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Manin Matrices
In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave like matrices whose elements commute. In particular there is natural definition of the determinant for them and most linear algebra theorems like Cramer's rule, Cayley–Hamilton theorem, etc. hold true for them. Any matrix with commuting elements is a Manin matrix. These matrices have applications in representation theory in particular to Capelli's identity, Yangian and quantum integrable systems. Manin matrices are particular examples of Manin's general construction of "non-commutative symmetries" which can be applied to any algebra. From this point of view they are "non-commutative endomorphisms" of polynomial algebra ''C'' 'x''1, ...''x''n Taking (q)-(super)-commuting variables one will get (q)-(super)-analogs of Manin matrices, which are closely related to quantum groups. Manin w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuri Manin
Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book ''Computable and Uncomputable''. Life and career Manin gained a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor Shafarevich. He is now a Professor at the Max-Planck-Institut für Mathematik in Bonn, and a professor emeritus at Northwestern University. Manin's early work included papers on the arithmetic and formal groups of abelian varieties, the Mordell conjecture in the function field case, and algebraic differential equations. The Gauss–Manin connection is a basic ingredient of the study of cohomology in families of algebraic varieties. He wrote a book on cubic surfaces and cubic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, and , of a group , is the element : . This element is equal to the group's identity if and only if and commute (from the definition , being equal to the identity if and only if ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of ''G'' generated by all commutators is closed and is called the ''derived group'' or the ''commutator subgroup'' of ''G''. Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group. The definition of the commutator above is used throughout this article, but many other group theorists define the commutator as :. Identities (group theory) Commutator identities are an important tool in group theory. The expr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Enveloping Algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Capelli Identity
In mathematics, Capelli's identity, named after , is an analogue of the formula det(''AB'') = det(''A'') det(''B''), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra \mathfrak_n. It can be used to relate an invariant ''ƒ'' to the invariant Ω''ƒ'', where Ω is Cayley's Ω process. Statement Suppose that ''x''''ij'' for ''i'',''j'' = 1,...,''n'' are commuting variables. Write ''E''ij for the polarization operator :E_ = \sum_^n x_\frac. The Capelli identity states that the following differential operators, expressed as determinants, are equal: : \begin E_+n-1 & \cdots &E_& E_ \\ \vdots& \ddots & \vdots&\vdots\\ E_ & \cdots & E_+1&E_ \\ E_ & \cdots & E_& E_ +0\end = \begin x_ & \cdots & x_ \\ \vdots& \ddots & \vdots\\ x_ & \cdots & x_ \end \begin \frac & \cdots &\frac \\ \vdots& \ddots & \vdots\\ \frac & \cdots &\frac \end. Both sides are differential operators. The determinant on the left has non-com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Complement
In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows. Suppose ''p'', ''q'' are nonnegative integers, and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' × ''p'', and ''q'' × ''q'' matrices of complex numbers. Let :M = \left begin A & B \\ C & D \end\right/math> so that ''M'' is a (''p'' + ''q'') × (''p'' + ''q'') matrix. If ''D'' is invertible, then the Schur complement of the block ''D'' of the matrix ''M'' is the ''p'' × ''p'' matrix defined by :M/D := A - BD^C. If ''A'' is invertible, the Schur complement of the block ''A'' of the matrix ''M'' is the ''q'' × ''q'' matrix defined by :M/A := D - CA^B. In the case that ''A'' or ''D'' is singular, substituting a generalized inverse for the inverses on ''M/A'' and ''M/D'' yields the generalized Schur complement. The Schur complement is named after Issai Schur who used it to prove Schur's lemma, although it had been used previous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial ''P'' in one variable, they allow expressing the sums of the ''k''-th powers of all roots of ''P'' (counted with their multiplicity) in terms of the coefficients of ''P'', without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity. Mathematical statement Formulation in terms of symmetric polynomials Let ''x''1, ..., ''x''''n'' be variables, denote for ''k'' ≥ 1 by ''p''''k''(''x''1, ..., ''x''''n'') the ''k''-th power ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjugate Matrix
In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a different concept, the adjoint operator which is the conjugate transpose of the matrix. The product of a matrix with its adjugate gives a diagonal matrix (entries not on the main diagonal are zero) whose diagonal entries are the determinant of the original matrix: :\mathbf \operatorname(\mathbf) = \det(\mathbf) \mathbf, where is the identity matrix of the same size as . Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its adjugate by its determinant. Definition The adjugate of is the transpose of the cofactor matrix of , :\operatorname(\mathbf) = \mathbf^\mathsf. In more detail, suppose is a unital commutative ring and is an matrix with entries from . The -''minor'' of , denoted , is the determ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MacMahon Master Theorem
In mathematics, MacMahon's master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph ''Combinatory analysis'' (1916). It is often used to derive binomial identities, most notably Dixon's identity. Background In the monograph, MacMahon found so many applications of his result, he called it "a master theorem in the Theory of Permutations." He explained the title as follows: "a Master Theorem from the masterly and rapid fashion in which it deals with various questions otherwise troublesome to solve." The result was re-derived (with attribution) a number of times, most notably by I. J. Good who derived it from his multilinear generalization of the Lagrange inversion theorem. MMT was also popularized by Carlitz who found an exponential power series version. In 1962, Good found a short proof of Dixon's identity from MMT. In 1969, Cartier and Foata found a new proof of MMT by combining alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominique Foata
Dominique Foata (born October 12, 1934) is a mathematician who works in enumerative combinatorics. With Pierre Cartier and Marcel-Paul Schützenberger he pioneered the modern approach to classical combinatorics, that lead, in part, to the current blossoming of algebraic combinatorics. His pioneering work on permutation statistics, and his combinatorial approach to special functions, are especially notable. Foata gave an invited talk at the International Congress of Mathematicians in Warsaw (1983). Among his honors are the Scientific Prize of the Union des Assurances de Paris (September 1985). With Adalbert Kerber and Volker Strehl he founded the mathematics journal '' Séminaire Lotharingien de Combinatoire''. He is also one of the contributors of the pseudonymous collective M. Lothaire. In 1985, Foata received the Prix Paul Doistau–Émile Blutet. He was born in Damascus while it was under French mandate. Selected publications Books * with Pierre Cartier : ''Problèmes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Cartier (mathematician)
Pierre Émile Cartier (born 10 June 1932) is a French mathematician. An associate of the Bourbaki group and at one time a colleague of Alexander Grothendieck, his interests have ranged over algebraic geometry, representation theory, mathematical physics, and category theory. He studied at the École Normale Supérieure in Paris under Henri Cartan and André Weil. Since his 1958 thesis on algebraic geometry he has worked in a number of fields. He is known for the introduction of the Cartier operator in algebraic geometry in characteristic ''p'', and for work on duality of abelian varieties and on formal groups. He is the eponym of Cartier divisors and Cartier duality. From 1961 to 1971 he was a professor at the University of Strasbourg. In 1970 he was an Invited Speaker at the International Congress of Mathematicians in Nice. He was awarded the 1978 Prize Ampère of the French Academy of Sciences. In 2012 he became a fellow of the American Mathematical Society. Publications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
