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

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[1] 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[2][3] 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

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Definition

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For a fixed field   an  -algebra[1] is a  -graded vector space

 

such that for   there exist degree  ,  -linear maps

 

which satisfy a coherence condition:

 ,

where  .

Understanding the coherence conditions

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The coherence conditions are easy to write down for low degrees[1]pgs 583–584.

For   this is the condition that

 ,

since   giving   and  . These two inequalities force   in the coherence condition, hence the only input of it is from  . Therefore   represents a differential.

Unpacking the coherence condition for   gives the degree   map  . In the sum there are the inequalities

 

of indices giving   equal to  . Unpacking the coherence sum gives the relation

 ,

which when rewritten with

  and  

as the differential and multiplication, it is

 ,

which is the Leibniz rule for differential graded algebras.

In this degree the associativity structure comes to light. Note if   then there is a differential graded algebra structure, which becomes transparent after expanding out the coherence condition and multiplying by an appropriate factor of  , the coherence condition reads something like

 

Notice that the left hand side of the equation is the failure for   to be an associative algebra on the nose. One of the inputs for the first three   maps are coboundaries since   is the differential, so on the cohomology algebra   these elements would all vanish since  . This includes the final term   since it is also a coboundary, giving a zero element in the cohomology algebra. From these relations we can interpret the   map as a failure for the associativity of  , meaning it is associative only up to homotopy.

d=4 and higher order terms

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Moreover, the higher order terms, for  , the coherent conditions give many different terms combining a string of consecutive   into some   and inserting that term into an   along with the rest of the  's in the elements  . When combining the   terms, there is a part of the coherence condition which reads similarly to the right hand side of  , namely, there are terms

 

In degree   the other terms can be written out as

 

showing how elements in the image of   and   interact. This means the homotopy of elements, including one that's in the image of   minus the multiplication of elements where one is a homotopy input, differ by a boundary. For higher order  , these middle terms can be seen how the middle maps   behave with respect to terms coming from the image of another higher homotopy map.

Diagrammatic interpretation of axioms

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There is a nice diagrammatic formalism of algebras which is described in Algebra+Homotopy=Operad[4] explaining how to visually think about this higher homotopies. This intuition is encapsulated with the discussion above algebraically, but it is useful to visualize it as well.

Examples

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Associative algebras

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Every associative algebra   has an  -infinity structure by defining   and   for  . Hence  -algebras generalize associative algebras.

Differential graded algebras

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Every differential graded algebra   has a canonical structure as an  -algebra[1] where   and   is the multiplication map. All other higher maps   are equal to  . Using the structure theorem for minimal models, there is a canonical  -structure on the graded cohomology algebra   which preserves the quasi-isomorphism structure of the original differential graded algebra. One common example of such dga's comes from the Koszul algebra arising from a regular sequence. This is an important result because it helps pave the way for the equivalence of homotopy categories

 

of differential graded algebras and  -algebras.

Cochain algebras of H-spaces

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One of the motivating examples of  -algebras comes from the study of H-spaces. Whenever a topological space   is an H-space, its associated singular chain complex   has a canonical  -algebra structure from its structure as an H-space.[3]

Example with infinitely many non-trivial mi

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Consider the graded algebra   over a field   of characteristic   where   is spanned by the degree   vectors   and   is spanned by the degree   vector  .[5][6] Even in this simple example there is a non-trivial  -structure which gives differentials in all possible degrees. This is partially due to the fact there is a degree   vector, giving a degree   vector space of rank   in  . Define the differential   by

 

and for  

 

where   on any map not listed above and  . In degree  , so for the multiplication map, we have   And in   the above relations give

 

When relating these equations to the failure for associativity, there exist non-zero terms. For example, the coherence conditions for   will give a non-trivial example where associativity doesn't hold on the nose. Note that in the cohomology algebra   we have only the degree   terms   since   is killed by the differential  .

Properties

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Transfer of A structure

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One of the key properties of  -algebras is their structure can be transferred to other algebraic objects given the correct hypotheses. An early rendition of this property was the following: Given an  -algebra   and a homotopy equivalence of complexes

 ,

then there is an  -algebra structure on   inherited from   and   can be extended to a morphism of  -algebras. There are multiple theorems of this flavor with different hypotheses on   and  , some of which have stronger results, such as uniqueness up to homotopy for the structure on   and strictness on the map  .[7]

Structure

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Minimal models and Kadeishvili's theorem

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One of the important structure theorems for  -algebras is the existence and uniqueness of minimal models – which are defined as  -algebras where the differential map   is zero. Taking the cohomology algebra   of an  -algebra   from the differential  , so as a graded algebra,

 ,

with multiplication map  . It turns out this graded algebra can then canonically be equipped with an  -structure,

 ,

which is unique up-to quasi-isomorphisms of  -algebras.[8] In fact, the statement is even stronger: there is a canonical  -morphism

 ,

which lifts the identity map of  . Note these higher products are given by the Massey product.

Motivation

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This theorem is very important for the study of differential graded algebras because they were originally introduced to study the homotopy theory of rings. Since the cohomology operation kills the homotopy information, and not every differential graded algebra is quasi-isomorphic to its cohomology algebra, information is lost by taking this operation. But, the minimal models let you recover the quasi-isomorphism class while still forgetting the differential. There is an analogous result for A∞-categories by Maxim Kontsevich and Yan Soibelman, giving an A∞-category structure on the cohomology category   of the dg-category consisting of cochain complexes of coherent sheaves on a non-singular variety   over a field   of characteristic   and morphisms given by the total complex of the Cech bi-complex of the differential graded sheaf  [1]pg 586-593. In this was, the degree   morphisms in the category   are given by  .

Applications

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There are several applications of this theorem. In particular, given a dg-algebra, such as the de Rham algebra  , or the Hochschild cohomology algebra, they can be equipped with an  -structure.

Massey structure from DGA's

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Given a differential graded algebra   its minimal model as an  -algebra   is constructed using the Massey products. That is,

 

It turns out that any  -algebra structure on   is closely related to this construction. Given another  -structure on   with maps  , there is the relation[9]

 ,

where

 .

Hence all such  -enrichments on the cohomology algebra are related to one another.

Graded algebras from its ext algebra

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Another structure theorem is the reconstruction of an algebra from its ext algebra. Given a connected graded algebra

 ,

it is canonically an associative algebra. There is an associated algebra, called its Ext algebra, defined as

 ,

where multiplication is given by the Yoneda product. Then, there is an  -quasi-isomorphism between   and  . This identification is important because it gives a way to show that all derived categories are derived affine, meaning they are isomorphic to the derived category of some algebra.

See also

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References

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  1. ^ a b c d e Aspinwall, Paul (2009). Dirichlet branes and mirror symmetry. American Mathematical Society. ISBN 978-0-8218-3848-8. OCLC 939927173.
  2. ^ Stasheff, Jim (2018-09-04). "L and A structures: then and now". arXiv:1809.02526 [math.QA].
  3. ^ a b Stasheff, James Dillon (1963). "Homotopy Associativity of H-Spaces. II". Transactions of the American Mathematical Society. 108 (2): 293–312. doi:10.2307/1993609. ISSN 0002-9947. JSTOR 1993609.
  4. ^ Vallette, Bruno (2012-02-15). "Algebra+Homotopy=Operad". arXiv:1202.3245 [math.AT].
  5. ^ Allocca, Michael; Lada, Thomas. "A Finite Dimensional A-infinity algebra example" (PDF). Archived (PDF) from the original on 28 Sep 2020.
  6. ^ Daily, Marilyn; Lada, Tom (2005). "A finite dimensional $L_\infty$ algebra example in gauge theory". Homology, Homotopy and Applications. 7 (2): 87–93. doi:10.4310/HHA.2005.v7.n2.a4. ISSN 1532-0073.
  7. ^ Burke, Jesse (2018-01-26). "Transfer of A-infinity structures to projective resolutions". arXiv:1801.08933 [math.KT].
  8. ^ Kadeishvili, Tornike (2005-04-21). "On the homology theory of fibre spaces". arXiv:math/0504437.
  9. ^ Buijs, Urtzi; Moreno-Fernández, José Manuel; Murillo, Aniceto (2019-02-19). "A-infinity structures and Massey products". arXiv:1801.03408 [math.AT].