|
Deligne's Conjecture On Hochschild Cohomology
In deformation theory, a branch of mathematics, Deligne's conjecture is about the operadic structure on Hochschild cochain complex. Various proofs have been suggested by Dmitry Tamarkin, Alexander A. Voronov, James E. McClure and Jeffrey H. Smith, Maxim Kontsevich and Yan Soibelman, and others, after an initial input of construction of homotopy algebraic structures on the Hochschild complex. It is of importance in relation with string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera .... See also * Piecewise algebraic space References {{reflist Further reading * https://ncatlab.org/nlab/show/Deligne+conjecture * https://mathoverflow.net/questions/374/delignes-conjecture-the-little-discs-operad-one Algebraic topology String theory Conjectures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation Theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces. Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of ''isolated solutions'', in that varying a solution may not be possible, ''or'' does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operad
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one defines an ''algebra over O'' to be a set together with concrete operations on this set which behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations. History Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1968 and by J. Peter May in 1972. Martin Markl, Steve Shnider, and Jim Stasheff write in their book on operads:"Operads in Algebra, Topology and Physics": Martin Markl, Steve Shnider, Jim Stasheff ... [...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''-alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dmitry Tamarkin
Dmitry (); Church Slavic form: Dimitry or Dimitri (); ancient Russian forms: D'mitriy or Dmitr ( or ) is a male given name common in Orthodox Christian culture Christian culture generally includes all the cultural practices which have developed around the religion of Christianity. There are variations in the application of Christian beliefs in different cultures and traditions. Christian culture has i ..., the Russian version of Demetrios (, ). The meaning of the name is "devoted to, dedicated to, or follower of Demeter" (Δημήτηρ, ''Dēmētēr''), "mother-earth", the Greek mythology, Greek goddess of agriculture. Short forms of the name from the 13th–14th centuries are Mit, Mitya, Mityay, Mit'ka or Miten'ka (, or ); from the 20th century (originated from the Church Slavic form) are Dima, Dimka, Dimochka, Dimulya, Dimusha, Dimon etc. (, etc.) St. Dimitri's Day The feast of the martyr Saint Demetrius, Saint Demetrius of Thessalonica is celebrated on Saturday befor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
