In mathematics, an algebra such as ( R , + , ⋅ ) {\displaystyle (\mathbb {R} ,+,\cdot )} has multiplication ⋅ {\displaystyle \cdot } whose associativity is well-defined on the nose. This means for any real numbers a , b , c ∈ R {\displaystyle a,b,c\in \mathbb {R} } we have
a ⋅ ( b ⋅ c ) − ( a ⋅ b ) ⋅ c = 0 {\displaystyle a\cdot (b\cdot c)-(a\cdot b)\cdot c=0} . But, there are algebras R {\displaystyle R} which are not necessarily associative, meaning if a , b , c ∈ R {\displaystyle a,b,c\in R} then
a ⋅ ( b ⋅ c ) − ( a ⋅ b ) ⋅ c ≠ 0 {\displaystyle a\cdot (b\cdot c)-(a\cdot b)\cdot c\neq 0}
in general. There is a notion of algebras, called A ∞ {\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. The study of A ∞ {\displaystyle A_{\infty }} -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 A ∞ {\displaystyle A_{\infty }} -algebra ( A ∙ , m i ) {\displaystyle (A^{\bullet },m_{i})} is a Z {\displaystyle \mathbb {Z} } -graded vector space over a field k {\displaystyle k} with a series of operations m i {\displaystyle m_{i}} on the i {\displaystyle i} -th tensor powers of A ∙ {\displaystyle A^{\bullet }} . The m 1 {\displaystyle m_{1}} corresponds to a chain complex differential, m 2 {\displaystyle m_{2}} is the multiplication map, and the higher m i {\displaystyle m_{i}} are a measure of the failure of associativity of the m 2 {\displaystyle m_{2}} . When looking at the underlying cohomology algebra H ( A ∙ , m 1 ) {\displaystyle H(A^{\bullet },m_{1})} , the map m 2 {\displaystyle m_{2}} should be an associative map. Then, these higher maps m 3 , m 4 , … {\displaystyle m_{3},m_{4},\ldots } should be interpreted as higher homotopies, where m 3 {\displaystyle m_{3}} is the failure of m 2 {\displaystyle m_{2}} to be associative, m 4 {\displaystyle m_{4}} is the failure for m 3 {\displaystyle m_{3}} 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.
Definition
Definition For a fixed field k {\displaystyle k} an A ∞ {\displaystyle A_{\infty }} -algebra is a Z {\displaystyle \mathbb {Z} } -graded vector space
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