About: Homotopy associative algebra

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In mathematics, an algebra such as has multiplication whose associativity is well-defined on the nose. This means for any real numbers we have . But, there are algebras which are not necessarily associative, meaning if then They are ubiquitous in homological mirror symmetry because of their necessity in defining the structure of the Fukaya category of D-branes on a Calabi–Yau manifold who have only a homotopy associative structure.

Attributes	Values
rdfs:label	Homotopy associative algebra (en)
rdfs:comment	In mathematics, an algebra such as has multiplication whose associativity is well-defined on the nose. This means for any real numbers we have . But, there are algebras which are not necessarily associative, meaning if then They are ubiquitous in homological mirror symmetry because of their necessity in defining the structure of the Fukaya category of D-branes on a Calabi–Yau manifold who have only a homotopy associative structure. (en)
dct:subject	Algebra Homotopy theory Homological algebra Algebraic geometry Homotopical algebra
Wikipage page ID	64508219 (xsd:integer)
Wikipage revision ID	1096406275 (xsd:integer)
Link from a Wikipage to another Wikipage	Product rule Mirror symmetry conjecture De Rham cohomology Derived algebraic geometry Derived noncommutative algebraic geometry Maxim Kontsevich Calabi–Yau manifold Fukaya category Koszul algebra Algebraic variety Algebra D-brane Homotopy theory H-space Jim Stasheff Associahedron Associative property Homological algebra Algebraic geometry Homotopical algebra Hochschild homology Homological mirror symmetry Homotopical algebra Homotopy Homotopy Lie algebra Regular sequence Associative algebra A∞-operad Inequality (mathematics) Chain complex Massey product Singular homology Čech cohomology Yan Soibelman Yoneda product Derived categories Smooth variety dbr:Arxiv:1202.3245 dbr:A∞-category
Link from a Wikipage to an external page	https://escholarship.org/uc/item/7v313232 http://www.claymath.org/library/monographs/cmim04.pdf
sameAs	Homotopy associative algebra Homotopy associative algebra
dbp:wikiPageUsesTemplate	dbt:Cite_arXiv dbt:Cite_book dbt:Cite_journal dbt:Cite_thesis dbt:Refbegin dbt:Refend
has abstract	In mathematics, an algebra such as has multiplication whose associativity is well-defined on the nose. This means for any real numbers we have . But, there are algebras which are not necessarily associative, meaning if then in general. There is a notion of algebras, called -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 -algebras is a subset of homotopical algebra, where there is a homotopical notion of associative algebras through a differential graded algebra with a multiplication operation and a series of higher homotopies giving the failure for the multiplication to be associative. Loosely, an -algebra is a -graded vector space over a field with a series of operations on the -th tensor powers of . The corresponds to a chain complex differential, is the multiplication map, and the higher are a measure of the failure of associativity of the . When looking at the underlying cohomology algebra , the map should be an associative map. Then, these higher maps should be interpreted as higher homotopies, where is the failure of to be associative, is the failure for to be higher associative, and so forth. Their structure was originally discovered by Jim Stasheff while studying A∞-spaces, but this was interpreted as a purely algebraic structure later on. These are spaces equipped with maps that are associative only up to homotopy, and the A∞ structure keeps track of these homotopies, homotopies of homotopies, and so forth. They are ubiquitous in homological mirror symmetry because of their necessity in defining the structure of the Fukaya category of D-branes on a Calabi–Yau manifold who have only a homotopy associative structure. (en)
prov:wasDerivedFrom	wikipedia-en:Homotopy_associative_algebra?oldid=1096406275&ns=0
page length (characters) of wiki page	21805 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Homotopy_associative_algebra
is Link from a Wikipage to another Wikipage of	Mirror symmetry conjecture Fukaya category Homotopical algebra Homotopy Lie algebra Differential graded algebra A∞-operad A-infinity algebra
is Wikipage redirect of	A-infinity algebra
is foaf:primaryTopic of	wikipedia-en:Homotopy_associative_algebra

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