Homotopy associative algebra explained
In mathematics, an algebra such as
has multiplication
whose
associativity is well-defined on the nose. This means for any real numbers
we have
a ⋅ (b ⋅ c)-(a ⋅ b) ⋅ c=0
.But, there are algebras
which are not necessarily associative, meaning if
then
a ⋅ (b ⋅ c)-(a ⋅ b) ⋅ c ≠ 0
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
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[1] [2] 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
an
-algebra
[3] is a
-graded vector space
such that for
there exist degree
,
-linear maps
md\colon(A\bullet) ⊗ \toA\bullet
which satisfy a coherence condition:
\sum
\begin{matrix1\leqp\leqd\\
0\leqq\leqd-p
\end{matrix}
}(-1)\alphamd-p+1(ad,\ldots,ap+q+1,mp(ap+q,\ldots,aq+1),aq,\ldots,a1)=0
,where
\alpha=
| deg(a1)+ … +\deg(aq)-q |
(-1) | |
.
Understanding the coherence conditions
The coherence conditions are easy to write down for low degreespgs 583–584.
d=1
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.
d=2
Unpacking the coherence condition for
gives the degree
map
. In the sum there are the inequalities
\begin{matrix}
1\leqp\leq2\\
0\leqq\leq2-p
\end{matrix}
of indices giving
equal to
. Unpacking the coherence sum gives the relation
m2(a2,m1(a1))+
m2(m1(a2),a1)+m1(m2(a1,a2))=0
,which when rewritten with
and
as the differential and multiplication, it is
d(a2 ⋅ a1)=
d(a2) ⋅ a1+a2 ⋅ d(a1)
,which is the
Leibniz rule for differential graded algebras.
d=3
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
\begin{align}
m2(m2(a ⊗ b) ⊗ c)-m2(a ⊗ m2(b ⊗ c))=&\pmm3(m1(a) ⊗ b ⊗ c)\\
&\pmm3(a ⊗ m1(b) ⊗ c)\\
&\pmm3(a ⊗ b ⊗ m1(c))\\
&\pmm1(m3(a ⊗ b ⊗ c)).
\end{align}
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
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
\begin{align}
&\pmmd(ad,\ldots,a2,m1(a1))\ &\pm … \\
&\pmmd(m1(ad),ad-1,\ldots,a1)\\
&\pmm1(md(ad,\ldots,a1)).
\end{align}
In degree
the other terms can be written out as
\begin{align}
&\pmm3(m2(a4,a3),a2,a1)\\
&\pmm3(a4,m2(a3,a2),a1)\\
&\pmm3(a4,a3,m2(a2,a1))\\
&\pmm2(m3(a4,a3,a2),a1)\\
&\pmm2(a4,m3(a3,a2,a1)),
\end{align}
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
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
Associative algebras
Every associative algebra
has an
-infinity structure by defining
and
for
. Hence
-algebras generalize associative algebras.
Differential graded algebras
Every differential graded algebra
has a canonical structure as an
-algebra 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
Ho(dga)\simeqHo(Ainfty-alg)
of differential graded algebras and
-algebras.
Cochain algebras of H-spaces
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.
Example with infinitely many non-trivial mi
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
\begin{align}
m1(v0)=w\\
m1(v1)=w,
\end{align}
and for
\begin{align}
md(v1 ⊗ w ⊗ ⊗ v1 ⊗ w ⊗ )
1&0\leqk\leqd-2\\
md(v1 ⊗ w ⊗ ⊗ v2)&=sd+1v1\\
md(v1 ⊗ w ⊗ )&=sd+1w,
\end{align}
where
on any map not listed above and
. In degree
, so for the multiplication map, we have
\begin{align}
m2(v1,v1)&=-v1\\
m2(v1,v2)&=v1\\
m2(v1,w)&=w.
\end{align}
And in
the above relations give
\begin{align}
m3(v1,v1,w)&=v1\\
m3(v1,w,v1)&=-v1\\
m3(v1,w,v2)&=-v1\\
m3(v1,w,w)&=-w.
\end{align}
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
Transfer of A∞ structure
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
f\colonB\bullet\toA\bullet
,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
Minimal models and Kadeishvili's theorem
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,
(HA\bullet,0,[m2],m3,m4,\ldots)
,which is unique up-to quasi-isomorphisms of
-algebras.
[8] In fact, the statement is even stronger: there is a canonical
-morphism
(HA\bullet,0,[m2],m3,m4,\ldots)\toA\bullet
,which lifts the identity map of
. 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 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
l{Hom}\bullet(l{F}\bullet,l{G}\bullet)
pg 586-593. In this was, the degree
morphisms in the category
are given by
Ext(l{F}\bullet,l{G}\bullet)
.
Applications
(\Omega\bullet(X),d,\wedge)
, or the
Hochschild cohomology algebra, they can be equipped with an
-structure.
Massey structure from DGA's
Given a differential graded algebra
its minimal model as an
-algebra
(HA\bullet,0,[m2],m3,m4,\ldots)
is constructed using the Massey products. That is,
\begin{align}
m3(x3,x2,x1)&=\langlex3,x2,x1\rangle\\
m4(x4,x3,x2,x1)&=\langlex4,x3,x2,x1\rangle\\
& … &
\end{align}
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] mn(x1,\ldots,xn)=\langlex1,\ldots,xn\rangle+\Gamma
,where
.Hence all such
-enrichments on the cohomology algebra are related to one another.
Graded algebras from its ext algebra
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
| \bullet(k |
\operatorname{Ext} | |
| A,k |
A)
,where multiplication is given by the
Yoneda product. Then, there is an
-quasi-isomorphism between
and
| \bullet(k |
\operatorname{Ext} | |
| 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 to the derived category of some algebra.
See also
References
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