3Hochschild homology and cohomology
III Algebras
3.4 Gerstenhaber algebra
We now want to understand the equations (
†
) better. To do so, we consider the
graded vector space
HH
·
(A, A) =
∞
M
A=0
HH
n
(A, A),
as a whole. It turns out this has the structure of a Gerstenhaber algebra
The first structure to introduce is the cup product. They are a standard tool
in cohomology theories. We will write
S
n
(A, A) = Hom
k
(A
⊗n
, A) = Hom
A-A
(A
⊗(n+2)
,
A
A
A
).
The Hochschild chain complex is then the graded chain complex S
·
(A, A).
Definition (Cup product). The cup product
^: S
m
(A, A) ⊗ S
n
(A, A) → S
m+n
(A, A)
is defined by
(f ^ g)(a
1
⊗ · · · ⊗ a
m
⊗ b
1
⊗ · · · ⊗ b
n
) = f(a
1
⊗ · · · ⊗ a
m
) · g(b
1
⊗ · · · ⊗ b
n
),
where a
i
, b
j
∈ A.
Under this product, S
·
(A, A) becomes an associative graded algebra.
Observe that
δ(f ^ g) = δf ^ g + (−1)
mn
f ^ δg.
So we say
δ
is a (left) graded derivation of the graded algebra
S
·
(
A, A
). In
homological (graded) algebra, we often use the same terminology but with
suitable sign changes which depends on the degree.
Note that the cocycles are closed under
^
. So cup product induces a product
on
HH
·
(
A, A
). If
f ∈ S
m
(
A, A
) and
g ∈ S
n
(
A, A
), and both are cocycles, then
(−1)
m
(g ^ f − (−1)
mn
(f ^ g)) = δ(f ◦ g),
where f ◦ g is defined as follows: we set
f ◦
i
g(a
1
⊗ · · · ⊗ a
i−1
⊗ b
1
⊗ · · · ⊗ b
n
⊗ a
i+1
· · · ⊗ a
m
)
= f(a
1
⊗ · · · ⊗ a
i−1
⊗ g(b
1
⊗ · · · ⊗ b
n
) ⊗ a
i+1
⊗ · · · ⊗ a
m
).
Then we define
f ◦ g =
m
X
i=1
(−1)
(n−1)(i−1)
f ◦
i
g.
This product
◦
is not an associative product, but is giving a pre-Lie structure to
S
·
(A, A).
Definition (Gerstenhaber bracket). The Gerstenhaber bracket is
[f, g] = f ◦ g − (−1)
(n+1)(m+1)
g ◦ f
This defines a graded Lie algebra structure on the Hochschild chain complex,
but notice that we have a degree shift by 1. It is a grade Lie algebra on
S
·
+1
(A, A).
Of course, we really should define what a graded Lie algebra is.
Definition (Graded Lie algebra). A graded Lie algebra is a vector space
L =
M
L
i
with a bilinear bracket [ · , · ] : L × L → L such that
– [L
i
, L
j
] ⊆ L
i+j
;
– [f, g] − (−1)
mn
[g, f]; and
– The graded Jacobi identity holds:
(−1)
mp
[[f, g], h] + (−1)
mn
[[g, h], f] + (−1)
np
[[h, f], g] = 0
where f ∈ L
m
, g ∈ L
n
, h ∈ L
p
.
In fact,
S
·
+1
(
A, A
) is a differential graded Lie algebra under the Gerstenhaber
bracket.
Lemma. The cup product on HH
·
(A, A) is graded commutative, i.e.
f ^ g = (−1)
mn
(g ^ f).
when f ∈ HH
m
(A, A) and g ∈ HH
n
(A, A).
Proof. We previously “noticed” that
(−1)
m
(g ^ f − (−1)
mn
(f ^ g)) = δ(f ◦ g),
Definition
(Gerstenhaber algebra)
.
A Gerstenhaber algebra is a graded vector
space
H =
M
H
i
with
H
·
+1
a graded Lie algebra with respect to a bracket [
· , ·
] :
H
m
× H
n
→
H
m+n−1
, and an associative product
^
:
H
m
× H
n
→ H
m+n
which is graded
commutative, such that if
f ∈ H
m
, then [
f, ·
] acts as a degree
m −
1 graded
derivation of ^:
[f, g ^ h] = [f, g] ^ h + (−1)
(m−1)
ng ^ [f, h]
if g ∈ H
n
.
This is analogous to the definition of a Poisson algebra. We’ve seen that
HH
·
(A, A) is an example of a Gerstenhaber algebra.
We can look at what happens in low degrees. We know that
H
0
is a
commutative k-algebra, and ^: H
0
× H
1
→ H
1
is a module action.
Also,
H
1
is a Lie algebra, and [
· , ·
] :
H
1
× H
0
→ H
0
is a Lie module
action, i.e.
H
0
gives us a Lie algebra representation of
H
1
. In other words,
the corresponding map [
· , ·
] :
H
1
→ End
k
(
H
0
) gives us a map of Lie algebras
H
1
→ Der(H
0
).
The prototype Gerstenhaber algebra is the exterior algebra
V
DerA
for a
commutative algebra A (with A in degree 0).
Explicitly, to define the exterior product over
A
, we first consider the tensor
product over A of two A-modules V and W , defined by
V ⊗
A
W =
V ⊗
k
W
hav ⊗ w − v ⊗ awi
The exterior product is then
V ∧
A
V =
V ⊗
A
V
hv ⊗ v : v ∈ V i
.
The product is given by the wedge, and the Schouten bracket is given by
[λ
1
∧ · · · ∧ λ
m
, λ
0
1
∧ · · · ∧ λ
0
n
]
= (−1)
(m−1)(n−1)
X
i,j
(−1)
i+j
[λ
i
, λ
j
] λ
1
∧ · · · λ
m
| {z }
ith missing
∧ λ
0
1
∧ · · · ∧ λ
0
n
| {z }
jth missing
.
For any Gerstenhaber algebra
H
=
L
H
i
, there is a canonical homomorphism
of Gerstenhaber algebras
^
H
0
H
1
→ H.
Theorem
(Hochschild–Kostant–Ronsenberg (HKR) theorem)
.
If
A
is a “smooth”
commutative k-algebra, and char k = 0, then the canonical map
^
A
(DerA) → HH
∗
(A, A)
is an isomorphism of Gerstenhaber algebras.
We will not say what “smooth” means, but this applies if
A
=
k
[
X
1
, · · · , X
n
],
or if
k
=
C
or
R
and
A
is appropriate functions on smooth manifolds or algebraic
varieties.
In the 1960’s, this was stated just for the algebra structure, and didn’t think
about the Lie algebra.
Example. Let A = k[X, Y ], with char k = 0. Then HH
0
(A, A) = A and
HH
1
(A, A) = DerA
∼
=
p(X, Y )
∂
∂y
+ q(X, Y )
∂
∂Y
: p, q ∈ A
.
So we have
HH
2
(A, A) = DerA ∧
A
DerA,
which is generated as an A-modules by
∂
∂X
∧
∂
∂Y
. Then
HH
i
(A, A) = 0 for all i ≥ 3
We can now go back to talk about star products. Recall when we considered
possible star products on
V ⊗
k
k
[[
t
]], where
V
is the underlying vector space of
the algebra
A
. We found that associativity of the star product was encapsulated
by some equations (†
λ
). Collectively, these are equivalent to the statement
Definition (Maurer–Cartan equation). The Maurer–Cartan equation is
δf +
1
2
[f, f]
Gerst
= 0
for the element
f =
X
t
λ
F
λ
,
where F
0
(a, b) = ab.
When we write [
· , ·
]
Gerst
, we really mean the
k
[[
t
]]-linear extension of the
Gerstenhaber bracket.
If we want to think of things in cohomology instead, then we are looking
at things modulo coboundaries. For the graded Lie algebra
V
·
+1
(
DerA
), the
Maurer–Cartan elements, i.e. solutions of the Maurer–Cartan equation, are the
formal Poisson structures. They are formal power series of the form
Π =
X
t
i
π
i
for π
i
∈ DerA ∧ DerA, satisfying
[Π, Π] = 0.
There is a deep theorem of Kontsevich from the early 2000’s which implies
Theorem (Kontsevich). There is a bijection
equivalence classes
of star products
←→
n
classes of formal
Poisson structures
o
This applies for smooth algebras in
char
0, and in particular for polynomial
algebras A = k[X
1
, · · · , X
n
].
This is a difficult theorem, and the first proof appeared in 2002.
An unnamed lecturer once tried to give a Part III course with this theorem as
the punchline, but the course ended up lasting 2 terms, and they never reached
the punchline.