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Homotopy Associative Algebra
In mathematics, an algebra such as (\R,+,\cdot) has multiplication \cdot whose associativity is well-defined on the nose. This means for any real numbers a,b,c\in \R we have :a\cdot(b\cdot c) - (a\cdot b)\cdot c = 0. But, there are algebras R which are not necessarily associative, meaning if a,b,c\in R then :a\cdot(b\cdot c) - (a\cdot b)\cdot c \neq 0 in general. There is a notion of algebras, called A_\infty-algebras, which still have a property on the multiplication which still acts like the first relation, meaning associativity holds, but only holds up to a homotopy, which is a way to say after an operation "compressing" the information in the algebra, the multiplication is associative. This means although we get something which looks like the second equation, the one of inequality, we actually get equality after "compressing" the information in the algebra. The study of A_\infty-algebras is a subset of homotopical algebra, where there is a homotopical notion of associative algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Sequence
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ... which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection. Definitions For a commutative ring ''R'' and an ''R''-Module (mathematics), module ''M'', an element ''r'' in ''R'' is called a non-zero-divisor on ''M'' if ''r m'' = 0 implies ''m'' = 0 for ''m'' in ''M''. An ''M''-regular sequence is a sequence :''r''1, ..., ''r''''d'' in ''R'' such that ''r''''i'' is a not a zero-divisor on ''M''/(''r''1, ..., ''r''''i''-1)''M'' for ''i'' = 1, ..., ''d''. Some authors also require that ''M''/(''r''1, ..., ''r''''d'')''M'' is not zero. Intuitively, to say that '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoneda Product
In algebra, the Yoneda product (named after Nobuo Yoneda) is the pairing between Ext groups of modules: :\operatorname^n(M, N) \otimes \operatorname^m(L, M) \to \operatorname^(L, N) induced by :\operatorname(N, M) \otimes \operatorname(M, L) \to \operatorname(N, L),\, f \otimes g \mapsto g \circ f. Specifically, for an element \xi \in \operatorname^n(M, N) , thought of as an extension :\xi : 0 \rightarrow N \rightarrow E_0 \rightarrow \cdots \rightarrow E_ \rightarrow M \rightarrow 0 , and similarly :\rho : 0 \rightarrow M \rightarrow F_0\rightarrow \cdots \rightarrow F_ \rightarrow L \rightarrow 0 \in \operatorname^m(L, M), we form the Yoneda (cup) product :\xi \smile \rho : 0 \rightarrow N \rightarrow E_0 \rightarrow \cdots \rightarrow E_ \rightarrow F_0 \rightarrow \cdots \rightarrow F_ \rightarrow L \rightarrow 0 \in \operatorname^(L, N). Note that the middle map E_ \rightarrow F_0 factors through the given maps to M. We extend this definition to include m, n = 0 using th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hochschild Homology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, and extended to algebras over more general rings by . Definition of Hochschild homology of algebras Let ''k'' be a field, ''A'' an associative ''k''-algebra, and ''M'' an ''A''-bimodule. The enveloping algebra of ''A'' is the tensor product A^e=A\otimes A^o of ''A'' with its opposite algebra. Bimodules over ''A'' are essentially the same as modules over the enveloping algebra of ''A'', so in particular ''A'' and ''M'' can be considered as ''Ae''-modules. defined the Hochschild homology and cohomology group of ''A'' with coefficients in ''M'' in terms of the Tor functor and Ext functor by : HH_n(A,M) = \operatorname_n^(A, M) : HH^n(A,M) = \operatorname^n_(A, M) Hochschild complex Let ''k'' be a ring, ''A'' an associative ''k''-algebra th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Rham Cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship. Definition The de Rham complex is the cochain complex of differential forms on some smooth manifold , with the exterior derivative as the differential: :0 \to \Omega^0(M)\ \stackrel\ \Omega^1(M)\ \stackrel\ \Omega^2(M)\ \stackrel\ \Omega^3(M) \to \cd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Čech Cohomology
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech. Motivation Let ''X'' be a topological space, and let \mathcal be an open cover of ''X''. Let N(\mathcal) denote the nerve of the covering. The idea of Čech cohomology is that, for an open cover \mathcal consisting of sufficiently small open sets, the resulting simplicial complex N(\mathcal) should be a good combinatorial model for the space ''X''. For such a cover, the Čech cohomology of ''X'' is defined to be the simplicial cohomology of the nerve. This idea can be formalized by the notion of a good cover. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of ''X'', ordered by refinement. This is the approach adopted below. Construction Let ''X'' be a topological space, and l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Variety
In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology. Definition First, let ''X'' be an affine scheme of finite type over a field ''k''. Equivalently, ''X'' has a closed immersion into affine space ''An'' over ''k'' for some natural number ''n''. Then ''X'' is the closed subscheme defined by some equations ''g''1 = 0, ..., ''g''''r'' = 0, where each ''gi'' is in the polynomial ring ''k'' 'x''1,..., ''x''''n'' The affine scheme ''X'' is smooth of dimension ''m'' over ''k'' if ''X'' has dimension at least ''m'' in a neighborhood of each point, and the matrix of derivatives (∂''g''''i''/∂''x''''j'') has rank at least ''n''−''m'' everywhere on ''X''. (It follows that ''X'' has dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yan Soibelman
Iakov (Yan) Soibelman (Russian: Яков Семенович Сойбельман) born 15 April 1956 ( Kiev, USSR) is a Russian American mathematician, professor at Kansas State University (Manhattan, USA), member of thKyiv Mathematical Society(Ukraine), founder of Manhattan Mathematical Olympiad. Scientific work Yan Soibelman is a specialist in theory of quantum groups, representation theory and symplectic geometry. He introduced the notion of quantum Weyl group, studied representation theory of the algebras of functions on compact quantum groups, and meromorphic braided monoidal categories. His long term collaboration with Maxim Kontsevich is devoted to various aspects of homological mirror symmetry, a proof of Deligne conjecture about operations on the cohomological Hochschild complex, a direct construction of Calabi-Yau varieties based on SYZ conjecture and non-archimedean geometry, and more recently to Donaldson-Thomas theory. Together with Kontsevich he laid the foundat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxim Kontsevich
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami. He received the Henri Poincaré Prize in 1997, the Fields Medal in 1998, the Crafoord Prize in 2008, the Shaw Prize and Fundamental Physics Prize in 2012, and the Breakthrough Prize in Mathematics in 2014. Academic career and research He was born into the family of Lev Kontsevich, Soviet orientalist and author of the Kontsevich system. After ranking second in the All-Union Mathematics Olympiads, he attended Moscow State University but left without a degree in 1985 to become a researcher at the Institute for Information Transmission Problems in Moscow. While at the institute he published papers that caught the interest of the Max Planck Institute in Bonn and was invited for three ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Massey Product
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist. Massey triple product Let a,b,c be elements of the cohomology algebra H^*(\Gamma) of a differential graded algebra \Gamma. If ab=bc=0, the Massey product \langle a,b,c\rangle is a subset of H^n(\Gamma), where n=\deg(a)+\deg(b)+\deg(c)-1. The Massey product is defined algebraically, by lifting the elements a,b,c to equivalence classes of elements u,v,w of \Gamma, taking the Massey products of these, and then pushing down to cohomology. This may result in a well-defined cohomology class, or may result in indeterminacy. Define \bar u to be (-1)^u. The cohomology class of an element u of \Gamma will be denoted by /math>. The Massey triple product of three cohomology classes is defined by : \langle rangle = \. The Massey product of three cohomology classes is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
