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geometry > algebraic geometry > homotopy associative algebra

topology > algebraic topology > homotopical algebra > homotopy associative algebra

... > abstract algebra > algebraic structure > homological algebra > homotopical algebra > homotopy associative algebra

topology > algebraic topology > homotopy theory > homotopy associative algebra

Terme préférentiel

homotopy associative algebra

Définition(s)

In mathematics, an algebra such as ${\\displaystyle (\\mathbb {R} ,+,\\cdot )}$ has multiplication ${\\displaystyle \\cdot }$ whose associativity is well-defined on the nose. This means for any real numbers ${\\displaystyle a,b,c\\in \\mathbb {R} }$ we have
${\\displaystyle a\\cdot (b\\cdot c)-(a\\cdot b)\\cdot c=0}$ .
But, there are algebras ${\\displaystyle R}$ which are not necessarily associative, meaning if ${\\displaystyle a,b,c\\in R}$ then
${\\displaystyle a\\cdot (b\\cdot c)-(a\\cdot b)\\cdot c\ eq 0}$
in general. There is a notion of algebras, called ${\\displaystyle A_{\\infty }}$ -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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Homotopy_associative_algebra)

Concept(s) générique(s)

Traductions

algèbre associative d'homotopie

français

URI

http://data.loterre.fr/ark:/67375/PSR-RS9D4FLP-X

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RDF/XML TURTLE JSON-LD Date de création 23/08/2023, dernière modif. 23/08/2023