Homotopy associative algebra

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

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

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

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

which satisfy a coherence condition:

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

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

Understanding the coherence conditions

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

d=1

For $d=1$ this is the condition that

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

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

d=2

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

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

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

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

which when rewritten with

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

as the differential and multiplication, it is

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

which is the Liebniz Law for differential graded algebras.

d=3

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

Examples

Differential graded algebras

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

Structure

Minimal models

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

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

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

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

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

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

which lifts the identity map of $A^{\bullet }$. 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 $(A^{\bullet },d)$ its minimal model as an $A_{\infty }$-algebra $(HA^{\bullet },0,[m_{2}],m_{3},m_{4},\ldots )$ is constructed using the Massey products. That is,

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

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

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

where

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

Graded algebras from its ext algebra

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

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

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

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

where multiplication is given by the Yoneda product. Then, there is an $A_{\infty }$-quasiisomorphism between $(A,0,m_{2},0,\ldots )$ and ${\text{Ext}}_{A}^{\bullet }(k_{A},k_{A})$. 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

• A-infinity structure on Ext-algebras
• Dirichlet Branes and Mirror Symmetry - page 593 for an example of an $A_{\infty }$-category with non-trivial $m_{3}$.
• 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