Homotopy associative algebra

In homotopical algebra there is a homotopical notion of associative algebras called ${\displaystyle A_{\infty }}$-algebras. Loosely, an ${\displaystyle A_{\infty }}$-algebra[1] ${\displaystyle (A^{\bullet },m_{i})}$ is a ${\displaystyle \mathbb {Z} }$-graded vector space over a field ${\displaystyle k}$ with a series of operations ${\displaystyle m_{i}}$ on the ${\displaystyle i}$-th tensor powers of ${\displaystyle A^{\bullet }}$. The ${\displaystyle m_{1}}$ corresponds to a chain complex differential, ${\displaystyle m_{2}}$ is the multiplication map, and the higher ${\displaystyle m_{i}}$ are a measure of the failure of associativity of the ${\displaystyle m_{2}}$. Their structure was originally discovered by Jim Stasheff[2] while studying A∞-spaces, but this was interpreted as a purely algebraic structure later on.

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

For a fixed field ${\displaystyle k}$ an ${\displaystyle A_{\infty }}$-algebra[1] is a ${\displaystyle \mathbb {Z} }$-graded vector space

${\displaystyle A=\bigoplus _{p\in \mathbb {Z} }A^{p}}$

such that for ${\displaystyle d\geq 1}$ there exist degree ${\displaystyle 2-d}$, ${\displaystyle k}$-linear maps

${\displaystyle m_{d}:(A^{\bullet })^{\otimes d}\to A^{\bullet }}$

which satisfy a coherence condition:

${\displaystyle \sum _{\begin{matrix}1\leq p\leq d\\0\leq q\leq d-p\end{matrix}}(-1)^{\alpha }m_{d-p+1}(a_{d},\ldots ,a_{p+q+1},m_{p}(a_{p+q},\ldots ,a_{q+1}),a_{q},\ldots ,a_{1})=0}$

where ${\displaystyle \alpha =(-1)^{{\text{deg}}(a_{1})+\cdots +{\text{deg}}(a_{q})-q}}$.

Understanding the coherence conditions

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

d=1

For ${\displaystyle d=1}$ this is the condition that

${\displaystyle m_{1}(m_{1}(a_{1}))=0}$

since ${\displaystyle 1\leq p\leq 1}$ giving ${\displaystyle p=1}$ and ${\displaystyle 0\leq q\leq d-1}$. These two inequalities force ${\displaystyle m_{d-p+1}=m_{1}}$ in the coherence condition, hence the only input of it is from ${\displaystyle m_{1}(a_{1})}$. Therefore ${\displaystyle m_{1}}$ represents a differential.

d=2

Unpacking the coherence condition for ${\displaystyle d=2}$ gives the degree ${\displaystyle 0}$ map ${\displaystyle m_{2}}$. In the sum there are the inequalities

${\displaystyle {\begin{matrix}1\leq p\leq 2\\0\leq q\leq 2-p\end{matrix}}}$

of indices giving ${\displaystyle (p,q)}$ equal to ${\displaystyle (1,0),(1,1),(2,0)}$. Unpacking the coherence sum gives the relation

${\displaystyle m_{2}(a_{2},m_{1}(a_{1}))+(-1)^{\deg(a_{1})-1}m_{2}(m_{1}(a_{2}),a_{1})+m_{1}(m_{2}(a_{1},a_{2}))=0}$

which when rewritten with

${\displaystyle (-1)^{\deg a}m_{1}(a)=d(a)}$ and ${\displaystyle (-1)^{\deg a_{1}}m_{2}(a_{2},a_{1})=a_{2}\cdot a_{1}}$

as the differential and multiplication, it is

${\displaystyle d(a_{2}\cdot a_{1})=(-1)^{\deg(a_{1})}d(a_{2})\cdot a_{1}+a_{2}\cdot d(a_{1})}$

which is the Liebniz Law for differential graded algebras.

d=3

In this degree the associativity structure comes to light. If ${\displaystyle m_{3}=0}$ then there is a differential graded algebra structure.

Examples

Differential graded algebras

Every differential graded algebra ${\displaystyle (A^{\bullet },d)}$ has a canonical structure as an ${\displaystyle A_{\infty }}$-algebra[1] where ${\displaystyle m_{1}=d}$ and ${\displaystyle m_{2}}$ is the multiplication map. All other higher maps ${\displaystyle m_{i}}$ are equal to ${\displaystyle 0}$. Using the structure theorem for minimal models, there is a canonical ${\displaystyle A_{\infty }}$-structure on the graded cohomology algebra ${\displaystyle HA^{\bullet }}$ which preserves the quasi-isomorphism structure of the original differential graded algebra.

Structure

Minimal models

One of the important structure theorems for ${\displaystyle A_{\infty }}$-algebras is the existence and uniqueness of minimal models. These are ${\displaystyle A_{\infty }}$-algebras where the differential map ${\displaystyle m_{1}=0}$. Taking the cohomology algebra ${\displaystyle HA^{\bullet }}$ of an ${\displaystyle A_{\infty }}$-algebra ${\displaystyle A^{\bullet }}$ from the differential ${\displaystyle m_{1}}$, so as a graded algebra

${\displaystyle HA^{\bullet }={\frac {{\text{ker}}(m_{1})}{m_{1}(A^{\bullet })}}}$

with multiplication map ${\displaystyle [m_{2}]}$. It turns out this graded algebra can then canonically be equipped with an ${\displaystyle A_{\infty }}$-structure,

${\displaystyle (HA^{\bullet },0,[m_{2}],m_{3},m_{4},\ldots )}$

which is unique up-to quasi-isomorphisms of ${\displaystyle A_{\infty }}$-algebras[3]. In fact, the statement is even stronger: there is a canonical ${\displaystyle A_{\infty }}$-morphism

${\displaystyle (HA^{\bullet },0,[m_{2}],m_{3},m_{4},\ldots )\to A^{\bullet }}$

which lifts the identity map of ${\displaystyle A^{\bullet }}$. Note these higher products are given by the Massey product.

Motivation

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 the 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.

Massey structure from DGA's

Given a differential graded algebra ${\displaystyle (A^{\bullet },d)}$ its minimal model as an ${\displaystyle A_{\infty }}$-algebra ${\displaystyle (HA^{\bullet },0,[m_{2}],m_{3},m_{4},\ldots )}$ is constructed using the Massey products. That is,

${\displaystyle {\begin{aligned}m_{3}(x_{3},x_{2},x_{1})&=\langle x_{3},x_{2},x_{1}\rangle \\m_{4}(x_{4},x_{3},x_{2},x_{1})&=\langle x_{4},x_{3},x_{2},x_{1}\rangle \\&\cdots &\end{aligned}}}$

It turns out that any ${\displaystyle A_{\infty }}$-algebra structure on ${\displaystyle HA^{\bullet }}$ is closely related to this construction. Given another ${\displaystyle A_{\infty }}$-structure on ${\displaystyle HA^{\bullet }}$ with maps ${\displaystyle m_{i}'}$, there is the relation[4]

${\displaystyle m_{n}(x_{1},\ldots ,x_{n})=\langle x_{1},\ldots ,x_{n}\rangle +\Gamma }$

where

${\displaystyle \Gamma \in \sum _{j=1}^{n-1}{\text{Im}}(m_{j})}$

Graded algebras from its ext algebra

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

${\displaystyle A=k_{A}\oplus A_{1}\oplus A_{2}\oplus \cdots }$

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

${\displaystyle {\text{Ext}}_{A}^{\bullet }(k_{A},k_{A})}$

where multiplication is given by the Yoneda product. Then, there is an ${\displaystyle A_{\infty }}$-quasiisomorphism between ${\displaystyle (A,0,m_{2},0,\ldots )}$ and ${\displaystyle {\text{Ext}}_{A}^{\bullet }(k_{A},k_{A})}$. This identification is important because it gives a way to show that all derived categories are derived affine, meaning they are isomorphic the derived category of some algebra.

References

1. Aspinwall, Paul, 1964- (2009). Dirichlet branes and mirror symmetry. American Mathematical Society. ISBN 978-0-8218-3848-8. OCLC 939927173.CS1 maint: multiple names: authors list (link)
2. Stasheff, Jim (2018-09-04). "$L_\infty$ and $A_\infty$ structures: then and now". Cite journal requires |journal= (help)
3. Kadeishvili, Tornike (2005-04-21). "On the homology theory of fibre spaces". arXiv:math/0504437.
4. Buijs, Urtzi; Moreno-Fernández, José Manuel; Murillo, Aniceto (2019-02-19). "A-infinity structures and Massey products". arXiv:1801.03408 [math].

• A-infinity structure on Ext-algebras
• Dirichlet Branes and Mirror Symmetry - page 593 for an example of an ${\displaystyle A_{\infty }}$-category with non-trivial ${\displaystyle m_{3}}$.
• Constructible Sheaves and the Fukaya Category
• Homological mirror symmetry for the quartic surface

See also

• Mirror symmetry conjecture
• Homological mirror symmetry
• Homotopy Lie algebra
• Derived algebraic geometry