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In [[Homotopical algebra]] there is a homotopical notion of [[Associative algebra|associative algebras]] called <math>A_\infty</math>-algebras. Loosely, an <math>A_\infty</math>-algebra <math>(A^\bullet, m_i)</math> is a <math>\mathbb{Z}</math>-graded vector space over a field <math>k</math> with a series of operations <math>m_i</math> on the <math>i</math>-th tensor powers of <math>A^\bullet</math>. The <math>m_1</math> corresponds to a [[Chain complex|chain complex differential]], <math>m_2</math> is the multiplication map, and the higher <math>m_i</math> are a measure of the failure of associativity of the <math>m_2</math>. Their structure was originally discovered by [[Jim Stasheff]]<ref>Liquid error: wrong number of arguments (given 1, expected 2)</ref> while studying [[A∞-operad|A∞-spaces]], but this was interpreted as an 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-brane|A-branes]] on a [[Calabi–Yau manifold|Calabi-Yau manifold]] who have only a homotopy associative structure.
== Definition ==
For a fixed field <math>k</math> an <math>A_\infty</math>-algebra<ref>Liquid error: wrong number of arguments (given 3, expected 1)</ref> is a <math>\mathbb{Z}</math>-graded vector space<blockquote><math>A = \bigoplus_{p \in \mathbb{Z}}A^p</math></blockquote>such that for <math>d \geq 1</math> there exist degree <math>2 - d</math>, <math>k</math>-linear maps<blockquote><math>m_d:(A^{\bullet})^{\otimes d} \to A^\bullet</math></blockquote>which satisfy a coherence condition:<blockquote><math>\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</math></blockquote>where <math>\alpha = (-1)^{\text{deg}(a_1) + \cdots + \text{deg}(a_q) - q}</math>.
=== Understanding the coherence conditions ===
The coherence conditions are easy to write down for low degrees.
==== d=1 ====
For <math>d = 1</math> this is the condition that<blockquote><math>m_1(m_1(a_1)) = 0 </math></blockquote>since <math>1 \leq p \leq 1</math> giving <math>p = 1 </math> and <math>0 \leq q \leq d - 1 </math>. These two inequalities force <math>m_{d-p+1} = m_{1} </math> in the coherence condition, hence the only input of it is from <math>m_1(a_1) </math>. Therefore <math>m_1 </math> represents a differential.
==== d=2 ====
Unpacking the coherence condition for <math>d=2 </math> gives the degree <math>0 </math> map <math>m_2 </math>. In the sum there are the inequalities<blockquote><math>\begin{matrix}
1 \leq p \leq 2 \\
0 \leq q \leq 2-p
\end{matrix} </math></blockquote>of indices giving <math>(p,q) </math> equal to <math>(1,0),(1,1),(2,0) </math>. Unpacking the coherence sum gives the relation<blockquote><math>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 </math></blockquote>which when rewritten with<blockquote><math>(-1)^{\deg a}m_1(a) = d(a) </math> and <math>(-1)^{\deg a_1}m_2(a_2,a_1) = a_2\cdot a_1 </math></blockquote>as the differential and multiplication, it is<blockquote><math>d(a_2\cdot a_1) = (-1)^{\deg(a_1)}d(a_2)\cdot a_1 +a_2\cdot d(a_1) </math> </blockquote>which is the Liebniz Law for differential graded algebras.
==== d=3 ====
In this degree the associativity structure comes to light. If <math>m_3=0 </math> then there is a differential graded algebra structure.
== Examples ==
=== Differential graded algebras ===
Every differential graded algebra <math>(A^\bullet, d)</math> has a canonical structure as an <math>A_\infty</math>-algebra where <math>m_1 = d</math> and <math>m_2</math> is the multiplication map. All other higher maps <math>m_i</math> are equal to <math>0</math>.
== References ==
* [[arxiv:math/0606144|A-infinity structure on Ext-algebras]]
* [https://ift.tt/2aY3sYm Dirichlet Branes and Mirror Symmetry] - page 593 for an example of an <math>A_\infty</math>-category with non-trivial <math>m_3</math>.
* [[arxiv:math/0604379|Constructible Sheaves and the Fukaya Category]]
== See also ==
* [[Homotopy Lie algebra]]
* [[Derived algebraic geometry]]
They are ubiquitous in [[homological mirror symmetry]] because of their necessity in defining the structure of the [[Fukaya category]] of [[D-brane|A-branes]] on a [[Calabi–Yau manifold|Calabi-Yau manifold]] who have only a homotopy associative structure.
== Definition ==
For a fixed field <math>k</math> an <math>A_\infty</math>-algebra<ref>Liquid error: wrong number of arguments (given 3, expected 1)</ref> is a <math>\mathbb{Z}</math>-graded vector space<blockquote><math>A = \bigoplus_{p \in \mathbb{Z}}A^p</math></blockquote>such that for <math>d \geq 1</math> there exist degree <math>2 - d</math>, <math>k</math>-linear maps<blockquote><math>m_d:(A^{\bullet})^{\otimes d} \to A^\bullet</math></blockquote>which satisfy a coherence condition:<blockquote><math>\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</math></blockquote>where <math>\alpha = (-1)^{\text{deg}(a_1) + \cdots + \text{deg}(a_q) - q}</math>.
=== Understanding the coherence conditions ===
The coherence conditions are easy to write down for low degrees.
==== d=1 ====
For <math>d = 1</math> this is the condition that<blockquote><math>m_1(m_1(a_1)) = 0 </math></blockquote>since <math>1 \leq p \leq 1</math> giving <math>p = 1 </math> and <math>0 \leq q \leq d - 1 </math>. These two inequalities force <math>m_{d-p+1} = m_{1} </math> in the coherence condition, hence the only input of it is from <math>m_1(a_1) </math>. Therefore <math>m_1 </math> represents a differential.
==== d=2 ====
Unpacking the coherence condition for <math>d=2 </math> gives the degree <math>0 </math> map <math>m_2 </math>. In the sum there are the inequalities<blockquote><math>\begin{matrix}
1 \leq p \leq 2 \\
0 \leq q \leq 2-p
\end{matrix} </math></blockquote>of indices giving <math>(p,q) </math> equal to <math>(1,0),(1,1),(2,0) </math>. Unpacking the coherence sum gives the relation<blockquote><math>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 </math></blockquote>which when rewritten with<blockquote><math>(-1)^{\deg a}m_1(a) = d(a) </math> and <math>(-1)^{\deg a_1}m_2(a_2,a_1) = a_2\cdot a_1 </math></blockquote>as the differential and multiplication, it is<blockquote><math>d(a_2\cdot a_1) = (-1)^{\deg(a_1)}d(a_2)\cdot a_1 +a_2\cdot d(a_1) </math> </blockquote>which is the Liebniz Law for differential graded algebras.
==== d=3 ====
In this degree the associativity structure comes to light. If <math>m_3=0 </math> then there is a differential graded algebra structure.
== Examples ==
=== Differential graded algebras ===
Every differential graded algebra <math>(A^\bullet, d)</math> has a canonical structure as an <math>A_\infty</math>-algebra where <math>m_1 = d</math> and <math>m_2</math> is the multiplication map. All other higher maps <math>m_i</math> are equal to <math>0</math>.
== References ==
* [[arxiv:math/0606144|A-infinity structure on Ext-algebras]]
* [https://ift.tt/2aY3sYm Dirichlet Branes and Mirror Symmetry] - page 593 for an example of an <math>A_\infty</math>-category with non-trivial <math>m_3</math>.
* [[arxiv:math/0604379|Constructible Sheaves and the Fukaya Category]]
== See also ==
* [[Homotopy Lie algebra]]
* [[Derived algebraic geometry]]
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