all x in A . In most natural examples, one also has that the involution is isometry isometric , i.e. x x , B algebras A B algebra is a Banach algebra in which the involution satisfies the following further property x x x sup 2 sup for all x in A . By a theorem of Gelfand and Naimark, given a B algebra A there exists a Hilbert space H and an isometric homomorphism from A into the algebra B H of all bounded linear operators on H . Thus every B algebra is isometrically isomorphic to a C algebra . Because of this, the term B algebra is rarely used in current terminology, and has been replaced by the overloading of the term C algebra . See also Algebra over a field Associative algebra algebra C algebra . References cite book author G. F. Simmons title Introduction to Topology and Modern Analysis publisher McGraw Hill year 1963 isbn 0 07 085695 8 Category Banach algebras Category C algebras it B algebra ms B algebra DEFAULTSORT B Algebra ... 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 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 Algebras Category Boolean algebra Category Topology zh algebra stub ... more details
The following outline is provided as an overview of and topical guide to algebraAlgebra &ndash one of the main ... mathematics relation and quantity . Algebra studies the effects of addition add ing and multiplication ..., algebra also covers symbols , variables, and Set mathematics set element mathematics elements ... fields . Overview Main Algebra Arithmetic Equation s An algebraic equation is an equation involving ... to be integer s. Polynomials Variable mathematics Variables Branches Pre algebra Elementary algebra Abstract algebra Linear algebra Universal algebra History Main History of algebra General algebra concepts Algebra &ndash Cubic equation &ndash Fundamental theorem of algebra &ndash Linear ... &ndash Boolean algebraAlgebra of sets Talk Algebra of sets Algebraic normal form Talk Algebraic normal ... Talk George Boole Boolean algebra structure Talk Boolean algebra structure Boolean algebras canonically ... Boolean domain Boolean function Talk Boolean function Boolean algebra logic Talk Boolean algebra logic ... Boole s syllogistic Canonical form Boolean algebra Talk normal form Boolean algebra Characteristic ... algebra Talk Complete Boolean algebra Consensus theorem Talk Consensus theorem Augustus De Morgan De ... system Talk formal system Free Boolean algebra Talk free Boolean algebra Heyting algebra Talk Heyting algebra Indicator function Talk Indicator function Interior algebra Talk interior algebra William ... Karnaugh map Talk Karnaugh map Laws of Form Talk Laws of Form Lindenbaum Tarski algebra Talk Lindenbaum Tarski algebra Logic gate Talk logic gate Logical connective Talk logical connective Logical ... negation operator Talk Minimal negation operator Monadic Boolean algebra Talk monadic Boolean algebra ... algebra Talk topological Boolean algebra Truth table Talk truth table Two element Boolean algebra Talk Two element Boolean algebra John Venn Venn, John Talk John Venn Venn diagram Talk Venn diagram Zeroth order logic Talk Zeroth order logic See also Portal Algebra Table of mathematical symbols External ... 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 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 tensor 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 ... algebra can be identified. In any characteristic, there is a symmetrization map from the symmetric algebra to the symmetric tensors, given by math v 1 cdots v k mapsto sum sigma in S n v ... in the tensor algebra and the quotient to the symmetric algebra is multiplication by math k math ... of graded vector spaces, and one can identify symmetric tensors with elements of the symmetric algebra ... theory of the symmetric group in characteristic 0, over an algebraically closed field, the group algebra ... 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 . Note that the first only has 0 as a projection. If K is a compact Hausdorff space, then C K is a projectionless algebra if and only if K is connected. References unreferenced date September 2008 Category C algebras algebra stub ... more details
Algebra Universalis ISSN 0002 5240 print edition ISSN 1420 8911 electronic edition is an international scientific journal focused on universal algebra and lattice theory . The journal, founded in 1971, is currently published by Springer Verlag . Honorary editors in chief of the journal include Alfred Tarski and currently Bjarni J nsson . External links http www.springerlink.com content 101190 Algebra Universalis on Springer.com Category Universal algebra Category Mathematics journals Category Publications established in 1971 Category Springer academic journals math journal stub ... more details
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