https://doi.org/10.4134/JKMS.j180762 pISSN: 0304-9914 / eISSN: 2234-3008
COHOMOLOGY RING OF THE TENSOR PRODUCT OF POISSON ALGEBRAS
Can Zhu
Abstract. In this paper, we study the Poisson cohomology ring of the tensor product of Poisson algebras. Explicitly, it is proved that the Pois- son cohomology ring of tensor product of two Poisson algebras is isomor- phic to the tensor product of the respective Poisson cohomology ring of these two Poisson algebras as Gerstenhaber algebras.
1. Introduction
Given a Poisson algebra, its Poisson cohomology, as introduced by Lich- nerowicz in [12], provides important information about the Poisson structure.
The Poisson cohomology HP ∗ (R) of a Poisson algebra R equipped with the wedge product and the Schouten bracket is a Gerstenhaber algebra [10]. It plays a crucial role in the study of deformations of the Poisson structure, just as the Hochschild cohomology ring for the deformation theory of associative algebras [5].
The tensor product of Poisson structure appeared in Poisson Hopf algebras [3]. The tensor product of Poisson structure makes the category Poi( k), the category of Poisson algebras over k, to be a tensor category [2]. Cherkashin researched the cohomologies, deformations and homotopies of tensor product of Poisson algebras in [1]. In this present paper, we study the Poisson cohomology ring of the tensor product of Poisson algebras. Let (R, π R ) and (T, π T ) be two Poisson algebras over k. Then there is a Poisson bracket π, called the product Poisson bracket, on R ⊗ T by
π(a ⊗ b, c ⊗ d) := π R (a, c) ⊗ bd + ac ⊗ π T (b, d)
for all a, c ∈ R and b, d ∈ T . The Poisson algebra (R ⊗ T, π) is called the Poisson tensor product of R and T . Our main result is the following (Theorem 3.3):
Received November 8, 2018; Revised March 16, 2019; Accepted July 25, 2019.
2010 Mathematics Subject Classification. Primary 17B63, 18G60.
Key words and phrases. Poisson algebra, tensor product, Gerstenhaber algebra.
This work was financially supported by NSFC #11771085 and #11871125, and by Shang- hai Science and Technology Program #19ZR1434600.
c
2020 Korean Mathematical Society
113
Theorem. Let R and T be two Poisson algebras. There is an isomorphism of Gerstenhaber algebras
HP ∗ (R ⊗ T ) ∼ = HP ∗ (R) ⊗ HP ∗ (T ).
Note that the tensor product of two Gerstenhaber algebras in the above theorem is defined by Le and Zhou in [11]. They also proved that, given two associative algebras A and B over a field k, at least one of which is finite dimensional, the Hochschild cohomology of the tensor product algebra A ⊗ k B is isomorphic to the tensor product of the respective Hochschild cohomologies of A and of B, as Gerstenhaber algebras. Hence, our result is a generalization of this to Poisson framework.
The paper is organized as follows. In Section 2, we recall some preliminary definitions and results mainly on Poisson algebra and cohomology ring, Ger- stenhaber algebra, tensor product of Poisson algebras, etc.. The main result is proved in Section 3. Throughout, k is a field and all algebras are k-algebras unless specialized and unadorned ⊗ means ⊗ k .
2. Preliminaries
In this section, we recall some definitions and results which is essential for our purpose.
2.1. Poisson algebra and cohomology
Definition. A commutative k-algebra R equipped with a bilinear map {−, −} : R × R → R is called a Poisson algebra if
(1) (R, {−, −}) is a Lie algebra;
(2) {−, −} is a derivation in each argument with respect to multiplication of R.
Usually, the Poisson bracket is denoted by π. Then, the Poisson algebra is denoted by (R, π), or (R, {−, −}). Next, we recall the notion of Poisson cohomology. Denote by X k (R) the space of all skew-symmetric k-linear maps R ⊗k → R that are derivations in each arguments. Here, a map f : R k → R is called skew-symmetric if f (a 1 , . . . , a k ) = sgn(σ)f (a σ1, . . . , a σk) for any permutation σ ∈ S k , where sgn(σ) denotes its sign and σ i means σ(i). Then there is a cochain complex (X ∗ (R), δ π ∗ ) ([7, 12]), where δ π k : X k (R) → X k+1 (R) is defined by
) for any permutation σ ∈ S k , where sgn(σ) denotes its sign and σ i means σ(i). Then there is a cochain complex (X ∗ (R), δ π ∗ ) ([7, 12]), where δ π k : X k (R) → X k+1 (R) is defined by
δ π k (P )(y 0 , y 1 , . . . , y k ) := X
0≤i≤k
(−1) i {y i , P (y 0 , y 1 , . . . , y b i , . . . , y k )}
+ X
0≤i<j≤k
(−1) i+j P ({y i , y j }, y 0 , y 1 , . . . , y b i , . . . , y b j , . . . , y k )
for all P ∈ X k (R). It is not hard to check that δ k π (P ) belongs to X k+1 (R) and
that δ k+1 π δ π k = 0. The k-th cohomology of this complex is denoted by HP k (R)
and is called the k-th Poisson cohomology of the Poisson algebra R.
Recall that Ω 1 (R) is the module of K¨ ahler differentials of R by definition the R-module and for each p ∈ N, the K¨ahler p-forms is the R-module Ω p (R) :=
V p
Ω 1 (R), where V is the wedge product over R; for p = 0, Ω 0 (R) = R. As an R-module, Ω p (R) is generated by the elements of the form dx 1 ∧ · · · ∧ dx p , where x 1 , . . . , x p ∈ R. There is a natural isomorphism of R-modules
(1) X k (R) ∼ = Hom R (Ω k (R), R),
which is given by P 7→ ˆ P , where ˆ P (xdx 1 ∧ · · · ∧ dx k ) := xP (x 1 , . . . , x k ) for all P ∈ X k (R) and x, x 1 , . . . , x k ∈ R.
2.2. Shuffles
As a matter of convenience, we recall the notion of a shuffle and make some more preliminary on it. For p, q ∈ N and Σ = {i 1 , i 2 , . . . , i p+q } a subset of N with i 1 < i 2 < · · · < i p+q , a permutation σ on Σ is called a (p, q)-shuffle on Σ if σ(i 1 ) < · · · < σ(i p ) and σ(i p+1 ) < · · · < σ(i p+q ). The set of all (p, q)-shuffle is denoted by S Σ p,q . Unless specialized, a (p, q)-shuffle is defined on the set {1, 2, . . . , p + q}. The set of all (p, q)-shuffle is just denoted by S p,q briefly.
Let A 1 = {2, 3, 4, . . . , i + j}. Then, there is a map χ : S i−1,j A1 [
S i,j−1 A1 → S i,j
τ 7→ τ
where τ is defined as follows; if τ ∈ S i−1,j A1 , then
τ (l) :=
( 1, l = 1,
τ (l), l ∈ {2, 3, . . . , i + j};
if τ ∈ S i,j−1 A1 , then
τ (l) :=
τ (l + 1), l ∈ {1, 2, . . . , i}, 1, l = i + 1,
τ (l), l ∈ {i + 2, . . . , i + j}.
Proposition 2.1. Keep the notation as above. The map χ
(2) χ : S i−1,j A1 [
S i,j−1 A1 → S i,j
is a bijection. Further, sgn(τ ) :=
( sgn(τ ), τ ∈ S i−1,j A1 ; (−1) i sgn(τ ), τ ∈ S i,j−1 A1 .
.
Proof. Note that for an (i, j)-shuffle ν on the set {1, 2, . . . , i+j}, we have either
ν(1) = 1 or ν(i + 1) = 1.
For i, j, k, l ∈ N, we define an (i, j, k, l)-shuffle by a permutation σ on the set {1, 2, . . . , i + j + k + l}, such that
σ(1) < σ(2) < · · · < σ(i),
σ(i + 1) < σ(i + 2) < · · · < σ(i + j),
σ(i + j + 1) < σ(i + j + 2) < · · · < σ(i + j + k) and σ(i + j + k + 1) < σ(i + j + k + 2) < · · · < σ(i + j + k + l).
The set of all (i, j, k, l)-shuffle is denoted by S i,j,k,l . For each (i + j, k + l)-shuffle σ, consider the set
S σ = n (τ, η)
τ ∈ S i,j A2, η ∈ S k,l A3o ,
o ,
where A 2 = {σ(1), σ(2), . . . , σ(i+j)} and A 3 = {σ(i+j +1), . . . , σ(i+j +k +l)}
For a given (i + j, k + l)-shuffle σ and a pair (τ, η) ∈ S σ , define a permutation δ by
(3) δ(n) :=
( τ σ(n), 1 ≤ n ≤ i + j;
ησ(n), i + j + 1 ≤ n ≤ i + j + k + l.
Then δ is an (i, j, k, l)-shuffle. Thus, we get a map % : S
σ∈S
i+j,k+lS σ → S i,j,k,l by sending the triple (σ, τ, η) to δ which defined above.
Proposition 2.2. Keeping the notation as above, the map % is a bijection and sgn(δ) = sgn(τ ) sgn(σ) sgn(η).
Proof. The proof is direct.
For a (p, q)-shuffle σ, consider the permutation σ 0 by
σ 0 (i) :=
( σ(i + p), 1 ≤ i ≤ q;
σ(i − q), q + 1 ≤ i ≤ q + p.
Then, σ 0 is a (q, p)-shuffle. Thus, we get a map ϑ : S p,q → S q,p , σ 7→ σ 0 . Proposition 2.3. Keeping the notation as above, the map ϑ is a bijection and sgn(σ) = (−1) pq sgn(σ 0 ).
Proof. The proof is direct.
Remark 2.4. The similar results hold for (i, j, k, l)-shuffles. For example, there is a bijection between S i,j,k,l and S k,j,i,l by send τ to τ 0 in a similar way.
Moreover, sgn(τ ) = (−1) k(i+j)+ij sgn(τ 0 ).
For an (i, j, k − 1, l)-shuffle σ and each integer p with i + 1 ≤ p ≤ i + j, define an integer q by
q :=
i + j, if σ(p) < σ(i + j + 1);
t, if σ(t) < σ(p) < σ(t + 1)
for some t with i + j + 1 ≤ t ≤ j + j + k − 1;
i + j + k − 1, if σ(i + j + k − 1) < σ(p).
Then, define a permutation τ by
τ (l) :=
σ(l), if 1 ≤ l ≤ p − 1;
σ(l + 1), if p ≤ l ≤ q − 1;
σ(p), if l = q;
σ(l), if q + 1 ≤ l ≤ i + j + k + l.
It is easy to see that τ is an (i, j − 1, k, l)-shuffle. That is, we get a map ς : {i + 1, i + 2, . . . , i + j} × S i,j,k−1,l → {i + j, . . . , i + j + k − 1} × S i,j−1,k,l
(p, σ) 7→ (q, τ ).
Proposition 2.5. Keeping the notation as above, the map ς is a bijection:
ς : {i+1, i+2, . . . , i+j}×S i,j,k−1,l → {i+j, i+j +1, . . . , i+j +k−1}×S i,j−1,k,l . And, sgn(τ ) = (−1) p+q+1 sgn(σ) if ς(p, σ) = (q, τ ).
Proof. It is easy to see that these two sets have same cardinality and that this
map is injective.
2.3. Gerstenhaber algebra
Definition. A Gerstenhaber algebra is a Z-graded k-vector space H • :=
L
n∈Z
H n equipped with two linear maps
∧ : H m × H n → H m+n , (a, b) 7→ a ∧ b and
[−, −] : H m × H n → H m+n−1 , (a, b) 7→ [a, b]
such that
(1) (H • , ∧) is a graded commutative algebra, i.e., a ∧ b = (−1) mn b ∧ a, ∀a ∈ H m , b ∈ H n ;
(2) (H • , [−, −]) is a graded Lie algebra of degree −1, i.e., [a, b] = −(−1) (m−1)(n−1) [b, a], ∀a ∈ H m , b ∈ H n and
(−1) (m−1)(l−1) [[a, b], c] + cyclic permutation of a, b and c = 0
for all a ∈ H m , b ∈ H n , c ∈ H l ;
(3) two operations satisfy the compatible condition:
[a ∧ b, c] = a ∧ [b, c] + (−1) (m−1)n b ∧ [a, c], ∀a ∈ H m , b ∈ H n , c ∈ H.
Definition. Let (H • , ∧ H , [−, −] H ) and (K • , ∧ K , [−, −] K ) be two Gersten- haber algebras over k. Then H is said to isomorphic to K as Gerstenhaber algebras if there is a k-linear bijection map Φ : H → K of degree 0 with the following properties
Φ(a ∧ H b) = Φ(a) ∧ K Φ(b), (4)
Φ([a, b] H ) = [Φ(a), Φ(b)] K . (5)
Let R be a Poisson algebra and X ∗ (R) := L
i X i (R). The wedge product ∧ on X ∗ (R):
(P ∧ Q)(a 1 , . . . , a m+n )
:= X
θ∈S
m,nsgn(θ)P (a θ(1) , . . . , a θ(m) )Q(a θ(m+1) , . . . , a θ(m+n) ) and the Schouten bracket on X ∗ (R)
[P, Q] S := P ◦ Q − (−1) (m−1)(n−1) Q ◦ P, where the product ◦ is defined by:
(P ◦ Q)(a 1 , . . . , a m+n−1 )
:= X
θ∈S
n,m−1sgn(θ)P (Q(a θ(1) , . . . , a θ(n) ), a θ(n+1) , . . . , a θ(m+n−1) ) for all P ∈ X m (R) and Q ∈ X n (R), make the triple (X ∗ (R), ∧, [−, −] S ) to be a Gerstenhaber algebra. Further, both the wedge product and Schouten bracket induce products in Poisson cohomology, which still denote by ∧ and [−, −] S . Moreover, the triple (HP ∗ (R), ∧, [−, −] S ) is a Gerstenhaber algebra [10, Proposition 4.9].
2.4. Tensor product of Poisson algebras
Let (R, {−, −}) and (T, {−, −}) be two Poisson algebras. Then R ⊗ T is a commutative associative algebra by
(a ⊗ b) · (c ⊗ d) := ac ⊗ bd.
Proposition 2.6 ([1, 3, 10]). There exists a unique Poisson bracket {a ⊗ b, c ⊗ d} := {a, c} ⊗ bd + ac ⊗ {b, d}
such that (R⊗T, ·, {−, −}) is a Poisson algebra, making the canonical inclusions R → R ⊗ T and T → R ⊗ T are Poisson algebra homomorphism.
The Poisson algebra (R ⊗ T, ·, {−, −}) is called the Poisson tensor product
of R and T , while the Poisson bracket {−, −} is called the product Poisson
structure.
3. The main result and its proof
Let R and T be two Poisson algebras. In this section, we firstly recall that the tensor product of two Gerstenhaber algebras is still a Gerstenhaber algebra.
Then, we prove that HP ∗ (R) ⊗ HP ∗ (T ) and HP ∗ (R ⊗ T ) are isomorphic as Gerstenhaber algebras.
Proposition-Definition 3.1 ([11, Proposition-Definition 2.2]). Let (H • , ∧ H , [−, −] H ) and (K • , ∧ K , [−, −] K ) be two Gerstenhaber algebras over k. Then there is a new Gerstenhaber algebra (L • , ∧, [−, −]) over k given as follows
(1) L n := ⊕ i+j=n H i ⊗ K j as a k-vector space for each n ∈ Z;
(2) (a ⊗ b) ∧ (a 0 ⊗ b 0 ) := (−1) |a0||b| (a ∧ H a 0 ) ⊗ (b ∧ K b 0 );
(3) [a ⊗ b, a 0 ⊗ b 0 ] := (−1) (|a|+|b|−1)|b0| [a, a 0 ] H ⊗ (b ∧ K b 0 ) + (−1) |a|(|a
0|+|b
0|−1) (a ∧ H a 0 ) ⊗ [b, b 0 ] K .
The Gerstenhaber algebra (L • , ∧, [−, −]) is called the tensor product of the two Gerstenhaber algebras H • and K • , and denote it by H • ⊗ K • .
In order to prove the main result of this section, we retrieve the relation between the K¨ ahler differentials of R ⊗ T and that of R and T . We rewrite the following result here for convenience though it is standard.
Lemma 3.2. Let R and T be two commutative algebras. Then Ω ∗ (R) ⊗ Ω ∗ (T ) ' Ω ∗ (R ⊗ T )
as complexes.
Proof. For each p, q, the map
ε : (a 0 da 1 ∧ · · · ∧ da p ) ⊗ (b 0 db 1 ∧ · · · ∧ db q ) →
(a 0 ⊗ b 0 )d(a 1 ⊗ 1) ∧ · · · ∧ d(a p ⊗ 1) ∧ d(1 ⊗ b 1 ) ∧ · · · ∧ d(1 ⊗ b q ) is well-defined and induces a morphism of R ⊗ T -modules (still denoted it by ε),
(6) ε : Ω p (R) ⊗ Ω q (T ) → Ω p+q (R ⊗ T ).
Moreover, it is bijective and a morphism of complexes. Theorem 3.3. Let R and T be two Poisson algebras. There is an isomorphism of Gerstenhaber algebras
HP ∗ (R ⊗ T ) ∼ = HP ∗ (R) ⊗ HP ∗ (T ).
Proof. The left hand side of the above isomorphism is computed by the complex X ∗ (R⊗T ), while the right hand side is computed by the complex X ∗ (R)⊗X ∗ (T ).
We will take in three steps to prove that there is an isomorphism (7) X ∗ (R) ⊗ X ∗ (T ) ' X ∗ (R ⊗ T )
as Gerstenhaber algebras
Step 1: construct an isomorphism of complexes Φ : X ∗ (R) ⊗ X ∗ (T ) → X ∗ (R ⊗ T );
Step 2: prove that Φ preserves the wedge product
Φ((f 1 ⊗ g 1 ) ∧ (f 2 ⊗ g 2 )) = Φ(f 1 ⊗ g 1 ) ∧ Φ(f 2 ⊗ g 2 ) for all f 1 ∈ X i1(R), f 2 ∈ X i2(R), g 1 ∈ X j1(T ) and g 2 ∈ X j2(T );
(R), g 1 ∈ X j1(T ) and g 2 ∈ X j2(T );
(T );
Step 3: prove that Φ preserves the Schouten product
Φ[f 1 ⊗ g 1 , f 2 ⊗ g 2 ] S = [Φ(f 1 ⊗ g 1 ), Φ(f 2 ⊗ g 2 )] S . Then by taking cohomology we get the desired conclusion.
Step 1: In [1], a morphism between the above complexes was constructed in the following way. Note that X k (R) = Hom R (Ω k (R), R) by equation (1) for each k. For f ∈ Hom R (Ω p (R), R) and g ∈ Hom T (Ω q (T ), T ), define
Φ pq : Hom R (Ω p (R), R) ⊗ Hom T (Ω q (T ), T ) → Hom R⊗T (Ω p+q (R ⊗ T ), R ⊗ T ) by
Φ pq (f ⊗ g) : (d(a 1 ⊗ b 1 )∧ · · · ∧ d(a p+q ⊗ b p+q )) 7→
X
σ∈S
p,qsgn(σ)a σ(p+1) · · · a σ(p+q) f (da σ(1) ∧ · · · ∧ da σ(p) )
⊗ b σ(1) · · · b σ(p) g(db σ(p+1) ∧ · · · ∧ db σ(p+q) ).
Then Φ := L
p,q Φ pq is a morphism from the complex Hom R (Ω ∗ (R), R) ⊗ Hom T (Ω ∗ (T ), T ) to the complex Hom R⊗T (Ω ∗ (R ⊗ T ), R ⊗ T ).
Conversely, we firstly construct F : Hom R⊗T (Ω r (R ⊗ T ), R ⊗ T ) →
r
M
t=0
Hom R⊗T (Ω t (R) ⊗ Ω r−t (T ), R ⊗ T ) by F := Hom R⊗T (ε, R ⊗ T ). It is easy to see that F is an isomorphism. Recall that it is known that if M and P are finite generated modules, then the map
G : Hom A (M, N ) ⊗ Hom B (P, Q) → Hom A⊗B (M ⊗ P, N ⊗ Q) α ⊗ β 7−→ G α⊗β ,
where G α⊗β (m ⊗ p) := α(m) ⊗ β(p) for each m ∈ M, p ∈ P , is an isomorphism.
Thus, we get an isomorphism
G : Hom R (Ω p (R), R) ⊗ Hom T (Ω q (T ), T ) → Hom R⊗T (Ω p (R) ⊗ Ω q (T ), R ⊗ T ) since both Ω p (R) and Ω q (T ) are finite generated for all p, q. Define
Ψ = G −1 F : Hom R⊗T (Ω r (R ⊗ T ), R ⊗ T )
→
r
M
t=0
Hom R (Ω t (R), R) ⊗ Hom T (Ω r−t (T ), T ).
Therefore, Ψ is bijective and F Φ = G, i.e., ΨΦ = id. What we have done can be described as the following diagram:
Hom R⊗T (Ω r (R ⊗ T ), R ⊗ T )
Ψ=G
−1F //
F ''
L r
t=0 Hom R (Ω t (R), R) ⊗ Hom T (Ω r−t (T ), T )
Φ
oo
tt G
L r
t=0 Hom R⊗T (Ω t (R) ⊗ Ω r−t (T ), R ⊗ T )
.
Henceforth, Φ is an isomorphism of complexes.
Step 2: Next, we prove that the isomorphism Φ preserves the wedge product (8) Φ((f 1 ⊗ g 1 ) ∧ (f 2 ⊗ g 2 )) = Φ(f 1 ⊗ g 1 ) ∧ Φ(f 2 ⊗ g 2 )
for all f 1 ∈ X i1(R), f 2 ∈ X i2(R), g 1 ∈ X j1(T ) and g 2 ∈ X j2(T ). To simplify the notation, set I = i 1 + i 2 , J = j 1 + j 2 , I 0 = i 1 + j 1 and J 0 = i 2 + j 2 .
(R), g 1 ∈ X j1(T ) and g 2 ∈ X j2(T ). To simplify the notation, set I = i 1 + i 2 , J = j 1 + j 2 , I 0 = i 1 + j 1 and J 0 = i 2 + j 2 .
(T ). To simplify the notation, set I = i 1 + i 2 , J = j 1 + j 2 , I 0 = i 1 + j 1 and J 0 = i 2 + j 2 .
Recall that
(f 1 ⊗ g 1 ) ∧ (f 2 ⊗ g 2 ) = (−1) i2j
1(f 1 ∧ f 2 ) ⊗ (g 1 ∧ g 2 ).
Hence, the left hand side of (8) is the map by sending (a 1 ⊗ b 1 ) ⊗ · · · ⊗ (a I+J ⊗ b I+J ) to
X
σ∈S
I,J(−1) i2j
1sgn(σ)a σ(I+1) · · · a σ(I+J ) f 1 ∧ f 2 (a σ(1) ⊗ · · · ⊗ a σ(I) )
⊗b σ(1) · · · b σ(I) g 1 ∧ g 2 (b σ(I+1) ⊗ · · · ⊗ b σ(I+J ) ).
(E1)
By the definition of the wedge product on X ∗ (R), f 1 ∧ f 2 (a σ(1) ⊗ · · · ⊗ a σ(I) )
= X
τ ∈S
A4i1,i2
sgn(τ )f 1 (a τ (1) ⊗ · · · ⊗ a τ (i1) )f 2 (a τ (i
1+1) ⊗ · · · ⊗ a τ (I) ),
where the set A 4 = {σ(1), σ(2), . . . , σ(I)}. Similarly, g 1 ∧ g 2 (b σ(I+1) ⊗ · · · ⊗ b σ(I+J ) )
= X
η∈S
j1,j2A5sgn(η)g 1 (b η(1) ⊗ · · · ⊗ b η(j1) )g 2 (b η(j
1+1) ⊗ · · · ⊗ b η(J ) ),
where the set A 5 = {σ(I + 1), . . . , σ(I + J )}. By Proposition 2.2,
(E1) = X
σ∈S
i1,i2,j1,j2(−1) i2j
1sgn(σ)a σ(I+1) · · · a σ(I+J ) f 1 (a σ(1) ⊗ · · · ⊗ a σ(i1) ) f 2 (a σ(i
1+1) ⊗ · · · ⊗ a σ(I) ) ⊗ b σ(1) · · · b σ(I)
) ) f 2 (a σ(i
1+1) ⊗ · · · ⊗ a σ(I) ) ⊗ b σ(1) · · · b σ(I)
g 1 (b σ(I+1) ⊗ · · · ⊗ b σ(I+j1) )g 2 (b σ(I+j
1+1) ⊗ · · · ⊗ b σ(I+J ) ).
(9)
For the right hand side of (8), Φ(f 1 ⊗ g 1 ) ∧ Φ(f 2 ⊗ g 2 ) is the map sending (a 1 ⊗ b 1 ) ⊗ · · · ⊗ (a I+J ⊗ b I+J ) to
X
σ∈S
I0 ,J 0sgn(σ)Φ(f 1 ⊗ g 1 )
(a σ(1) ⊗ b σ(1) ) ⊗ · · · ⊗ (a σ(I0) ⊗ b σ(I
0) )
⊗ Φ(f 2 ⊗ g 2 )
(a σ(I0+1) ⊗ b σ(I
0+1) ) ⊗ · · · ⊗ (a σ(I
0+J
0) ⊗ b σ(I
0+J
0) )
= X
σ∈S
I0 ,J 0sgn(σ) X
τ ∈S
A6i1,j1
sgn(τ )a τ (i1+1) · · · a τ (I
0) f 1 (a τ (1) ⊗ · · · ⊗ a τ (i
1) )
⊗ b τ (1) · · · b τ (i1) g 1 (b τ (i
1+1) ⊗ · · · ⊗ b τ (I
0) )
⊗ X
η∈S
A7i2,j2
sgn(η)a η(i2+1) · · · a η(J
0) f 2 (a η(1) ⊗ · · · ⊗ a η(i
2) )
⊗ b η(1) · · · b η(i2) g 2 (b η(i
2+1) ⊗ · · · ⊗ b η(J
0) )
= X
σ∈S
I0 ,J 0τ ∈S
A6i1,j1
, η∈S
A7i2,j2
sgn(σ) sgn(η) sgn(τ )
a τ (i1+1) · · · a τ (I
0) a η(i
2+1) · · · a η(J
0)
f 1 (a τ (1) ⊗ · · · ⊗ a τ (i1) )f 2 (a η(1) ⊗ · · · ⊗ a η(i
2) )
⊗
b τ (1) · · · b τ (i1)
b η(1) · · · b η(i2) g 1 (b τ (i
1+1) ⊗ · · · ⊗ b τ (I
0) )g 2 (b η(i
2+1) ⊗ · · · ⊗ b η(J
0) ) , (E2)
where the set A 6 = {σ(1), σ(2), . . . , σ(I 0 )} and the set A 7 = {σ(I 0 + 1), . . . , σ(I 0 +J 0 )}. By Proposition 2.2,
(E2) = X
δ∈S
i1,j1,i2,j2sgn(δ)
a δ(i1+1) · · · a δ(I
0) a δ(I
0+i
2+1) · · · a δ(I
0+J
0) (10)
f 1 (a δ(1) ⊗ · · · ⊗ a δ(i1) )f 2 (a δ(I
0+1) ⊗ · · · ⊗ a δ(I
0+i
2) )
⊗
b δ(1) · · · b δ(i1) b δ(I
0+1) · · · b δ(I
0+i
2)
g 1 (b δ(i1+1) ⊗ · · · ⊗ b δ(I
0) )g 2 (b δ(I
0+i
2+1) ⊗ · · · ⊗ b δ(I
0+J
0) ) . Finally, interchanging S i
1,i
2,j
1,j
2 and S i1,j
1,i
2,j
2 by using the argument in Re- mark 2.4 we get that
,j
1,i
2,j
2by using the argument in Re- mark 2.4 we get that
RHS of (10) = RHS of (9).
That is, two sides of (8) define the same map. Henceforth, Φ preserves the wedge product.
Step 3: We will prove that the isomorphism Φ preserves the Schouten product
(11) Φ[f 1 ⊗ g 1 , f 2 ⊗ g 2 ] = [Φ(f 1 ⊗ g 1 ), Φ(f 2 ⊗ g 2 )]
for all f 1 ∈ X i1(R), f 2 ∈ X i2(R), g 1 ∈ X j1(T ) and g 2 ∈ X j2(T ). Recall that [f 1 ⊗ g 1 , f 2 ⊗ g 2 ] = (−1) (I0−1)j
2[f 1 , f 2 ] ⊗ g 1 ∧ g 2 + (−1) i1(J
0−1) f 1 ∧ f 2 ⊗ [g 1 , g 2 ].
(R), g 1 ∈ X j1(T ) and g 2 ∈ X j2(T ). Recall that [f 1 ⊗ g 1 , f 2 ⊗ g 2 ] = (−1) (I0−1)j
2[f 1 , f 2 ] ⊗ g 1 ∧ g 2 + (−1) i1(J
0−1) f 1 ∧ f 2 ⊗ [g 1 , g 2 ].
(T ). Recall that [f 1 ⊗ g 1 , f 2 ⊗ g 2 ] = (−1) (I0−1)j
2[f 1 , f 2 ] ⊗ g 1 ∧ g 2 + (−1) i1(J
0−1) f 1 ∧ f 2 ⊗ [g 1 , g 2 ].
(J
0−1) f 1 ∧ f 2 ⊗ [g 1 , g 2 ].
Hence, the left hand side of (11) is the map sending (a 1 ⊗ b 1 ) ⊗ · · · ⊗ (a I+J −1 ⊗ b I+J −1 ) to
(−1) (I0−1)j
2Φ([f 1 , f 2 ] ⊗ g 1 ∧ g 2 ) (a 1 ⊗ b 1 ) ⊗ · · · ⊗ (a I+J −1 ⊗ b I+J −1 ) +(−1) i1(J
0−1) Φ(f 1 ∧ f 2 ⊗ [g 1 , g 2 ]) (a 1 ⊗ b 1 ) ⊗ · · · ⊗ (a I+J −1 ⊗ b I+J −1 ).
(J
0−1) Φ(f 1 ∧ f 2 ⊗ [g 1 , g 2 ]) (a 1 ⊗ b 1 ) ⊗ · · · ⊗ (a I+J −1 ⊗ b I+J −1 ).
Since
Φ([f 1 , f 2 ] ⊗ g 1 ∧ g 2 ) (a 1 ⊗ b 1 ) ⊗ · · · ⊗ (a I+J −1 ⊗ b I+J −1 )
= X
σ∈S
I−1,Jsgn(σ)a σ(I) · · · a σ(I+J −1) [f 1 , f 2 ] a σ(1) ⊗ · · · ⊗ a σ(I−1)
⊗ b σ(1) · · · b σ(I−1) g 1 ∧ g 2 b σ(I) ⊗ · · · ⊗ b σ(I+J −1)
= X
σ∈S
I−1,Jsgn(σ)a σ(I) · · · a σ(I+J −1) X
τ ∈S
A8i2,i1−1sgn(τ )f 1 (f 2 (a τ (1) ⊗ · · · ⊗ a τ (i2) )
⊗ a τ (i2+1) ⊗ · · · ⊗ a τ (I−1) )
− (−1) (i1−1)(i
2−1) X
η∈S
i1,i2−1A9sgn(η)f 2 (f 1 (a η(1) ⊗ · · · ⊗ a η(i1) )
⊗ a η(i1+1) ⊗ · · · ⊗ a η(I−1) )
⊗
b σ(1) · · · b σ(I−1) X
δ∈S
j1,j2A10sgn(δ)g 1 (b δ(I) ⊗ · · · ⊗ b δ(I+j1−1) )g 2 (b δ(I+j
1) ⊗ · · · ⊗ b δ(I+J −1) ) ,
where the set A 8 = {σ(1), σ(2), . . . , σ(I − 1)}, the set A 9 = {σ(1), σ(2), . . ., σ(I − 1)} and the set A 10 = {σ(I), . . . , σ(I + J − 1)}. By using Proposition 2.2,
Φ([f 1 , f 2 ] ⊗ g 1 ∧ g 2 ) (a 1 ⊗ b 1 ) ⊗ · · · ⊗ (a I+J −1 ⊗ b I+J −1 )
= X
ν∈S
i2,i1−1,j1,j2sgn(ν)f 1 f 2 (a ν(1) ⊗ · · · ⊗ a ν(i2) ) ⊗ a ν(i
2+1) ⊗ · · · ⊗ a ν(I−1)
a ν(I) · · · a ν(I+J −1) ⊗ b ν(1) · · · b ν(I−1)
g 1 (b ν(I) ⊗ · · · ⊗ b ν(I+j1−1) )g 2 (b ν(I+j
1) ⊗ · · · ⊗ b ν(I+J 2−1) )
− X
ν∈S
i1,i2−1,j1,j2sgn(ν)(−1) (i1−1)(i
2−1) f 2 f 1 (a ν(1) ⊗ · · · ⊗ a ν(i
1) ) ⊗ a ν(i
1+1)
⊗ · · · ⊗ a ν(I−1) · a ν(I) · · · a ν(I+J −1) ⊗ b ν(1) · · · b ν(I−1) g 1 (b ν(I) ⊗ · · · ⊗ b ν(I+j1−1) )g 2 (b ν(I+j
1) ⊗ · · · ⊗ b ν(I+J −1) ).
It is easy to see that the term
f 1 f 2 (a ν(1) ⊗ · · · ⊗ a ν(i2) ) ⊗ a ν(i
2+1) ⊗ · · · ⊗ a ν(I−1) · a ν(I) · · · a ν(I+J −1)
⊗ b ν(1) · · · b ν(I−1) · g 1 (b ν(I) ⊗ · · · ⊗ b ν(I+j1−1) )g 2 (b ν(I+j
1) ⊗ · · · ⊗ b ν(I+J 2−1) ) is completely determined by the (i 2 , i 1 − 1, j 1 , j 2 )-shuffle ν, and we write it as f 1 (f 2 (· · · ) · · · ) · · · ⊗ · · · g 1 (· · · )g 2 (· · · ) briefly. Similarly for the second term in the last equation. Moreover,
Φ[f 1 ⊗ g 1 , f 2 ⊗ g 2 ] (a 1 ⊗ b 1 ) ⊗ · · · ⊗ (a I+J −1 ⊗ b I+J −1 ) (12)
= X
ν∈S
i2,i1−1,j1,j2Kf 1 (f 2 (· · · ) · · · ) · · · ⊗ · · · g 1 (· · · )g 2 (· · · )
− X
ν∈S
i1,i2−1,j1,j2Lf 2 (f 1 (· · · ) · · · ) · · · ⊗ · · · g 1 (· · · )g 2 (· · · )
+ X
ν∈S
i1,i2,j2,j1−1M f 1 (· · · )f 2 (· · · ) · · · ⊗ · · · g 1 (g 2 (· · · ) · · · )
− X
ν∈S
i1,i2,j1,j2−1N f 1 (· · · )f 2 (· · · ) · · · ⊗ · · · g 2 (g 1 (· · · ) · · · ),
where the constants K= sgn(ν)(−1) (I0−1)j
2, L = sgn(ν)(−1) (I0−1)j
2+(i
1−1)(i
2−1) , M = sgn(ν)(−1) i
1(J
0−1) and N = sgn(ν)(−1) i
1(J
0−1)+(j
1−1)(j
2−1) .
−1)j
2+(i
1−1)(i
2−1) , M = sgn(ν)(−1) i
1(J
0−1) and N = sgn(ν)(−1) i
1(J
0−1)+(j
1−1)(j
2−1) .
Now, let us compute the right hand side of (11),
[Φ(f 1 ⊗ g 1 ), Φ(f 2 ⊗ g 2 )] (a 1 ⊗ b 1 ) ⊗ · · · ⊗ (a I+J −1 ⊗ b I+J −1 )
= X
σ∈S
J 0 ,I0 −1sgn(σ)Φ(f 1 ⊗ g 1 )
Φ(f 2 ⊗ g 2 ) (a σ(1) ⊗ b σ(1) ) ⊗ · · · ⊗ (a σ(J0) ⊗ b σ(J
0) )
⊗ (a σ(J0+1) ⊗ b σ(J
0+1 )) ⊗ · · · ⊗ (a σ(I
0+J
0−1) ⊗ b σ(I
0+J
0−1) )
− X
σ∈S
I0 ,J 0 −1sgn(σ)(−1) (I0−1)(J
0−1) Φ(f 2 ⊗ g 2 )
Φ(f 1 ⊗ g 1 ) (a σ(1) ⊗ b σ(1) ⊗ · · · ⊗
(a σ(I0) ⊗ b σ(I
0) ) ⊗ (a σ(I
0+1) ⊗ b σ(I
0+1) ) ⊗ · · · ⊗ (a σ(I
0+J
0−1) ⊗ b σ(I
0+J
0−1) ) , and denote it by U + V briefly.
Denote Φ(f 2 ⊗ g 2 ) (a σ(1) ⊗ b σ(1) ) ⊗ · · · ⊗ (a σ(J0) ⊗ b σ(J
0) ) by a 0 ⊗ b 0 briefly, that is,
a 0 ⊗ b 0 = X
ν∈S
A11i2,j2
sgn(ν)f 2 (a ν(1) ⊗ · · · ⊗ a ν(i2) )a ν(i
2+1) · · · a ν(J
0)
(13)
⊗ b ν(1) · · · b ν(i2) g 2 (b ν(i
2+1) ⊗ · · · ⊗ b ν(J
0) ), where the set A 11 = {σ(1), . . . , σ(J 0 )}.
Then,
Φ(f 1 ⊗ g 1 )
Φ(f 2 ⊗ g 2 ) (a σ(1) ⊗ b σ(1) ) ⊗ · · · ⊗ (a σ(J0) ⊗ b σ(J
0) )⊗
(a σ(J0+1) ⊗ b σ(J
0+1) ) ⊗ · · · ⊗ (a σ(I
0+J
0−1) ⊗ b σ(I
0+J
0−1) )
= Φ(f 1 ⊗ g 1 )
(a 0 ⊗ b 0 ) ⊗ (a σ(J0+1) ⊗ b σ(J
0+1) ) ⊗ · · ·
⊗ (a σ(I0+J
0−1) ⊗ b σ(I
0+J
0−1) )
= X
τ ∈S
A12i1,j1
sgn(τ )f 1 (a τ (1) ⊗ · · · ⊗ a τ (i1) )a τ (i
1+1) · · ·
a τ (I0) ⊗ b τ (1) · · · b τ (i
1) g 1 (b τ (i
1+1) ⊗ · · · ⊗ b τ (I
0) ), (E3)
where the set A 12 = {0, σ(J 0 + 1), . . . , σ(I 0 + J 0 − 1)}. By Proposition 2.1,
(E3) = X
η∈S
i1−1,j1A13sgn(η)f 1 (a 0 ⊗ a η(1) ⊗ · · · ⊗ a η(i1−1) )a η(i
1) · · · a η(I
0−1)
⊗ b 0 b η(1) · · · b η(i1−1) g 1 (b η(i
1) ⊗ · · · ⊗ b η(I
0−1) )
+ X
δ∈S
i1,j1−1A13(−1) i1sgn(δ)f 1 (a δ(1) ⊗ · · · ⊗ a δ(i1) )a 0 a δ(i
1+1) · · · a δ(I
0−1)
) )a 0 a δ(i
1+1) · · · a δ(I
0−1)
⊗ b δ(1) · · · b δ(i1) g 1 (b 0 ⊗ b δ(i
1+1) ⊗ · · · ⊗ b δ(I
0−1) ), (E4)
where the set A 13 = {σ(J 0 + 1), . . . , σ(I 0 + J 0 − 1)}. Combining with (13), we have
(E4) = X
ν∈S
A11i2,j2
η∈S
A13i1−1,j1sgn(η) sgn(ν)f 1
f 2 a ν(1) ⊗ · · · ⊗ a ν(i2) a ν(i
2+1) · · · a ν(J
0)
⊗ a η(1) ⊗ · · · ⊗ a η(i1−1)
a η(i1) · · · a η(I
0−1) ⊗ b ν(1) · · · b ν(i
2)
g 2 (b ν(i2+1) ⊗ · · · ⊗ b ν(J
0) )b η(1) · · · b η(i
1−1) g 1 (b η(i
1) ⊗ · · · ⊗ b η(I
0−1) )
+ X
ν∈S
A11i2,j2
δ∈S
A13i1,j1−1
