In mathematics , multilinearalgebra extends the methods of linear algebra . Just as linear algebra is built on the concept of a vector space vector and develops the theory of vector space s, multilinearalgebra builds on the concepts of p vector s and multivector s with Grassmann algebra . Origin In a vector ... of the determinant is the most immediate application. Multilinearalgebra also has applications ... , and re published in 1862, his work was slow to find acceptance as ordinary linear algebra provided sufficient challenges to comprehension. The topic of multilinearalgebra is applied in some studies ... in multivariate calculus, and their manipulation is done with exterior algebra . After some preliminary work by Elwin Bruno Christoffel , a major advance in multilinearalgebra came in the work of Gregorio ... form of multilinearalgebra that Marcel Grossman and Michele Besso introduced to Albert Einstein ... of Mercury, established multilinearalgebra and tensor s as important mathematics. Use ... abstractly. The Nicolas Bourbaki Bourbaki group s treatise MultilinearAlgebra was especially influential &mdash in fact the term multilinearalgebra was probably coined there. Fact date April 2008 One reason at the time was a new area of application, homological algebra . The development of algebraic ... problems in terms of multilinearalgebra, there is a clear and well defined best solution the constraints ... of old and new viewpoints. Topics in multilinearalgebra The subject matter of multilinearalgebra ... to it tensor dual space bilinear operator inner product multilinear map Exterior algebra Cramer s rule ... of view of applications Some of the ways in which multilinearalgebra concepts are applied classical ... Mathematics footer Category Multilinearalgebra ca lgebra multilineal es lgebra multilineal .... This practical reference led to the use of the word tensor to describe the elements of the multilinear space. The extra structure in a multilinear space has led it to play an important role in various ... more details
In multilinearalgebra , a multilinear form is a Map mathematics map of the type math f V n to bold K , math where V is a vector space over the field mathematics field K , that is separately linear in each its n variables. ref MathWorld title Multilinear Form urlname MultilinearForm ref For n 2, i.e. only two variables, one calls a bilinear form . An important type of multilinear forms are alternating multilinear forms which have the additional property of changing their sign under exchange of two arguments. When K has characteristic algebra characteristic other than 2, this is equivalent to saying that math f dots,x, dots,x, dots 0 , math i.e. the form vanishes if supplied the same argument twice. The exceptional case of characteristic 2 requires more care. Special cases of these are determinant forms and differential form s. See also Homogeneous polynomial References Reflist DEFAULTSORT Multilinear Form Category Multilinearalgebra cs Multiline rn forma de Multilinearform es Forma multilineal fr Forme multilin aire pt Fun o n linear zh ... more details
In algebra, a multilinear polynomial is a polynomial that is linear in each of its variables. In other words, no variable occurs to a power of 2 or higher or alternatively, each monomial is a constant times a product of distinct variables. They are important in the study of polynomial identity testing , because if a multilinear polynomial is zero on a set of vectors that Linear span span the space, it will be zero everywhere. The Degree of a polynomial degree of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial. ref A. Giambruno, Mikhail Zaicev. Polynomial Identities and Asymptotic Methods. AMS Bookstore, 2005 ISBN 978 0 82183829 7. Section 1.3. ref References references Category Polynomials ... more details
In linear algebra , a multilinear map is a function mathematics function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function math f colon V ... function of math v i math . ref Lang. Algebra. Springer 3rd edition January 8, 2002 ref A multilinear map of two variables is a bilinear map . More generally, a multilinear map of k variables is called a k linear map . If the codomain of a multilinear map is the field of scalars, it is called a multilinear form . Multilinear maps and multilinear forms are fundamental objects of study in multilinearalgebra . If all variables belong to the same space, one can consider symmetric function symmetric ... Multilinear projection References references Category Multilinearalgebra de Multilineare Abbildung ... Any bilinear map is a multilinear map. For example, any inner product on a vector space is a multilinear ... is an antisymmetric multilinear function of the columns or rows of a square matrix . If math F colon ... m to mathbb R n math . The Multilinear subspace learning Tensor to vector projection .28TVP.29 tensor to vector projection in multilinear subspace learning is a multilinear map as well. Coordinate representation Let math f colon V 1 times cdots times V n to W text , math be a multilinear map between ... has dimension math d math . If we choose a basis linear algebra basis math textbf e i1 , ldots, textbf ... math completely determine the multilinear function math f math . In particular, if math textbf v i ... Relation to tensor products There is a natural one to one correspondence between multilinear maps ... by the formula math F v 1 otimes cdots otimes v n f v 1, ldots,v n . math Multilinear functions on n × n matrices One can consider multilinear functions, on an n × n matrix over a commutative ... A be such a matrix and math a i math , 1 i n be the rows of A . Then the multilinear function D can ... × 2 matrices math D A A 1,1 A 2,2 A 1,2 A 2,1 , math Properties A multilinear map has a value ... more details
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 is the branch of mathematics ... , 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 the concept ... 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 ... rings and field mathematics fields , studied in the area of mathematics called abstract 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 ... segments to which letters had been had been attached, and the geometric proofs in al Khwarizmi 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 ... and Diophantus ref http library.thinkquest.org 25672 diiophan.htm Diophantus, Father of Algebra ref ... 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 ... point to the fact that the algebra found in Al Jabr is slightly more elementary than the algebra .... ref supported by geometric proofs, while treating algebra as an independent discipline in its ... more details
, SIAM Review 51 3 2009 455 500. ref . They both employ multilinearalgebra tools. The difference ...Multilinear subspace learning MSL aims to learn a specific small part of a large space of multidimensional ... haiping Publication SurveyMSL PR2011.pdf title A Survey of Multilinear Subspace Learning for Tensor ... tensor space or vector space is named as Multilinear subspace learning Multilinear projection multilinear projection ref name MSLsurvey ref name mlprjICCV cite conference first M.A.O. last Vasilescu first2 D. last2 Terzopoulos title Multilinear Projection for Appearance Based Recognition in the Tensor ... 2007 doi 10.1109 ICCV.2007.4409067 ref . Multilinear subspace learning Algorithms MSL methods are higher ... 10.1109 TNN.2007.901277 MPCA Multilinear principal component analysis of tensor objects , IEEE ... . Multilinear projection A multilinear subspace is defined through a multilinear projection ... citation.cfm?id 354398 A multilinear singular value decomposition , SIAM Journal of Matrix ... to a scalar is an elementary multilinear projection EMP . In EMP, a tensor is projected to a point ... problem. The solution obtained is local optimum . Algorithms Multilinear extension of principal component analysis PCA Multilinear principal component analysis Multilinear Principal Component Analysis MPCA ref name MPCA Uncorrelated Multilinear Principal Component Analysis UMPCA ref H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, http dx.doi.org 10.1109 TNN.2009.2031144 Uncorrelated multilinear principal component analysis for unsupervised multilinear subspace learning , IEEE Trans. Neural Netw., vol. 20, no. 11, pp. 1820 1836, Nov. 2009. ref Multilinear extension of linear discriminant .... Pattern Anal. Mach. Intell., vol. 29, no. 10, pp. 1700 1715, Oct. 2007. ref Uncorrelated Multilinear ... 10.1109 TNN.2008.2004625 Uncorrelated multilinear discriminant analysis with regularization and aggregation ..., Jan. 2009. ref Pedagogical resources Survey http dx.doi.org 10.1016 j.patcog.2011.01.004 A survey of multilinear ... more details
Multilinear principal component analysis MPCA ref name MPCA H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, http www.dsp.utoronto.ca haiping Publication MPCA TNN08 rev2010.pdf MPCA Multilinear principal component analysis of tensor objects , IEEE Trans. Neural Netw., vol. 19, no. 1, pp. 18 39, Jan. 2008. ref is a mathematical procedure that uses multiple orthogonal transformations to convert a set of multidimensional objects into another set of multidimensional objects of lower dimensions. There is one orthogonal transformation for each dimension mode . This transformation aims to capture as high a variance as possible, accounting for as much of the variability in the data as possible, subject to the constraint of mode wise orthogonality. MPCA is a multilinear extension of principal component analysis PCA and it is a basic algorithm in multilinear subspace learning . Its origin can be traced back to the Tucker decomposition ref Cite journal author Ledyard R Tucker title Some mathematical notes on three mode factor analysis journal Psychometrika volume 31 issue 3 month September year ... citation.cfm?id 354398 A multilinear singular value decomposition , SIAM Journal of Matrix Analysis ... MPCA performs feature extraction by determining a Multilinear subspace learning Multilinear projection multilinear projection that captures most of the original tensorial input variations. As in PCA ... www.dsp.utoronto.ca haiping Publication UMPCA TNN09.pdf Uncorrelated multilinear principal component analysis for unsupervised multilinear subspace learning , IEEE Trans. Neural Netw., vol. 20, no. 11 ... uncorrelated multilinear features. Feature selection MPCA produces tensorial features. For conventional ... url http www.dsp.utoronto.ca haiping Publication SurveyMSL PR2011.pdf title A Survey of Multilinear .... ref Non negative MPCA NMPCA ref Y. Panagakis, C. Kotropoulos, G. R. Arce, Non negative multilinear ... RMPCA ref K. Inoue, K. Hara, K. Urahama, Robust multilinear principal component analysis , Proc. IEEE ... more details
1989 isbn 3 540 64243 9 Category Algebras Category Multilinearalgebra Category Polynomials Category ...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 ... square and denoted Sym sup 2 sup V . It has a universal property with respect to symmetric multilinear 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 ... more details
In abstract algebra , an alternative algebra is an algebra over a field algebra in which multiplication ... math math yx x y xx math for all x and y in the algebra. Every associative algebra is obviously alternative, but so too are some strictly Algebra over a field Non associative algebras nonassociative algebras ... is alternating form alternating . The associator is a multilinear map trilinear map given by math x,y,z xy z x yz math By definition a multilinear map is alternating if it vanishes whenever two of it arguments are equal. The left and right alternative identities for an algebra are equivalent to math ... called flexible identity math xy x x yx . math The associator of an alternative algebra is therefore alternating. Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of left alternative identity math x xy xx y math right .... The converse holds so long as the characteristic algebra characteristic of the base field is not 2. Properties Artin s theorem states that in an alternative algebra the subalgebra generated by any two elements is associative . Conversely, any algebra for which this is true is clearly alternative. It follows ... in an alternative algebra. A generalization of Artin s theorem states that whenever three elements math x,y,z math in an alternative algebra associate i.e. math x,y,z 0 math the subalgebra generated ... math a x ay axa y math math xa y a x aya math math ax ya a xy a math hold in any alternative algebra. In a unital alternative algebra, multiplicative inverses are unique whenever they exist. Moreover ... and forms a Moufang loop . This loop of units in an alternative ring or algebra is analogous to the group of units in an associative ring or algebra. Applications The projective plane over any alternative division ring is a Moufang plane . See also Zorn ring Maltsev algebra References Cite book first ... Alternative rings and algebras DEFAULTSORT Alternative Algebra Category Nonassociative algebras de ... more details
Stanis aw Lem s Love and Tensor Algebra Category Algebras Category Multilinearalgebra ar ...Unreferenced date March 2009 In mathematics , the tensor algebra of a vector space V , denoted T V or T sup sup V , is the algebra over a field algebra of tensor s on V of any rank with multiplication being the tensor product . It is the free algebra on V , in the sense of being left adjoint to the forgetful functor from algebras to vector spaces it is the most general algebra containing V , in the sense of the corresponding universal property see below . The tensor algebra also has two Coalgebra ... one, which yields a bialgebra , and can be extended with an antipode to a Hopf algebra structure. Note In this article, all algebras are assumed to be unital algebra unital and associative algebra ... algebra T V is naturally a graded algebra with T sup k sup V serving as the grade k subspace. This grading ... k . The construction generalizes in straightforward manner to the tensor algebra of any module mathematics ... The tensor algebra T V is also called the free algebra on the vector space V , and is functorial ... forgetful functor. In this case, it s the functor which sends each K algebra to its underlying vector space. Explicitly, the tensor algebra satisfies the following universal property , which formally expresses the statement that it is the most general algebra containing V Any linear transformation f V &rarr A from V to an algebra A over K can be uniquely extended to an algebra homomorphism from ... property of the tensor algebra Here i is the canonical inclusion of V into T V the unit of the adjunction . One can, in fact, define the tensor algebra T V as the unique algebra satisfying this property ... algebra is functorial in nature. That is, T is a functor from the K Vect , category of vector ... map from V to W extends uniquely to an algebra homomorphism from T V to T W . Non commutative polynomials If V has finite dimension n , another way of looking at the tensor algebra is as the algebra ... more details
in linear algebra. Linear algebra is a branch of mathematics that studies vector space s, also called ... mathematics matrices if a Basis linear algebra basis is given. Thus Matrix mathematics matrix theory is often considered as a part of linear algebra. Linear algebra is commonly restricted to the case ... dimensional case are traditionally covered in linear functional analysis . Linear algebra is central to modern mathematics and its applications. An elementary application of linear algebra is to find ... are ubiquitous in areas as diverse as abstract algebra and functional analysis . Linear algebra has ... algebra . matrix mathematics Matrices and tensors were introduced in the latter part of the 19th ... much to spread the subject of linear algebra beyond pure mathematics. The origin of many of these ideas ... structures of linear algebra are vector space s and linear map s between them. A vector space is a set ... algebra include linear combination , span linear algebra span , linear independence of vectors, a basis linear algebra basis of a vector space, and the dimension linear algebra dimension of a vector ... in linear algebra. This is a potentially useful remark, but a proper context needs to be set ... a basis linear algebra basis . ref The existence of a basis is straightforward for finitely generated ... finite dimension are isomorphic to each other. Generalizations and related topics Since linear algebra ... mathematics module theory, one replaces the field mathematics field of scalars by a ring. In multilinearalgebra , one considers multivariable linear transformations, that is, mappings that are linear ... product . Functional analysis mixes the methods of linear algebra with those of mathematical analysis . See also List of linear algebra topics Numerical linear algebra Eigenvectors Transformation ... of Linear Algebra http mathdl.maa.org images upload library 22 Ford DesmondFearnleySander.pdf ... 26, 2004 title Linear Algebra Done Right publisher Springer edition 2nd isbn 978 0387982588 ... 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
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 algebraAlgebra , named for the Persian mathematician Mu ammad ibn M s al Khw rizm al ... different types of mathematical structures. 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 ... understood to be algebra . As a mathematical structure Several different classes of algebraic structures ... include In ring theory 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 .... 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 , the dual notion. A different class of algebras consists of objects which ... 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 F algebra math F math algebra F coalgebra math F math coalgebra Other Algebra ... 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
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
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
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 , 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
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