- Email: [email protected]
cohomology of preprojective algebras of Dynkin quivers
Journal of Pure and Applied Algebra 214 (2010) 28–46
Contents lists available at ScienceDirect
Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa
The calculus structure of the Hochschild homology/cohomology of preprojective algebras of Dynkin quivers Ching-Hwa Eu Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
article
info
Article history: Received 28 September 2007 Received in revised form 1 April 2009 Available online 12 August 2009 Communicated by E.M. Friedlander MSC: Primary: 16E40 secondary: 16G20
abstract The Hochschild cohomology ring of any associative algebra, together with the Hochschild homology, forms a structure of calculus. This was proved in Daletski et al. (1990) [1]. In this paper, we compute the calculus structure for the preprojective algebras of Dynkin quivers over a field of characteristic zero, using the Batalin–Vilkovisky structure of the Hochschild cohomology. Together with the results of Crawley-Boevey et al. [2], where the Batalin–Vilkovisky structure is computed for non-ADE quivers (and the calculus can be easily computed from that), this work gives us a complete description of the calculus for any quiver. © 2009 Published by Elsevier B.V.
1. Introduction The Hochschild homology and cohomology spaces and the duality between them were established in [3]. The cup product structure of the Hochschild cohomology was computed in [4] for the quivers of type A and [5] for those of types D and E. We use the notations from these papers and give the calculus structure in terms of the bases which were defined there. First, we compute the Connes differential on Hochschild homology by using the Cartan identity. Since it turns out that this differential makes the Hochschild cohomology ring a Batalin–Vilkovisky algebra, this gives us an easy way to compute the Gerstenhaber bracket and the contraction map. Then we use the Cartan identity to compute the Lie derivative. 2. Preliminaries 2.1. Quivers and path algebras Let Q be a quiver of ADE type with vertex set I and |I | = r. We write a ∈ Q to say that a is an arrow in Q . We define Q ∗ to be the quiver obtained from Q by reversing all of its arrows. We call Q¯ = Q ∪ Q ∗ the double of Q . Let C be the adjacency matrix corresponding to the quiver Q¯ . The concatenation of arrows generates the nontrivial paths inside the quiver Q¯ . We define ei , i ∈ I to be the trivial path which starts and ends at i. The path algebra PQ¯ = CQ¯ of Q¯ over C is the C-algebra with the paths in Q¯ as the bases and the product xy of two paths x and y to be their concatenation if they are compatible and 0 if not. We define the Lie bracket [x, y] = xy − yx.
E-mail addresses: [email protected], [email protected]. 0022-4049/$ – see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.jpaa.2009.04.018
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
29
Let R = ⊕i∈I Cei . Then R is a commutative semisimple algebra, and PQ is naturally an R-bimodule. Notation 2.1.1. Throughout this paper, we write ‘‘⊗ = ⊗R ’’. 2.2. Frobenius algebras Let A be a finite-dimensional unital C-algebra. We call it Frobenius if there is a linear function f : A → C, such that the '
form (x, y) := f (xy) is nondegenerate, or, equivalently, if there exists an isomorphism φ : A → A∗ of left A-modules: given f , we can define φ(a)(b) = f (ba), and given φ , we define f = φ(1). If f˜ is another linear function satisfying the same properties as f from above, then f˜ (x) = f (xa) for some invertible a ∈ A. Indeed, we define the form {a, b} = f˜ (ab). Then {−, 1} ∈ A∗ , so there is an a ∈ A, such that φ(a) = {−, 1}. Then f˜ (x) = {x, 1} = φ(a)(x) = f (xa). 2.3. The Nakayama automorphism Given a Frobenius algebra A (with a function f inducing a bilinear form (−, −) from above), the automorphism η : A → A defined by the equation (x, y) = (y, η(x)) is called the Nakayama automorphism (corresponding to f ). 2.4. The preprojective algebra
P Given a∗ quiver Q , we define the preprojective algebra ΠQ to be the quotient of the path algebra PQ¯ by the relation a∈Q [a, a ] = 0. Given a path x, we write x∗ for the path obtained from x by reversing all arrows. From now on, we write A = ΠQ .
2.5. Graded spaces Let M = ⊕d≥0 M (d) be a Z+ -graded vector space, with finite-dimensional homogeneous subspaces. We denote by M [n] the same space with grading shifted by n. The graded dual space M ∗ is defined by the formula M ∗ (n) = M (−n)∗ . 2.6. Root system parameters Let w0 be the longest element of the Weyl group W of Q . Then we define ν to be the involution of I, such that w0 (αi ) = −αν(i) (where αi is the simple root corresponding to i ∈ I). It turns out that η(ei ) = eν(i) ([6]; see [4]). Let mi , i = 1, . . . , r, be the exponents of the root system attached to Q , enumerated in the increasing order. Let h = mr + 1 be the Coxeter number of Q . Let P be the permutation matrix corresponding to the involution ν . Let r+ = dim ker(P − 1) and r− = dim ker(P + 1). Thus, r− is half the number of vertices which are not fixed by ν , and r+ = r − r− . A is finite-dimensional, and the following Hilbert series is known from [7, Theorem 2.3.]: HA (t ) = (1 + Pt h )(1 − Ct + t 2 )−1 .
(2.6.1) top
We see that the top degree of A is h − 2, and for the top degree A dimensional submodules: Atop = A(h − 2) =
M
part we get the following decomposition in one-
ei A(h − 2)eν(i) .
i∈I
3. Hochschild cohomology and homology The Hochschild cohomology and homology spaces of A were computed in [3]. We recall the results: Definition 3.0.3. We define the spaces U = ⊕d
(2.6.2)
30
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
Theorem 3.0.4. For the Hochschild cohomology spaces, we have the following natural isomorphisms: HH 0 (A) = U [−2] ⊕ L[h − 2], HH 1 (A) = U [−2], HH 2 (A) = K [−2], HH 3 (A) = K ∗ [−2], HH 4 (A) = U ∗ [−2], HH 5 (A) = U ∗ [−2] ⊕ Y ∗ [−h − 2], HH 6 (A) = U [−2h − 2] ⊕ Y [−h − 2], and HH 6n+i (A) = HH i (A)[−2nh] ∀i ≥ 1. Theorem 3.0.5. The Hochschild homology spaces of A, as graded spaces, are as follows: HH0 (A) = R, HH1 (A) = U , HH2 (A) = U ⊕ Y [h], HH3 (A) = U ∗ [2h] ⊕ Y ∗ [h], HH4 (A) = U ∗ [2h]. HH5 (A) = K [2h], HH6 (A) = K [2h], and HH6n+i (A) = HHi (A)[2nh] ∀i ≥ 1. In [3], an isomorphism HH• (A) = HH 8−• (A)[2h + 2] was introduced. However, because of the periodicity of the Schofield resolution (with period 6), we get for every m ≥ 0 an isomorphism ∼
D : HH• (A) → HH 6m+2−• (A)[2mh + 2].
(3.0.6)
Notation 3.0.7. In [5], the basis elements zk ∈ U [−2] ⊂ HH 0 (A), ωk ∈ L[h − 2], θk ∈ U [−2] = HH 1 (A), fk ∈ K [−2] = HH 2 (A), hk ∈ K ∗ [−2], ζk ∈ U ∗ [−2] = HH 4 (A), ψk ∈ U ∗ [−2] ⊂ HH 5 (A), and εk ∈ Y ∗ [−h − 2] were introduced. Our notations are taken from [5] and are different from those in [4]: zk in [4] corresponds to z2k in our notations, zm corresponds to ω h−3 , 2
gk corresponds to θ2k ,
ψk corresponds to ζl−k and ζk corresponds to θl−k for l = (s)
For ck ∈ HH i (A), 0 ≤ i ≤ 5, we write ck (s)
n
h−2 h−3
Q = An , n odd, Q = An , n ev en
for the corresponding cocycle in HH i+6s . We write ck,t for a cycle in HHj+6t ,
0 ≤ j ≤ 5 which equals D−1 (ck ). We also introduced the maps α : K → K ∗ and β : Y ∗ → Y there. 4. The calculus structure of the preprojective algebra We recall the definition of the calculus. 4.1. Definition of calculus Definition 4.1.1 (Gerstenhaber Algebra). A graded vector space V • is a Gerstenhaber algebra if it is equipped with a graded commutative and associative product ∧ of degree 0 and a graded Lie bracket (the ‘‘Gerstenhaber bracket’’) [, ] of degree −1. These operations have to be compatible in the sense of the following Leibniz rule
[γ , γ1 ∧ γ2 ] = [γ , γ1 ] ∧ γ2 + (−1)k1 (k+1) γ1 ∧ [γ , γ2 ],
(4.1.2)
where γ ∈ V k and γ1 ∈ V k1 . We recall from [8] that Definition 4.1.3 (Precalculus). A precalculus is a pair of a Gerstenhaber algebra (V • , ∧, [, ]) and a graded vector space W • together with
• a module structure ι• : V • ⊗ W −• → W −• of the graded commutative algebra V • on W −• .
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
31
Table 1 Contraction map ιa (b). a
b
(s)
zk
ωk(s) θk(s) ( s)
fk
(s)
θ l, t
ω l, t
zl,t
ψl,t
εl,t
ζl,t
hl , t
fl,t
(zk θl )t −s
δk0 ωl,t −s
(zk ψl )t −s
δk0 hl,t −s
δk0 fl,t −s
0
0
0
0
0
0
δk0 εl,t −s δkl ψ0,t −s δk0 β(εl,t −s )
(zk ζl )t −s
0
(zk zl )t −s δl0 ωk,t −s (zl θk )t −s
0
0
δk0 α(fl,t )
δl0 α(fk,t −s−1 )
0
δl0 fk,t −s−1
δkl ψ0,t −s
(Mα )kl ζ0,t −s
0
δkl ψ0,t −s
f h−3 ,t −s−1
δk,h−3 δl, h−3 · 2 θk,t −s
δk,h−3 (l + 1)·
0
0
0
0
δk,h−3 (l + 1)· θh−3,t −s
0
(zk ψl )t −s δl,h−3 (k + 1)· θh−3,t −s−1
hk
0
0
δl0 hk,t −s−1
0
ζk(s)
(zl ψk )t −s−1
0
(zl ζk )t −s−1
δk,h−3 δl,h−3 · α(f h−3 ,t −s−1 )
0
2
(s)
εk
−δl0 β(εk,t −s )
(s)
ψk
0
δkl ψ0,t −s−1
δl,0 εk,t −s−1
0
(zk ψl )t −s−1
0
0
0
0
−(Mβ )k,l · ζ0,t −s−1 0
δl,h−3 (k + 1)· zl,t −s
δk, h−3 δl,h−3 · 2 θh−3,t −s δk,h−3 δl,h−3 · 2
δk,h−3 δl,h−3 · α(fh−3,t −s−1 )
zk,t −s
• an action L• : V •+1 ⊗ W −• → W −• of the graded Lie algebra V •+1 on W −• which are compatible in the sense of the following equations
ιa Lb − (−1)|a|(|b|+1) Lb ιa = ι[a,b] ,
(4.1.4)
La∧b = La ιb + (−1)|a| ιa Lb .
(4.1.5)
and
Definition 4.1.6 (Calculus). A calculus is a precalculus (V • , W • , [, ], ∧, ι• , L• ) with a degree 1 differential d on W • such that the Cartan identity,
La = dιa − (−1)|a| ιa d,
(4.1.7)
holds. Let A be an associative algebra. The contraction of the Hochschild cochain P ∈ C k (A, A) with the Hochschild chain (a0 , a1 , . . . , an ) is defined by IP (a0 , a1 , . . . , an ) =
(a0 P (a1 , . . . , ak ), ak+1 , . . . , an ) 0
n ≥ k, else.
(4.1.8)
We have Proposition 4.1.9 (Daletski et al. [1]). The contraction IP together with the Connes differential, the Gerstenhaber bracket, the cup product and the action of cochains on chains [9, (3.5), page 46] induce on the pair (HH • (A, A), HH• (A, A)) a structure of calculus. 5. Results about the calculus structure of the Hochschild cohomology/homology of preprojective algebras of Dynkin quivers We state the results in terms of the bases of HH • (A) and HH• (A) which were introduced in Notation 3.0.7: Theorem 5.0.10. The calculus structure is given by Tables 1–3 and the Connes differential B (see [10, 2.1.7]), given as follows The Connes differential B is given as follows:
+ sh zk,s , 2 1 B2+6s (ωk,s ) = + s hβ −1 (ωk,s ), B1+6s (θk,s ) =
1+
k
2
B2+6s (zk,s ) = 0, B3+6s (ψk,s ) =
(s + 1)h − 1 −
k 2
B3+6s (εk,s ) = 0, B4+6s = 0, B5+6s (hk,s ) = (s + 1)hα −1 (hk,s ), B6+6s = 0.
ζk,s ,
32
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
Table 2 Gerstenhaber bracket [a, b]. a
b
( s)
ωk(s)
ζl(t )
εl(t )
ψl(t )
0
−δk0 sh· α −1 (h(l s+t ) )
0
0
(
0
0
0
(−1 + (s − t )h)· δk0 h(l s+t )
−(2 +
ωl(t )
θl(t )
fl
0
−δk0 sh· β −1 (ωl(s+t ) )
(
k − sh)· 2 (zk zl )(s+t )
0
0
(t )
zk
hl
zl
(
θk(s)
(t )
(t )
l−k 2
+ (s − t )h)· −(1 + th)· δk0 fl(s+t ) (z θ )(s+t ) k l
(s)
fk
−(1 + sh)· δkl ζ0(s+t )
0
hk
l 2
(zk ζl )
h 2
+ 1 + th)·
δkl ζ0(s+t )
+ th)·
−(1 + th + (s+t )
(s+t )
0
δk0 εl
h 2
0 h+1
(s − t )h· (Mα−1 )kl · ψ0(s+t )
(s)
−(
k − sh)· 2 (zk ζl )(s+t )
+ th)· −( 2 δk, h−3 δl,h−3 ·
0
0
−(2 +
k+l 2
+ (t − s)h)·
(zk ψl )(s+t ) −(k + 1)· (1 + sh)·
δl,h−3 zh(s−+3t +1)
h−1 ((s − t )h − )· 2 δk, h−3 δl,h−3 · 2
2
(s+t +1)
zh−3
ζk(s)
)·
0
0
s+t +1) θh(− 3 h+1 −(sh + )· 2 δk,h−3 δl,h−3 ·
(s+t +1)
f h−3 2
εk(s)
0
ψk(s)
0
(s − t )h· δk,h−3 δl,h−3 · α(f h(s−+3t +1) ) 2
6. Batalin–Vilkovisky structure on Hochschild cohomology Recall the isomorphism (3.0.6). It translates the Connes differential B : HH• (A) → HH•+1 (A) on Hochschild homology into a differential ∆ : HH • (A) → HH •−1 (A) on Hochschild cohomology, i.e. we have the commutative diagram B
HH• (A)
HH•+1 (A)
−−−−→
Dy∼
∼ yD ∆
HH 6m+2−• (A)[2mh + 2] −−−−→ HH6m+1−• (A)[2mh + 2] Theorem 6.0.11 (BV Structure on Hochschild Cohomology). Gerstenhaber bracket we get the following equation:
[a, b] = ∆(a ∪ b) − ∆(a) ∪ b − (−1)|a| a ∪ ∆(b),
∆ makes HH • (A) a Batalin–Vilkovisky algebra, i.e. for the
∀a, b ∈ HH ∗ (A).
(6.0.12)
The isomorphism D intertwines contraction and cup product maps, i.e. we have
D(ιη c ) = η ∪ D(c ),
∀c ∈ HH• (A), η ∈ HH • (A).
(6.0.13)
Remark 6.0.14. Note that ∆ in Eq. (6.0.12) depends on which m HH 6m+2−• (A)[2mh + 2], where the Gerstenhaber bracket does not.
∼
∈ N we choose to identify D : HH• (A) →
Proof. We apply the functor HomAe (−, A ⊗C A) : Ae − mod M
→ 7→
Ae − mod, M∨
on the Schofield resolution: d∨ 1
d∨ 3
d∨ 2
d∨ 4
(A ⊗ A)∨ → (A ⊗ V ⊗ A)∨ → (A ⊗ A[2])∨ → (A ⊗ N [h])∨ → d∨ 4
d∨ 5
d∨ 6
d∨ 7
→ (A ⊗ V ⊗ N [h])∨ → (A ⊗ N [h + 2])∨ → (A ⊗ A[2h])∨ → . . . . An element in (A ⊗ A)∨ or (A ⊗ N )∨ is determined by the image of 1 ⊗ 1, An element in (A ⊗ V ⊗ A)∨ or (A ⊗ V ⊗ N )∨ by the images of 1 ⊗ a ⊗ 1 for all arrows a ∈ Q¯ .
(6.0.15)
ωk(s)
zk
( s)
ψk(s)
εk
( s)
ζk(s)
hk
(s)
fk
(s)
θk(s)
a
l
1
−((s + )h + 1)· 2 δkl ζ0,t −s−1
((s + )h + 1)· 2 δl0 εk,t −s−1
0
−δk0 sh β −1 (ωl,t −s )
0
(1 + + th)· 2 (zl ψk )t −s−1
(k − sh) (zk θl )t −s
(1 + th)· δl0 ωk,t −s
l
1
0
0
0
( + t )h· 2 δk0 ωl,t −s
1
ωl,t
−(2 + + sh)· 2 (zl ζk )t −s−1
k
(1 + th)· δl0 hk,t −s−1
(1 + + th)· 2 (zk θl )t −s −(1 + sh)· δl0 fk,t −s−1
θl,t
b
Table 3 Lie derivative La (b).
1 2
+ t − s )h·
β −1 (ωk,t −s )
δl0 (
0
((t − s)h − 1 − )· 2 (zl ζk )t −s−1
0
0
k−l
+ (t − s)h)· 2 (zk zl )t −s
k+l
δl0 (t − s)h· α −1 (hk,t −s−1 )
0
(1 +
zl,t l
0
( − sh)· 2 (zk ζl )t −s
2
+ (t − s))h· δk0 εl,t −s
δkl · (−1 + h + (t − s)h)· ζ0,t −s
0
0
h+1 (th + )· 2 δk,h−3 δl,h−3 · α(f h−3 ,t −s−1 ) k
0
0
0
0
(
1
εl,t
0
2
2
f h−3 ,t −s−1
((t + 1)h − 1 − )· 2 (zk ψl )t −s −(1 + sh)· δl,h−3 zh−3,t −s h+1 )· (th + 2 δk, h−3 δl,h−3 · 2 θh−3,t −s h+1 )· −(sh + 2 δk,h−3 δl,h−3 ·
ψl,t
2
0
0
2
f h−3 ,t −s−1
(t − s)h· δk,h−3 δl,h−3
0
0
zh−3,t −s
h−1 ((t − s)h + )· 2 δk, h−3 δl,h−3 ·
0
((t − s + 1)h − 1 − (zk ζl )t −s
ζl,t 2
l−k
)
2
0
(k − sh)· α −1 (hl,t −s )
(t + 1)h· δk,h−3 δl,h−3 · θh−3,t −s
0
zh−3,t −s
h+1 )· −(sh + 2 δk,h−3 δl, h−3 ·
(l + 1)·
0
0
zh−3,t −s
((t − s)h + 1 + δk,h−3
0
0
2
h−3
((t − s + 1)h − 1)· δkl ζ0,t −s
0
−(1 + sh)· δkl ζ0,t −s (t + 1 )h· (Mα−1 )lk · ψ0,t −s
(t − s + 1)h· δk0 fl,t −s
fl,t
(t + 1 )h· δk0 hl,t −s
hl,t
)·
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46 33
34
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
n
+1
Q = A,
Let us define σ = −1 Q = D, E . We make the following identifications:
(A ⊗ A)[−2mh] = (A ⊗ A[2mh])∨ : we identify x ⊗ y with thePmap that sends 1 ⊗ 1 to σ m y ⊗ x, (A ⊗ V ⊗ A)[−2mh − 2] = (A ⊗ V ⊗ A[2mh])∨ : we identify a∈Q¯ a xa ⊗ a∗ ⊗ ya with the map that sends 1 ⊗ a ⊗ 1 to −σ m ya ⊗ xa , (A ⊗ A)[−2mh − 2] = (A ⊗ A[2mh + 2])∨ : we identify x ⊗ y with the map that sends 1 ⊗ 1 to −σ m y ⊗ x, (A ⊗ N )[−(2m + 1)h] = (A ⊗ N [(2m + 1)h])∨ : we identify x ⊗ y with the P map that sends 1 ⊗ 1 to −σ m η(y) ⊗ x, ∨ (A ⊗ V ⊗ N )[−(2m + 1)h − 2] = (A ⊗ V ⊗ N [(2m + 1)h]) : we identify a∈Q¯ a xa ⊗ η(a∗ ) ⊗ ya with the map that sends 1 ⊗ a ⊗ 1 to σ m+1 η(ya ) ⊗ xa , (A ⊗ N )[−2(m + 1)h − 2] = (A ⊗ A[2(m + 1)h + 2])∨ : we identify x ⊗ y with the map that sends 1 ⊗ 1 to σ m+1 η(y) ⊗ x, so (6.0.15) becomes d∨ 1
d∨ 3
d∨ 2
d∨ 4
(A ⊗ A) → (A ⊗ V ⊗ A) → (A ⊗ A[−2]) → (A ⊗ N [−h]) → d∨ 6
d∨ 5
d∨ 4
d∨ 7
→ (A ⊗ V ⊗ N [−h − 2]) → (A ⊗ N [−h − 2]) → (A ⊗ A[−2h]) → . . . .
(6.0.16)
∨
We show under the identification from above, the differentials di corresponds to the differentials from the Schofield resolution, i.e. (6.0.16) can be rewritten in this form: d2 [−2]
d1 [−2]
(A ⊗ A) →
(A ⊗ V ⊗ A) → (A ⊗ A[−2])
d6 [−2h−2]
(A ⊗ N [−h])
→
d4 [−2h−2]
d5 [−2h−2]
→
→
(A ⊗ V ⊗ N [−h − 2])
d3 [−2h−2]
(A ⊗ N [−h − 2])
→
d6 [−2h−2]
→
(A ⊗ A[−2h])
d4 [−2h−2]
→
d2 [−2h−2]
→
....
(6.0.17)
It is enough to show this for the first period. d∨ 1 (x ⊗ y)(1 ⊗ a ⊗ 1) = (x ⊗ y) ◦ (a ⊗ 1 − 1 ⊗ a) = ay ⊗ x − y ⊗ xa, so d∨ 1 (x ⊗ y) =
X
a (xa ⊗ a∗ ⊗ y − x ⊗ a∗ ⊗ ay) =
a∈Q¯
d∨ 2
a (xa ⊗ a∗ ⊗ y + x ⊗ a ⊗ a∗ y) = d2 (x ⊗ y),
a∈Q¯
X
X
a xa ⊗ a∗ ⊗ ya (1 ⊗ 1) =
a∈Q¯
X
a xa ⊗ a∗ ⊗ ya ◦
X
¯
=
b b ⊗ b∗ ⊗ 1 + b 1 ⊗ b ⊗ b∗
¯
a∈Q b∈Q X (a a∗ ya ⊗ xa − a ya ⊗ xa a∗ ), a∈Q¯
so
d∨ 2
X
a xa ⊗ a∗ ⊗ ya =
a∈Q¯
X
a (xa a∗ ⊗ ya − xa ⊗ a∗ ya ) = d1 (a xa ⊗ a∗ ⊗ ya ),
a∈Q¯
! d3 (x ⊗ y)(1 ⊗ 1) = (x ⊗ y) ◦ ∨
X
∗
xi ⊗ xi
xi ∈B
=−
X
xi y ⊗ xx∗i = −
xi ∈B
X
η(x∗i )y ⊗ xxi ,
xi ∈B
so d∨ 3 (x ⊗ y) =
X
xxi ⊗ x∗i η(y) =
xi ∈B
X
xyxi ⊗ x∗i = d6 (x ⊗ y)
xi ∈B
d∨ 4 (x ⊗ y)(1 ⊗ a ⊗ 1) = (x ⊗ y) ◦ (a ⊗ 1 − 1 ⊗ a) = −aη(y) ⊗ x + η(y) ⊗ xη(a), so d∨ 4 (x ⊗ y) =
X
a σ (−x ⊗ η(a∗ ) ⊗ η(a)y + xη(a) ⊗ η(a∗ ) ⊗ y)
a∈Q¯
=
X (a xa ⊗ a∗ ⊗ y + a x ⊗ a ⊗ a∗ y) = d5 (x ⊗ y), a∈Q¯
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
d∨ 5
X
a xa ⊗ η(a∗ ) ⊗ ya (1 ⊗ a ⊗ 1) =
a∈Q¯
X
a xa ⊗ η(a∗ ) ⊗ ya ◦
X (b b ⊗ b∗ ⊗ 1 + b 1 ⊗ b ⊗ b∗ )
¯
= σ
a∈Q X
35
b∈Q¯
(−a η(ya ) ⊗ xa + a η(ya ) ⊗ xa η(a∗ )), ∗
a∈Q¯
so
d∨ 5
X
a xa ⊗ η(a∗ ) ⊗ ya =
a∈Q¯
X (−a xa ⊗ η(a∗ )ya + a xa η(a∗ ) ⊗ ya ) a∈Q¯
= d4
X
a xa ⊗ η(a∗ ) ⊗ ya ,
a∈Q¯
! X =σ xi η(y) ⊗ xη(x∗i ) xi ∈B xi ∈B X X x∗i ⊗ xyxi , x∗i η(y) ⊗ xxi = σ = σ
d6 (x ⊗ y)(1 ⊗ 1) = (x ⊗ y) ◦ ∨
X
∗
x⊗ xi
xi ∈B
xi ∈B
so d∨ 6 (x ⊗ y) =
X
xyxi ⊗ x∗i = d3 (x ⊗ y).
xi ∈B
Fix m ≥ 0. The map which shifts the degree by −2mh − 2 produces the following diagram which commutes by the computations above: d6m+2
A ⊗ A[2mh + 2] −−−−→
d6m+1
−−−−→ . . .
A ⊗ V ⊗ A[2h]
y
y d∨ 1
(A ⊗ A)
d∨ 2
−−−−→ (A ⊗ V ⊗ A)[−2] −−−−→ . . .
d2
. . . −−−−→
d1
A⊗V ⊗A
−−−−→
y
mult .
A⊗A
−−−−→ A
y
d∨ 6m+1
d∨ 6m+2
mult .
. . . −−−−→ (A ⊗ V ⊗ A)[−2mh − 2] −−−−→ (A ⊗ A)[−mh − 2] −−−−→ A Similarly to the proof of [11, Theorem 3.4.3.], this self-dual morphism of the Schofield resolution C • into the dual complex
(C • )∨ can be used to prove (6.0.13).
(6.0.12) follows, as in the proof of [11, Theorem 3.4.3.], from (6.0.13) and the calculus structure.
6.1. Computation of the calculus structure of the preprojective algebra Since the calculus structure is defined on Hochschild chains and cochains, we have to work with them on the resolution for computations. It turns out that we only have to compute Lθ0 directly, the rest can be deduced from formulas given by the calculus and the BV structure. d3
d2
d1
d0
· · · −−−−→ A ⊗ A[2] −−−−→ A ⊗ V ⊗ A −−−−→ A ⊗ A −−−−→
µ2 y µ1 y
b3
· · · −−−−→
A⊗4
b2
−−−−→
A⊗3
b1
A −−−−→ 0
b0
−−−−→ A⊗2 −−−−→ A −−−−→ 0
These maps ψi gives us a chain map between the Schofield and the bar resolution:
µ1 (1 ⊗ y ⊗ 1) = 1 ⊗ y ⊗ 1, X µ2 (1 ⊗ 1) = a 1 ⊗ a ⊗ a∗ ⊗ 1, a∈Q¯
µ3 (1 ⊗ 1) =
XX a∈Q¯ xi ∈B
a 1 ⊗ xi ⊗ a ⊗ a∗ ⊗ x∗i ,
36
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
and
µ3+i = µi
XX
xi ⊗ a ⊗ a∗ ⊗ x∗i .
a∈Q¯ xi ∈B
Now, we apply the functor − ⊗Ae A on the commutative diagram: d03
d02
d1
· · · −−−−→ AR [2] −−−−→ (V ⊗ A)R −−−−→ µ02 y µ01 y b03
b2
AR
−−−−→ 0
b1
· · · −−−−→ (A⊗3 )R −−−−→ (A⊗2 )R −−−−→ (A⊗1 )R −−−−→ 0 where
µ01 (x ⊗ y) = x ⊗ y, X µ02 (x) = a a ⊗ a∗ ⊗ x, a∈Q¯
µ3 ( x ) = 0
XX
a xi ⊗ a ⊗ a∗ ⊗ x∗i η(x),
a∈Q¯ xi ∈B
and
µ03+i = µ0i
XX
xi ⊗ a ⊗ a∗ ⊗ x∗i .
a∈Q¯ xi ∈B
Now, we compute Lθ0 : Lemma 6.1.1. For each x ∈ HHi (A),
Lθ0 (x) = x
deg(x) 2
.
(6.1.2)
Proof. Via µ0 , we already identified x ∈ HHi (A) with cycles in the Hochschild chain, but we still have to identify θ0 with an element in HomAe (A⊗3 , A): given any monomial b = b1 . . . bl , bi ∈ V , the map
τ (1 ⊗ b ⊗ 1) =
l X
b1 . . . bi−1 ⊗ bi ⊗ bi+1 . . . bl
i =1
makes the diagram d0
d1
A ⊗ V ⊗ A −−−−→ A ⊗ A −−−−→ A −−−−→ 0
x
τ
A⊗3
b0
b1
−−−−→ A⊗2 −−−−→ A −−−−→ 0
commute. Applying HomAe (__ ⊗ A), we get a map
τ ∗ : Homk (V ) → Homk (A), such that
(θ0 ◦ τ ∗ )(b1 . . . bl ) =
l X
b1 . . . bi−1 θ0 (bi )bi+1 . . . bl = s(b) · b,
i =1
where for b = b1 . . . bl , s(b) is the number of bi ∈ Q ∗ . Recall from [9, (3.5), page 46] that the Lie derivative of θ0 ◦ τ ∗ on Hochschild chains is defined by
Lθ0 ◦τ ∗ (a1 ⊗ · · · ⊗ ak ) =
k X
a1 ⊗ · · · ⊗ (θ0 ◦ τ ∗ )(ai ) ⊗ · · · ⊗ ak
i =1 k X = (s(a1 ) + · · · s(ak ))a1 ⊗ · · · ⊗ ak , i =1
and it can easily be checked that for each x ∈ HHi (A), Lθ0 ◦τ ∗ acts on µ0i (x), x ∈ HH i (A), by multiplication with
1 2
deg(x).
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
37
6.1.1. The contraction map From (6.0.13) we know that the contraction map on Hochschild homology is given by the cup product on Hochschild cohomology which was computed in [4,5]. Table 1 contains these results, rewritten in terms of the contraction maps. 6.1.2. The Connes differential We start with the computation of the Connes differential and recall the diagram from [3]: degree 0
y 2 ≤ deg ≤ h − 1
HH1 (A)
U
B1 y
∼y
HH2 (A)
2 ≤ deg ≤ h
U
B2 y h ≤ deg ≤ 2h − 2
HH3 (A)
U ∗ [2h]⊕Y ∗ [h]
B3 y
∼y
h + 1 ≤ deg ≤ 2h − 2 HH4 (A)
U ∗ [2h]
B4 y
y
0
HH5 (A)
2h
K ∗ [2h]
B5 y
∼y
HH6 (A)
2h
⊕ Y [h] ∼y
K [2h]
B6 y
y
0
2h + 2 ≤ deg ≤ 3h − 1 HH7 (A)
U [2h]
B7 y .. . Proposition 6.1.3. The Connes differential B is given as follows: (s)
k
+ sh zk(s) , 1 (s) B2+6s (ωk ) = + s hβ −1 (ωk(s) ), B1+6s (θk ) =
1+
2
2
(s)
B2+6s (zk ) = 0, (s)
B3+6s (ψk ) =
(s + 1)h − 1 −
k 2
ζk ,
(s)
B3+6s (εk ) = 0, B4+6s = 0, (s)
(s)
B5+6s (hk ) = (s + 1)hα −1 (hk ), B6+6s = 0. Proof. We use the Cartan identity (4.1.7) with a ∈ θ0 ,
Lθ0 = Bιθ0 + ιθ0 B,
(6.1.4)
where Lθ0 acts on x ∈ HHi by multiplication by (t )
(t )
(t )
(t )
1 2
deg(x) (see Lemma 6.1.1). The above identities for the Connes differential (t )
(t )
(t )
follow since ιθ0 acts on θk , ωk , ψk and hk by zero, and zk , β −1 (ωk ), ζk contraction with ιθ0 .
(s)
and α −1 (hk ) are their unique preimages the
38
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
6.1.3. The Gerstenhaber bracket We compute the brackets using the identification HH i (A) = HH6m+2−i (A)[−2mh − 2] for m 1 and the BV-identity (6.0.12). Brackets involving HH 6s (A): By degree argument these brackets are zero:
ωk(s) with HH 1+6t (A), HH 2+6t (A), HH 3+6t (A), HH 4+6t (A), and ψl ∈ HH 5+6t (A). From the BV-identity (6.0.12), we see (s) (t ) (t ) that brackets of zk with zl ∈ HH 6t (A), HH 2+6t (A), HH 4+6t (A) and εl ∈ HH 5+6t (A) are zero because ∆ acts by zero 6t 2+6t 4+6t on U [−2th − 2] ⊂ HH (A), HH (A) and HH (A). We compute the remaining brackets:
[zk(s) , ωl(t ) ] = ∆(zk(s) ∪ ωl(t ) ) − ∆(zk(s) ) ∪ωl(t ) − zk(s) ∪ ∆(ωl(t ) ) | {z } =0
= δk0 ∆(ωl ) − + (m − t )h zk(s) ∪ β −1 (ωl(t ) ) 2 h h + (m − s − t )h β −1 (ωl(s+t ) ) − + (m − t )h β −1 (ωl(s+t ) ) = δk0 (s+t )
h
2
2
= −δk0 shβ
−1
(s+t )
(ωl
),
[zk , θl ] = ∆(zk ∪ θl ) − ∆(zk(s) ) ∪θl(t ) − zk(s) ∪ ∆(θl(t ) ) | {z } (s)
(t )
(s)
(t )
=0
l = ∆((zk θl ) ) − 1 + + (m − t )h zk(s) zl(t ) 2 k+l l = 1+ + (m − s − t )h (zk zl )(s+t ) − 1 + + (m − t )h (zk zl )(s+t ) 2 2 k − sh (zk zl )(s+t ) , = (s+t )
2
[zk , hl ] = ∆(zk(s) ∪ h(l t ) ) − ∆(zk(s) ) ∪h(l t ) − zk(s) ∪ ∆(h(l t ) ) | {z } (s)
(t )
=0
(s+t )
= δk0 ∆(hl
) − (h + (m − t − 1)h)zk(s) ∪ α −1 (h(l t ) )
= δk0 (h + (m − s − t − 1)h)α −1 (h(l s+t ) ) − δk0 (h + (m − t − 1)h)α −1 (h(l s+t ) ) = −δk0 shα −1 (h(l s+t ) ), [zk(s) , ψl(t ) ] = ∆(zk(s) ∪ ψl(t ) ) − ∆(zk(s) ) ∪ψl(t ) − zk(s) ∪ ∆(ψl(t ) ) | {z } =0
t
(s) (t ) = ∆((zk ψl ) )− h−1− zk ζl 2 l−k l (s+t ) = (m − s − t )h − 1 − (zk ζl ) − (m − t )h − 1 − (zk ζl )(s+t ) 2 2 k = − sh (zk ζl )(s+t ) (s+t )
2
(s)
(t )
[ωk , εl ] = ∆(ωk(s) ∪ εl(t ) ) − ∆(ωk(s) ) ∪ εl(t ) − ωk(s) ∪ ∆εl(t ) | {z } =0
h
+ (m − s)h β −1 (ωk ) ∪ εl(t ) h = δkl (h − 1 + (m − s − t − 1)h)ζ0(s+t ) − δkl + (m − s)h ζ0 2 h = δkl − − 1 − th ζ0 . = ∆(δkl ψ0s+t ) −
2
2
(s)
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
Brackets involving HH 1+6s (A):
[θk(s) , θl(t ) ] = ∆(θk(s) ∪ θl(t ) ) − ∆(θk(s) ) ∪ θl(t ) + θk(s) ∪ ∆(θl(t ) ) | {z } =0
= − 1+ =
l−k 2
k 2
l + (m − s)h zk(s) θl(t ) + 1 + + (m − t )h θk(s) zl(t ) 2
+ (s − t )h (zk θl )(s+t )
[θk(s) , fl(t ) ] = ∆(θk(s) ∪ fl(t ) ) − ∆(θk(s) ) ∪ fl(t ) + θk(s) ∪ ∆(fl(t ) ) | {z } =0
= δk0 (∆(α(fl
(s+t )
)) − (1 + (m − s)h)fl
(s+t )
)
= δk0 (h + (m − s − t − 1)h)fl(s+t ) − (1 + (m − s)h)fl(s+t ) = −δk0 (1 + th)fl(s+t ) [θk(s) , h(l t ) ] = ∆(θk(s) ∪ h(l t ) ) − ∆(θk(s) ) ∪ h(l t ) + θk(s) ∪ ∆(h(l t ) ) | {z } =0
= − 1 + (m − s)h +
k
(s)
(t )
z k ∪ hl
2
+ (h + (m − t − 1)h)θk(s) ∪ α −1 (h(l t ) )
= −δk0 (1 + (m − s)h)h(l s+t ) + δk0 (m − t )hh(l s+t ) = δk0 (−1 + (s − t )h)h(l s+t ) [θk(s) , ζl(t ) ] = ∆(θk(s) ∪ ζl(t ) ) − ∆(θk(s) ) ∪ ζl(t ) + θk(s) ∪ ∆(ζl(t ) ) | {z } =0
= ∆((zk ψl )(s+t ) ) − 1 + =
h−1−
l−k 2
k 2
+ (m − s)h zk(s) ∪ ζl(t )
+ (m − s − t − 1)h (zk ζl )(s+t )
k − 1 + + (m − s)h (zk ζl )(s+t ) 2
= − 2+ (s)
(t )
l 2
+ th (zk ζl )(s+t )
(s)
[θk , ψl ] = ∆(θk ∪ ψl(t ) ) −∆(θk(s) ) ∪ ψl(t ) + θk(s) ∪ ∆(ψl(t ) ) {z } | =0
= − 1+ = − 2+
k 2
l + (m − s)h zk(s) ψl(t ) + h − 1 − + (m − t − 1)h θk(s) ζl(t ) 2
k+l 2
+ (t − s)h (zk ψl )(s+t ) ,
[θk(s) , εl(t ) ] = ∆(θk(s) ∪ εl(t ) ) − ∆(θk(s) ) ∪ εl(t ) + θk(s) ∪ ∆(εl(t ) ) | {z } =0
= δk0 ∆(β(εl(s+t ) )) − δk0 1 + (m − s)h + = δk0
h 2
k 2
(s) (t )
zk εl
+ (m − s − t − 1)h εl(s+t ) − (1 + (m − s)h)εl(s+t )
1 (s+t ) = −δk0 1 + t + h εl . 2
39
40
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
Brackets involving HH 2+6s (A): By degree argument, the bracket of HH 2+6s (A) with HH 2+6t (A) is zero.
[fk(s) , h(l t ) ] = ∆(fk(s) ∪ h(l t ) ) − ∆(fk(s) ) ∪h(l t ) − fk(s) ∪ ∆(h(l t ) ) | {z } =0
= ∆(δkl ψ0(s+t ) ) − (h + (m − t − 1)h)fk(s) ∪ α −1 (h(l t ) ) = δkl (h − 1 + (m − s − t − 1)h)ζ0 − δkl (m − t )hζ0 = −δkl (1 + sh)ζ0(s+t ) , [fk(s) , ζl(t ) ] = ∆(fk(s) ∪ ζl(t ) ) − ∆(fk(s) ) ∪ζl(t ) − fk(s) ∪ ∆(ζl(t ) ) | {z } | {z } = δl,h−3 (k + (s)
(t )
(s)
=0 (s+t ) 1)∆(zh−3 ) = (t ) (s)
=0
0,
[fk , ψl ] = ∆(fk ∪ ψl ) − ∆(fk ) ∪ψl(t ) − fk(s) ∪ ∆(ψl(t ) ) | {z } =0
(t )
l
+ (m − t − 1)h fk(s) ∪ ζl(t ) h−3 s+t +1) = δl,h−3 (k + 1)∆(θh(− ) − δ ( m − t ) h − 1 − (k + 1)zh(s−+3t +1) l,h−3 3 2 h−3 = δl,h−3 (k + 1) 1 + + (m − s − t − 1)h zh(s−+3t +1) 2 h−3 (k + 1)zh(s−+3t +1) − δl,h−3 (m − t )h − 1 − (s)
= ∆(fk ∪ ψl ) − h − 1 −
2
2
= −δl,h−3 (k + 1)(1 +
(s+t +1) sh)zh−3 , (s) (t )
[fk , εl ] = ∆ (fk(s) ) ∪ (εl(t ) ) −∆(fk ) ∪ εl − ∆fk(s) ∪ ∆(εl(t ) ) = 0. | | {z } {z } | {z }=0 (s)
(t )
=0
=0
Brackets involving HH 3+6s (A): We have
[h(ks) , h(l t ) ] = ∆(hk(s) ∪ h(l t ) ) − ∆(h(ks) ) ∪ h(l t ) + hk(s) ∪ ∆(h(l t ) ) | {z } =0
= −(h + (m − s − 1)h)α −1 (h(ks) ) ∪ h(l t ) + (h + (m − t − 1)h)h(ks) ∪ α −1 (h(l t ) ) = (s − t )hα −1 (h(ks) ) ∪ h(l t ) = (s − t )h(Mα−1 )kl ψ0(s+t ) , [h(ks) , ζl(t ) ] = ∆(h(ks) ∪ ζl(t ) ) − ∆(hk(s) ) ∪ ζl(t ) + hk(s) ∪ ∆(ζl(t ) ) | {z } =0
− (m − s)hα (h(ks) ) ∪ ζl(t ) h−3 = δk, h−3 δl,h−3 1+ + (m − s − t − 1)h zh(s−+3t +1) − (m − s)hzh(s−+3t +1) 2 2 h+1 = δk, h−3 δl,h−3 − − th zh(s−+3t +1) . =
s+t +1) δk, h−3 δl,h−3 ∆(θh(− ) 3 2
−1
2
2
We have
[h(ks) , ψl(t ) ] = ∆(hk(s) ∪ ψl(t ) ) − ∆(h(ks) ) ∪ ψl(t ) + h(ks) ∪ ∆(ψl(t ) ) | {z } =0
(s)
(t )
l
= −(m − s)hα (hk ) ∪ ψl + (m − t )h − 1 − 2 s+t +1) = −(m − s)hδk, h−3 δl,h−3 θh(− + (m − t )h − 1 − 3 2 h−1 s+t +1) = δl,h−3 δk, h−3 (s − t )h − θh(− 3 −1
2
2
(s)
(t )
hk ∪ ζ l
l 2
s+t +1) δk, h−3 θh(− 3 2
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
41
[h(ks) , εl(t ) ] = ∆(hk(s) ∪ εl(t ) ) − ∆(hk(s) ) ∪ εl(t ) − ∆h(ks) ∪ ∆(εl(t ) ) | {z } | {z } =0
=0
(s)
(t )
= −(m − s)hα (hk ) ∪ εl = 0. −1
Brackets involving HH 4+6s (A): (s) (t ) (s) (t ) (s) (t ) (s) (t ) The bracket [ζk , ζl ] = ∆(ζk ∪ ζl ) − ∆(ζk ) ∪ ζl − ζk ∪ ∆(ζl ) = 0 because ∆ is zero on HH 2+6s and HH 4+6s .
[ζk(s) , ψl(t ) ] = ∆(ζk(s) ∪ ψl(t ) ) − ∆(ζk(s) ) ∪ψl(t ) − ζk(s) ∪ ∆(ψl(t ) ), | {z } =0
l = δk,h−3 δl,h−3 ∆ α f h(−s+3 t +1) ζk(s) ∪ ζl(t ) − (m − t )h − 1 − 2 2 h−3 (s+t +1) (s+t +1) = δk,h−3 δl,h−3 (m − s − t − 1)hf h−3 − (m − t )h − 1 − f h−3 2 2 2 h+1 = δk,h−3 δl,h−3 −sh − f h−3 . 2
2
The bracket of HH 5+6s (A) with HH 5+6s (A):
[ψk(s) , ψl(t ) ] = ∆(ψk(s) ∪ ψl(t ) ) −∆(ψk(s) ) ∪ ψl(t ) + ψk(s) ∪ ∆(ψl(t ) ), | {z } =0
= − (m − s)h − 1 −
k
l ζk(s) ∪ ψl(t ) + (m − t )h − 1 − ψk(s) ∪ ζl(t )
2
2
(s+t +1)
= δk,h−3 δl,h−3 (s − t )hα(f h−3
).
2
6.1.4. The Lie derivative L We use the Cartan identity (4.1.7) to compute the Lie derivative. HH 1+6s (A)-Lie derivatives: From the Cartan identity, we see that
Lθ (s) = Bιθ (s) + ιθ (s) B. k
k
k
On θl,t , ωl,t , ψl,t and hl,t , the Connes differential acts by multiplication with
degree and taking the preimage under ιθ0 , and ιθ (s) acts on them by zero. B acts by zero on zl , εk , ζl,t and fl,t . Since B is degree preserving, this means that Lθ (s) acts on (t )
(t )
k
and then multiplication with
Lθ (s) (θl,t ) =
1+
k
Lθ (s) (zl,t ) =
l 2
2
k
Lθ (s) (ωl,t ) = δk0
2
k
Lθ (s) (εl,t ) = δk0 k
Lθ (s) (ψl,t ) =
k
Lθ (s) (ζl,t ) = k
1
1 2
+ (t − s)h (zk zl )t −s ,
+ t h ω l ,t − s , + ( t − s ) h εl ,t − s ,
(t + 1)h − 1 −
l
2
(t − s + 1)h − 1 −
Lθ (s) (hl,t ) = δk0 (t + 1)hhl,t −s , k
Lθ (s) (fl,t ) = δk0 (t − s + 1)hfl,t −s . k
1 2
+ th (zk θl )t −s ,
k+l
1+
k
(s)
their degree times zk , and on zl,t , εk,t , ζl,t and fl,t by multiplication with zk degree of their product. So we get the following formulas:
θl,t , ωl,t , ψl,t and hl,t by multiplication with 1 2
(s)
1 2
(zk ψl )t −s , l−k 2
(zk ζl )t −s ,
42
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
HH 2+6s (A)-Lie derivatives: We compute Lf (s) : k
Lf (s) (θl,t ) = B(ιf (s) (θl,t )) − ιf (s) (B(θl,t )) k
k
k
l = B(δl0 α(fk,t −s−1 )) − 1 + + th ιf (s) zl,t 2
k
= δl0 (t − s)hfk,t −s−1 − δl0 (1 + th)fk,t −s−1 = −δl0 (1 + sh)fk,t −s , Lf (s) (fl,t ) = B( ιf (s) (fl,t ) ) = 0, k | k {z } ∈HH4+6(t −s)
Lf (s) (zl,t ) = δl0 B(fk,t −s ) = 0, k
Lf (s) (ωl,t ) = B(ιf (s) ωl,t ) + ιf (s) B(ωl,t ) = k
| k {z }
+ t hιf (s) β −1 (ωl,t ) = 0,
1 2
k
k
=0
Lf (s) (εl,t ) = B(ιf (s) (εl,t )), k
| k {z } =0
Lf (s) (ψl,t ) = B(ιf (s) (ψl,t )) − ιf (s) B(ψl,t ) k
k
k
l = B(δl,h−3 (k + 1)θh−3,t −s ) − (t + 1)h − 1 − ι (s) (ζl,t ) 2 fk h−3 l = δl,h−3 (k + 1) 1 + + (t − s)h zh−3,t −s − δl,h−3 (k + 1) (t + 1)h − 1 − zh−3,t −s 2
2
= −δl,h−3 (k + 1)(1 + sh)zh−3,t −s , Lf (s) (ζl,t ) = B(ιf (s) (ζl,t )) = B(kδl,h−3 zh−3,t −s ) = 0, k
k
Lf (s) (hl,t ) = B(ιf (s) (hl,t )) − ιf (s) B(hl,t ) k
k
k
= B(δk,l ψ0,t −s ) − (t + 1)hιf (s) α −1 (hl,t ) k
= δkl ((t − s + 1)h − 1)ζ0,t −s − δkl (t + 1)hζ0,t −s = −δkl (sh + 1)ζ0,t −s . HH 3+6s (A)-Lie derivatives: We compute Lh(s) : k
(t )
(t )
Lh(s) (θl ) = B(ιh(s) (θl )) + ιh(s) B(θl,t ) =
| k {z }
k
1+
k
l 2
+ th ιh(s) zl,t k
=0
= δl0 (1 + th)hk,t −s−1 , Lh(s) (zl,t ) = B(δl0 hk,t −s−1 ) = δl0 (t − s)hα −1 (hk,t −s−1 ), k
Lh(s) (ωl,t ) = B ιh(s) (ωl,t ) + k
| k {z } =0
ιh(s) B(ωl,t ) | k {z }
= 0,
cup product in HH 3 (A)×HH 5 (A)
Lh(s) (εl,t ) = B ιh(s) εl,t = 0, k
| k{z } =0
(s)
Lh(s) (ψl,t ) = B(ιh(s) (ψl,t )) + hk B(ψl,t ) k
k
| {z } =0
=
(t + 1)h − 1 −
l 2
(s)
hk ζl,t = δk, h−3 δl,h−3 2
Lh(s) (ζl,t ) = B(ιh(s) (ζl,t )) = δk, h−3 B(δl,h−3 θh−3,t −s ) k
2
k
= δl,h−3 (t − s)h +
h−1 2
zh−3,t −s ,
th +
h+1 2
θh−3,t −s ,
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
Lh(s) (hl,t ) = B(ιh(s) (hl,t )) + ιh(s) B(hl,t ) = (t + 1)hιh(s) α −1 (hl,t ) k
| k {z }
k
k
=0
= (t + 1)h(Mα−1 )lk ψ0,t −s Lh(s) (fl,t ) = B(ιh(s) (fl,t )) = B(δkl ψ0,t −s ) = δkl ((t − s + 1)h − 1)ζ0,t −s . k
HH
k
(A)-Lie derivatives: We compute Lζ (s) : 4+6s
k
Lζ (s) (θl,t ) = Bιζ (s) (θl,t ) − ιζ (s) B(θl,t ) = B((zl ψk )t −s−1 ) − ιζ (s) k
k
= =
k
1+
k
k−l
l
l 2
+ th zl,t
(t − s)h − 1 − (zl ζk )t −s−1 − 1 + + th (zl ζk )t −s−1 2 2 k (zl ζk )t −s−1 , −sh − 2 − 2
Lζ (s) (ωl,t ) = 0, k
Lζ (s) (zl,t ) = Bιζ (s) (zl,t ) − ιζ (s) B(zl,t ) k
k
k
| {z } =0
= B((zl ζk )t −s−1 ) = 0, Lζ (s) (ψl,t ) = Bιζ (s) (ψl,t ) − ιζ (s) B(ψl,t ) k
k
k
l = δk,h−3 δl,h−3 B(α(f h−3 ,t −s−1 )) − ιζ (s) (t + 1)h − 1 − ζ l ,t 2 k 2 h−3 = δk,h−3 δl,h−3 (t − s)hf h−3 ,t −s−1 − (t + 1)h − 1 − f h−3 ,t −s−1 2 2 2 h+1 = δk,h−3 δl,h−3 −sh − f h−3 ,t −s−1 , 2
2
− ιζ (s) B(εl,t ) = 0, k | {z } =0 | k {z }=0 Lζ (s) (ζl,t ) = Bιζ (s) (ζl,t ) − ιζ (s) B(ζl,t ) k k k | {z } Lζ (s) (εl,t ) = Bιζ (s) (εl,t ) k
=0
= δkh−3 δl,h−3 B(f h−3 ,t −s−1 ) = 0, 2
Lζ (s) (hl,t ) = Bιζ (s) (hl,t ) − ιζ (s) B(hl,t ) k
k
k
= δl, h−3 δk,h−3 B(θh−3,t −s ) − (t + 1)hιζ (s) α −1 (hl,t ), 2 k h−3 = δl, h−3 δk,h−3 zh−3,t −s 1+ + (t − s)h − (t + 1)h 2 2 h+1 = δl, h−3 δk,h−3 zh−3,t −s − − sh , 2
2
Lζ (s) (fl,t ) = Bιζ (s) (fl,t ) − ιζ (s) B(fl,t ) k
k
k
| {z } =0
= (l + 1)δk,h−3 B(zh−3,t −s ) = 0. HH 5+6s (A)-Lie derivatives: We compute Lε(s) : k
Lε(s) (θl,t ) = B(ιε(s) (θl,t )) + ιε(s) B(θl,t ) k
k
k
l = B(−δl0 β(εk,t −s−1 )) + 1 + + th ιε(s) (zl,t ) k 2 1 = −δl0 + t − s − 1 hεk,t −s−1 + (1 + th)δl0 εk,t −s−1 2
43
44
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
=
s+
1
2
h + 1 δl0 εk,t −s−1 ,
Lε(s) (zl,t ) = B(ιε(s) (zl,t )) = B(εk,t −s−1 ) = 0, k
k
Lε(s) (ωl,t ) = B(ιε(s) (ωl,t )) + ιε(s) B(ωl,t ) k
k
k
+ t hιε(s) β −1 (ωl,t ) k 2 1 = δkl ((t − s)h − 1)ζ0,t −s−1 − δkl + t hζ0,t −s−1 2 1 = −δkl 1 + + s h ζ0,t −s−1 , = B(δkl ψ0,t −s−1 ) +
1
2
Lε(s) (ψl,t ) = B(ιε(s) (ψl,t )) + ιε(s) B(ψl,t ) k
| k {z }
k
=0
=
(t + 1)h − 1 −
l 2
ιε(s) ζl,t = 0, k
Lε(s) (εl,t ) = B(ιε(s) (εl,t )) = B(−(Mβ )kl ζ0,t −s−1 ) = 0, k
k
Lε(s) (ζl,t ) = B(ιε(s) ζl,t ) = 0, k
| k{z } =0
Lε(s) (hl,t ) = B(ιε(s) (hl,t )) + (t + 1)hιε(s) α −1 (hl,t ) = 0, k
k
k
Lε(s) (fl,t ) = B(ιε(s) (fl,t )) = 0. k
| k {z } =0
We compute Lψ (s) : k
Lψ (s) (θl,t ) = B ιψ (s) (θl,t ) +ιψ (s) B(θl,t ) k
| k {z }
k
=0
= ιψ (s) zl,t 1 + k
l
+ th = (zl ψk )t −s−1 1 +
2
l 2
+ th ,
Lψ (s) (zl,t ) = Bιψ (s) (zl,t ) + ιψ (s) B(zl,t ) k
k
k
| {z } =0
k−l = B((zl ψk )t −s−1 ) = (t − s)h − 1 − (zl ζk )t −s−1 , 2
Lψ (s) (ωl,t ) = B ιψ (s) (ωl,t ) +ιψ (s) B(ωl,t ) k
k
| k {z } =0
=
1 2
+ t hιψ (s) β −1 (ωl,t ) = 0, k
Lψ (s) (ψl,t ) = B ιψ (s) (ψl,t ) +ιψ (s) B(ψl,t ) k
| k {z }
k
=0
l (t + 1)h − 1 − ι (s) ζl,t 2 ψk h−3 α(f h−3 ,t −s−1 ), = δk,h−3 δl,h−3 (t + 1)h − 1 − 2 2 } | {z =
1 =th+ h+ 2
Lψ (s) (εl,t ) = B ιψ (s) (εl,t ) +ιψ (s) B(εl,t ) = 0 k
| k {z }
k
| {z } =0
=0
Lψ (s) (ζl,t ) = Bιψ (s) (ζl,t ) + ιψ (s) B(ζl,t ) k
k
k
| {z } =0
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
45
= δk,h−3 δl,h−3 B(α(f h−3 ,t −s−1 )) 2
= δk,h−3 δl,h−3 (t − s)hf h−3 ,t −s−1 , 2
Lψ (s) (hl,t ) = B ιψ (s) (hl,t ) +ιψ (s) B(hl,t ) = ιψ (s) α −1 (hl,t )(t + 1)h
| k {z }
k
k
k
=0
= δk,h−3 δl, h−3 (t + 1)hθh−3,t −s , 2
Lψ (s) (fl,t ) = Bψ (s) (fl,t ) + ιψ (s) B(fl,t ) k
k
k
| {z } =0
= (l + 1)δk,h−3 B(θh−3,t −s ) h−3 δk,h−3 zh−3,t −s . = (l + 1) 1 + (t − s)h + 2
HH
(A)-Lie derivatives: B acts on θl,t , ωl,t , ψl,t and hl,t by multiplication with 6+6s
degree and taking the preimage under ιθ0 . On zl,t , εl,t , ζl,t and fl,t , (s) B acts by zero. Since the spaces U, U , K , K , Y and Y are zk -invariant and zk has degree k − 2sh, Lz (s) acts on θl,t , ωl,t , ψl,t ∗
∗
1 2
∗
k
(s)
and hl,t by multiplication with 2k − sh and taking the preimage under ιθ0 and multiplication with zk , and on zl,t , εl,t , ζl,t and fl,t it acts by zero. We have the following formulas:
Lz (s) (θl,t ) =
k 2
k
− sh (zk θl )t −s ,
Lz (s) (zl,t ) = 0, k
Lz (s) (ωl,t ) = −δk0 shβ −1 (ωl,t −s ), k
Lz (s) (ψl,t ) =
k 2
k
− sh (zk ζl )t −s ,
Lz (s) (εl,t ) = 0, k
Lz (s) (ζl,t ) = 0, k
Lz (s) (hl,t ) =
k 2
k
− sh α −1 (hl,t −s ),
Lz (s) (fl,t ) = 0. k
Now we compute Lω(s) : k
We observe that ιω(s) (εl,t ) = δkl ψ0,t −s , ιω(s) (zl,t ) = δl0 ωk,t −s , and k
k
ιω(s) (θl,t ) = ιω(s) (ωl,t ) = ιω(s) (ψl,t ) = ιω(s) (ζl,t ) = ιω(s) (hl,t ) = ιω(s) (fl,t ) = 0. k
k
k
k
k
k
Then we have
Lω(s) (εl,t ) = Bιω(s) (εl,t ) = δkl B(ψ0,t −s ) k
k
= δkl ((t − s + 1)h − 1)ζ0,t −s , Lω(s) (zl,t ) = Bιω(s) (zl,t ) = δl0 B(ωk,t −s ) k k 1 = δl0 + t − s hβ −1 (ωk,t −s ), 2 l Lω(s) (θl,t ) = ιω(s) B(θl,t ) = 1 + + th ιω(s) zl,t k
2
k
k
= δl0 (1 + th)ωk,t −s , and
Lω(s) (ωl,t ) = Lω(s) (ψl,t ) = Lω(s) (ζl,t ) = Lω(s) (hl,t ) = Lω(s) (fl,t ) = 0. k
k
k
k
k
Acknowledgements C. Eu wants to thank his advisor P. Etingof for useful discussions and V. Dolgushev for his explanations about calculus.
46
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
References [1] [2] [3] [4] [5] [6] [7] [8]
Yu. Daletski, I. Gelfand, B. Tsygan, On a variant of noncommutative geometry, Soviet Math. Dokl. 40 (2) (1990) 422–426. W. Crawley-Boevey, P. Etingof, V. Ginzburg, Noncommutative geometry and quiver algebras, arXiv:math.AG/0502301. P. Etingof, C. Eu, Hochschild and cyclic homology of preprojective algebras of ADE quivers, arxiv:math.AG/0609006. K. Erdmann, N. Snashall, On Hochschild cohomology of preprojective algebras. I, II, J. Algebra 205 (2) (1998) 391–412. 413–434. C. Eu, The product in the Hochschild cohomology ring of preprojective algebras of Dynkin quivers, arxiv:math/0703568v1. A. Schofield, Wild algebras with periodic Auslander–Reiten translate, preprint. A. Malkin, V. Ostrik, M. Vybornov, Quiver varieties and Lusztig’s algebra, arXiv:math.AG/0403222. J. Cuntz, G. Skandalis, B. Tsygan, Cyclic homology in non-commutative geometry, in: Operator Algebras and Non-Commutative Geometry, II, in: Encyclopedia of Mathematical Sciences, vol. 121, Springer Verlag, Berlin, 2004. [9] V.A. Dolgushev, A proof of Tsygan’s formality conjecture for an arbitrary smooth manifold, Ph.D. Thesis, MIT. arxiv:math.QA/0504420. [10] J.-L. Loday, Cyclic homology, in: Grundreihen der mathematischen Wissenschaften, Vol. 301, in: A Series of Comprehensive Studies in Mathematics, Springer Verlag, 1992. [11] V. Ginzburg, Calabi–Yau algebras, arxiv:math/0612139v3.
Contents lists available at ScienceDirect
Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa
The calculus structure of the Hochschild homology/cohomology of preprojective algebras of Dynkin quivers Ching-Hwa Eu Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
article
info
Article history: Received 28 September 2007 Received in revised form 1 April 2009 Available online 12 August 2009 Communicated by E.M. Friedlander MSC: Primary: 16E40 secondary: 16G20
abstract The Hochschild cohomology ring of any associative algebra, together with the Hochschild homology, forms a structure of calculus. This was proved in Daletski et al. (1990) [1]. In this paper, we compute the calculus structure for the preprojective algebras of Dynkin quivers over a field of characteristic zero, using the Batalin–Vilkovisky structure of the Hochschild cohomology. Together with the results of Crawley-Boevey et al. [2], where the Batalin–Vilkovisky structure is computed for non-ADE quivers (and the calculus can be easily computed from that), this work gives us a complete description of the calculus for any quiver. © 2009 Published by Elsevier B.V.
1. Introduction The Hochschild homology and cohomology spaces and the duality between them were established in [3]. The cup product structure of the Hochschild cohomology was computed in [4] for the quivers of type A and [5] for those of types D and E. We use the notations from these papers and give the calculus structure in terms of the bases which were defined there. First, we compute the Connes differential on Hochschild homology by using the Cartan identity. Since it turns out that this differential makes the Hochschild cohomology ring a Batalin–Vilkovisky algebra, this gives us an easy way to compute the Gerstenhaber bracket and the contraction map. Then we use the Cartan identity to compute the Lie derivative. 2. Preliminaries 2.1. Quivers and path algebras Let Q be a quiver of ADE type with vertex set I and |I | = r. We write a ∈ Q to say that a is an arrow in Q . We define Q ∗ to be the quiver obtained from Q by reversing all of its arrows. We call Q¯ = Q ∪ Q ∗ the double of Q . Let C be the adjacency matrix corresponding to the quiver Q¯ . The concatenation of arrows generates the nontrivial paths inside the quiver Q¯ . We define ei , i ∈ I to be the trivial path which starts and ends at i. The path algebra PQ¯ = CQ¯ of Q¯ over C is the C-algebra with the paths in Q¯ as the bases and the product xy of two paths x and y to be their concatenation if they are compatible and 0 if not. We define the Lie bracket [x, y] = xy − yx.
E-mail addresses: [email protected], [email protected]. 0022-4049/$ – see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.jpaa.2009.04.018
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
29
Let R = ⊕i∈I Cei . Then R is a commutative semisimple algebra, and PQ is naturally an R-bimodule. Notation 2.1.1. Throughout this paper, we write ‘‘⊗ = ⊗R ’’. 2.2. Frobenius algebras Let A be a finite-dimensional unital C-algebra. We call it Frobenius if there is a linear function f : A → C, such that the '
form (x, y) := f (xy) is nondegenerate, or, equivalently, if there exists an isomorphism φ : A → A∗ of left A-modules: given f , we can define φ(a)(b) = f (ba), and given φ , we define f = φ(1). If f˜ is another linear function satisfying the same properties as f from above, then f˜ (x) = f (xa) for some invertible a ∈ A. Indeed, we define the form {a, b} = f˜ (ab). Then {−, 1} ∈ A∗ , so there is an a ∈ A, such that φ(a) = {−, 1}. Then f˜ (x) = {x, 1} = φ(a)(x) = f (xa). 2.3. The Nakayama automorphism Given a Frobenius algebra A (with a function f inducing a bilinear form (−, −) from above), the automorphism η : A → A defined by the equation (x, y) = (y, η(x)) is called the Nakayama automorphism (corresponding to f ). 2.4. The preprojective algebra
P Given a∗ quiver Q , we define the preprojective algebra ΠQ to be the quotient of the path algebra PQ¯ by the relation a∈Q [a, a ] = 0. Given a path x, we write x∗ for the path obtained from x by reversing all arrows. From now on, we write A = ΠQ .
2.5. Graded spaces Let M = ⊕d≥0 M (d) be a Z+ -graded vector space, with finite-dimensional homogeneous subspaces. We denote by M [n] the same space with grading shifted by n. The graded dual space M ∗ is defined by the formula M ∗ (n) = M (−n)∗ . 2.6. Root system parameters Let w0 be the longest element of the Weyl group W of Q . Then we define ν to be the involution of I, such that w0 (αi ) = −αν(i) (where αi is the simple root corresponding to i ∈ I). It turns out that η(ei ) = eν(i) ([6]; see [4]). Let mi , i = 1, . . . , r, be the exponents of the root system attached to Q , enumerated in the increasing order. Let h = mr + 1 be the Coxeter number of Q . Let P be the permutation matrix corresponding to the involution ν . Let r+ = dim ker(P − 1) and r− = dim ker(P + 1). Thus, r− is half the number of vertices which are not fixed by ν , and r+ = r − r− . A is finite-dimensional, and the following Hilbert series is known from [7, Theorem 2.3.]: HA (t ) = (1 + Pt h )(1 − Ct + t 2 )−1 .
(2.6.1) top
We see that the top degree of A is h − 2, and for the top degree A dimensional submodules: Atop = A(h − 2) =
M
part we get the following decomposition in one-
ei A(h − 2)eν(i) .
i∈I
3. Hochschild cohomology and homology The Hochschild cohomology and homology spaces of A were computed in [3]. We recall the results: Definition 3.0.3. We define the spaces U = ⊕d
(2.6.2)
30
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
Theorem 3.0.4. For the Hochschild cohomology spaces, we have the following natural isomorphisms: HH 0 (A) = U [−2] ⊕ L[h − 2], HH 1 (A) = U [−2], HH 2 (A) = K [−2], HH 3 (A) = K ∗ [−2], HH 4 (A) = U ∗ [−2], HH 5 (A) = U ∗ [−2] ⊕ Y ∗ [−h − 2], HH 6 (A) = U [−2h − 2] ⊕ Y [−h − 2], and HH 6n+i (A) = HH i (A)[−2nh] ∀i ≥ 1. Theorem 3.0.5. The Hochschild homology spaces of A, as graded spaces, are as follows: HH0 (A) = R, HH1 (A) = U , HH2 (A) = U ⊕ Y [h], HH3 (A) = U ∗ [2h] ⊕ Y ∗ [h], HH4 (A) = U ∗ [2h]. HH5 (A) = K [2h], HH6 (A) = K [2h], and HH6n+i (A) = HHi (A)[2nh] ∀i ≥ 1. In [3], an isomorphism HH• (A) = HH 8−• (A)[2h + 2] was introduced. However, because of the periodicity of the Schofield resolution (with period 6), we get for every m ≥ 0 an isomorphism ∼
D : HH• (A) → HH 6m+2−• (A)[2mh + 2].
(3.0.6)
Notation 3.0.7. In [5], the basis elements zk ∈ U [−2] ⊂ HH 0 (A), ωk ∈ L[h − 2], θk ∈ U [−2] = HH 1 (A), fk ∈ K [−2] = HH 2 (A), hk ∈ K ∗ [−2], ζk ∈ U ∗ [−2] = HH 4 (A), ψk ∈ U ∗ [−2] ⊂ HH 5 (A), and εk ∈ Y ∗ [−h − 2] were introduced. Our notations are taken from [5] and are different from those in [4]: zk in [4] corresponds to z2k in our notations, zm corresponds to ω h−3 , 2
gk corresponds to θ2k ,
ψk corresponds to ζl−k and ζk corresponds to θl−k for l = (s)
For ck ∈ HH i (A), 0 ≤ i ≤ 5, we write ck (s)
n
h−2 h−3
Q = An , n odd, Q = An , n ev en
for the corresponding cocycle in HH i+6s . We write ck,t for a cycle in HHj+6t ,
0 ≤ j ≤ 5 which equals D−1 (ck ). We also introduced the maps α : K → K ∗ and β : Y ∗ → Y there. 4. The calculus structure of the preprojective algebra We recall the definition of the calculus. 4.1. Definition of calculus Definition 4.1.1 (Gerstenhaber Algebra). A graded vector space V • is a Gerstenhaber algebra if it is equipped with a graded commutative and associative product ∧ of degree 0 and a graded Lie bracket (the ‘‘Gerstenhaber bracket’’) [, ] of degree −1. These operations have to be compatible in the sense of the following Leibniz rule
[γ , γ1 ∧ γ2 ] = [γ , γ1 ] ∧ γ2 + (−1)k1 (k+1) γ1 ∧ [γ , γ2 ],
(4.1.2)
where γ ∈ V k and γ1 ∈ V k1 . We recall from [8] that Definition 4.1.3 (Precalculus). A precalculus is a pair of a Gerstenhaber algebra (V • , ∧, [, ]) and a graded vector space W • together with
• a module structure ι• : V • ⊗ W −• → W −• of the graded commutative algebra V • on W −• .
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
31
Table 1 Contraction map ιa (b). a
b
(s)
zk
ωk(s) θk(s) ( s)
fk
(s)
θ l, t
ω l, t
zl,t
ψl,t
εl,t
ζl,t
hl , t
fl,t
(zk θl )t −s
δk0 ωl,t −s
(zk ψl )t −s
δk0 hl,t −s
δk0 fl,t −s
0
0
0
0
0
0
δk0 εl,t −s δkl ψ0,t −s δk0 β(εl,t −s )
(zk ζl )t −s
0
(zk zl )t −s δl0 ωk,t −s (zl θk )t −s
0
0
δk0 α(fl,t )
δl0 α(fk,t −s−1 )
0
δl0 fk,t −s−1
δkl ψ0,t −s
(Mα )kl ζ0,t −s
0
δkl ψ0,t −s
f h−3 ,t −s−1
δk,h−3 δl, h−3 · 2 θk,t −s
δk,h−3 (l + 1)·
0
0
0
0
δk,h−3 (l + 1)· θh−3,t −s
0
(zk ψl )t −s δl,h−3 (k + 1)· θh−3,t −s−1
hk
0
0
δl0 hk,t −s−1
0
ζk(s)
(zl ψk )t −s−1
0
(zl ζk )t −s−1
δk,h−3 δl,h−3 · α(f h−3 ,t −s−1 )
0
2
(s)
εk
−δl0 β(εk,t −s )
(s)
ψk
0
δkl ψ0,t −s−1
δl,0 εk,t −s−1
0
(zk ψl )t −s−1
0
0
0
0
−(Mβ )k,l · ζ0,t −s−1 0
δl,h−3 (k + 1)· zl,t −s
δk, h−3 δl,h−3 · 2 θh−3,t −s δk,h−3 δl,h−3 · 2
δk,h−3 δl,h−3 · α(fh−3,t −s−1 )
zk,t −s
• an action L• : V •+1 ⊗ W −• → W −• of the graded Lie algebra V •+1 on W −• which are compatible in the sense of the following equations
ιa Lb − (−1)|a|(|b|+1) Lb ιa = ι[a,b] ,
(4.1.4)
La∧b = La ιb + (−1)|a| ιa Lb .
(4.1.5)
and
Definition 4.1.6 (Calculus). A calculus is a precalculus (V • , W • , [, ], ∧, ι• , L• ) with a degree 1 differential d on W • such that the Cartan identity,
La = dιa − (−1)|a| ιa d,
(4.1.7)
holds. Let A be an associative algebra. The contraction of the Hochschild cochain P ∈ C k (A, A) with the Hochschild chain (a0 , a1 , . . . , an ) is defined by IP (a0 , a1 , . . . , an ) =
(a0 P (a1 , . . . , ak ), ak+1 , . . . , an ) 0
n ≥ k, else.
(4.1.8)
We have Proposition 4.1.9 (Daletski et al. [1]). The contraction IP together with the Connes differential, the Gerstenhaber bracket, the cup product and the action of cochains on chains [9, (3.5), page 46] induce on the pair (HH • (A, A), HH• (A, A)) a structure of calculus. 5. Results about the calculus structure of the Hochschild cohomology/homology of preprojective algebras of Dynkin quivers We state the results in terms of the bases of HH • (A) and HH• (A) which were introduced in Notation 3.0.7: Theorem 5.0.10. The calculus structure is given by Tables 1–3 and the Connes differential B (see [10, 2.1.7]), given as follows The Connes differential B is given as follows:
+ sh zk,s , 2 1 B2+6s (ωk,s ) = + s hβ −1 (ωk,s ), B1+6s (θk,s ) =
1+
k
2
B2+6s (zk,s ) = 0, B3+6s (ψk,s ) =
(s + 1)h − 1 −
k 2
B3+6s (εk,s ) = 0, B4+6s = 0, B5+6s (hk,s ) = (s + 1)hα −1 (hk,s ), B6+6s = 0.
ζk,s ,
32
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
Table 2 Gerstenhaber bracket [a, b]. a
b
( s)
ωk(s)
ζl(t )
εl(t )
ψl(t )
0
−δk0 sh· α −1 (h(l s+t ) )
0
0
(
0
0
0
(−1 + (s − t )h)· δk0 h(l s+t )
−(2 +
ωl(t )
θl(t )
fl
0
−δk0 sh· β −1 (ωl(s+t ) )
(
k − sh)· 2 (zk zl )(s+t )
0
0
(t )
zk
hl
zl
(
θk(s)
(t )
(t )
l−k 2
+ (s − t )h)· −(1 + th)· δk0 fl(s+t ) (z θ )(s+t ) k l
(s)
fk
−(1 + sh)· δkl ζ0(s+t )
0
hk
l 2
(zk ζl )
h 2
+ 1 + th)·
δkl ζ0(s+t )
+ th)·
−(1 + th + (s+t )
(s+t )
0
δk0 εl
h 2
0 h+1
(s − t )h· (Mα−1 )kl · ψ0(s+t )
(s)
−(
k − sh)· 2 (zk ζl )(s+t )
+ th)· −( 2 δk, h−3 δl,h−3 ·
0
0
−(2 +
k+l 2
+ (t − s)h)·
(zk ψl )(s+t ) −(k + 1)· (1 + sh)·
δl,h−3 zh(s−+3t +1)
h−1 ((s − t )h − )· 2 δk, h−3 δl,h−3 · 2
2
(s+t +1)
zh−3
ζk(s)
)·
0
0
s+t +1) θh(− 3 h+1 −(sh + )· 2 δk,h−3 δl,h−3 ·
(s+t +1)
f h−3 2
εk(s)
0
ψk(s)
0
(s − t )h· δk,h−3 δl,h−3 · α(f h(s−+3t +1) ) 2
6. Batalin–Vilkovisky structure on Hochschild cohomology Recall the isomorphism (3.0.6). It translates the Connes differential B : HH• (A) → HH•+1 (A) on Hochschild homology into a differential ∆ : HH • (A) → HH •−1 (A) on Hochschild cohomology, i.e. we have the commutative diagram B
HH• (A)
HH•+1 (A)
−−−−→
Dy∼
∼ yD ∆
HH 6m+2−• (A)[2mh + 2] −−−−→ HH6m+1−• (A)[2mh + 2] Theorem 6.0.11 (BV Structure on Hochschild Cohomology). Gerstenhaber bracket we get the following equation:
[a, b] = ∆(a ∪ b) − ∆(a) ∪ b − (−1)|a| a ∪ ∆(b),
∆ makes HH • (A) a Batalin–Vilkovisky algebra, i.e. for the
∀a, b ∈ HH ∗ (A).
(6.0.12)
The isomorphism D intertwines contraction and cup product maps, i.e. we have
D(ιη c ) = η ∪ D(c ),
∀c ∈ HH• (A), η ∈ HH • (A).
(6.0.13)
Remark 6.0.14. Note that ∆ in Eq. (6.0.12) depends on which m HH 6m+2−• (A)[2mh + 2], where the Gerstenhaber bracket does not.
∼
∈ N we choose to identify D : HH• (A) →
Proof. We apply the functor HomAe (−, A ⊗C A) : Ae − mod M
→ 7→
Ae − mod, M∨
on the Schofield resolution: d∨ 1
d∨ 3
d∨ 2
d∨ 4
(A ⊗ A)∨ → (A ⊗ V ⊗ A)∨ → (A ⊗ A[2])∨ → (A ⊗ N [h])∨ → d∨ 4
d∨ 5
d∨ 6
d∨ 7
→ (A ⊗ V ⊗ N [h])∨ → (A ⊗ N [h + 2])∨ → (A ⊗ A[2h])∨ → . . . . An element in (A ⊗ A)∨ or (A ⊗ N )∨ is determined by the image of 1 ⊗ 1, An element in (A ⊗ V ⊗ A)∨ or (A ⊗ V ⊗ N )∨ by the images of 1 ⊗ a ⊗ 1 for all arrows a ∈ Q¯ .
(6.0.15)
ωk(s)
zk
( s)
ψk(s)
εk
( s)
ζk(s)
hk
(s)
fk
(s)
θk(s)
a
l
1
−((s + )h + 1)· 2 δkl ζ0,t −s−1
((s + )h + 1)· 2 δl0 εk,t −s−1
0
−δk0 sh β −1 (ωl,t −s )
0
(1 + + th)· 2 (zl ψk )t −s−1
(k − sh) (zk θl )t −s
(1 + th)· δl0 ωk,t −s
l
1
0
0
0
( + t )h· 2 δk0 ωl,t −s
1
ωl,t
−(2 + + sh)· 2 (zl ζk )t −s−1
k
(1 + th)· δl0 hk,t −s−1
(1 + + th)· 2 (zk θl )t −s −(1 + sh)· δl0 fk,t −s−1
θl,t
b
Table 3 Lie derivative La (b).
1 2
+ t − s )h·
β −1 (ωk,t −s )
δl0 (
0
((t − s)h − 1 − )· 2 (zl ζk )t −s−1
0
0
k−l
+ (t − s)h)· 2 (zk zl )t −s
k+l
δl0 (t − s)h· α −1 (hk,t −s−1 )
0
(1 +
zl,t l
0
( − sh)· 2 (zk ζl )t −s
2
+ (t − s))h· δk0 εl,t −s
δkl · (−1 + h + (t − s)h)· ζ0,t −s
0
0
h+1 (th + )· 2 δk,h−3 δl,h−3 · α(f h−3 ,t −s−1 ) k
0
0
0
0
(
1
εl,t
0
2
2
f h−3 ,t −s−1
((t + 1)h − 1 − )· 2 (zk ψl )t −s −(1 + sh)· δl,h−3 zh−3,t −s h+1 )· (th + 2 δk, h−3 δl,h−3 · 2 θh−3,t −s h+1 )· −(sh + 2 δk,h−3 δl,h−3 ·
ψl,t
2
0
0
2
f h−3 ,t −s−1
(t − s)h· δk,h−3 δl,h−3
0
0
zh−3,t −s
h−1 ((t − s)h + )· 2 δk, h−3 δl,h−3 ·
0
((t − s + 1)h − 1 − (zk ζl )t −s
ζl,t 2
l−k
)
2
0
(k − sh)· α −1 (hl,t −s )
(t + 1)h· δk,h−3 δl,h−3 · θh−3,t −s
0
zh−3,t −s
h+1 )· −(sh + 2 δk,h−3 δl, h−3 ·
(l + 1)·
0
0
zh−3,t −s
((t − s)h + 1 + δk,h−3
0
0
2
h−3
((t − s + 1)h − 1)· δkl ζ0,t −s
0
−(1 + sh)· δkl ζ0,t −s (t + 1 )h· (Mα−1 )lk · ψ0,t −s
(t − s + 1)h· δk0 fl,t −s
fl,t
(t + 1 )h· δk0 hl,t −s
hl,t
)·
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46 33
34
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
n
+1
Q = A,
Let us define σ = −1 Q = D, E . We make the following identifications:
(A ⊗ A)[−2mh] = (A ⊗ A[2mh])∨ : we identify x ⊗ y with thePmap that sends 1 ⊗ 1 to σ m y ⊗ x, (A ⊗ V ⊗ A)[−2mh − 2] = (A ⊗ V ⊗ A[2mh])∨ : we identify a∈Q¯ a xa ⊗ a∗ ⊗ ya with the map that sends 1 ⊗ a ⊗ 1 to −σ m ya ⊗ xa , (A ⊗ A)[−2mh − 2] = (A ⊗ A[2mh + 2])∨ : we identify x ⊗ y with the map that sends 1 ⊗ 1 to −σ m y ⊗ x, (A ⊗ N )[−(2m + 1)h] = (A ⊗ N [(2m + 1)h])∨ : we identify x ⊗ y with the P map that sends 1 ⊗ 1 to −σ m η(y) ⊗ x, ∨ (A ⊗ V ⊗ N )[−(2m + 1)h − 2] = (A ⊗ V ⊗ N [(2m + 1)h]) : we identify a∈Q¯ a xa ⊗ η(a∗ ) ⊗ ya with the map that sends 1 ⊗ a ⊗ 1 to σ m+1 η(ya ) ⊗ xa , (A ⊗ N )[−2(m + 1)h − 2] = (A ⊗ A[2(m + 1)h + 2])∨ : we identify x ⊗ y with the map that sends 1 ⊗ 1 to σ m+1 η(y) ⊗ x, so (6.0.15) becomes d∨ 1
d∨ 3
d∨ 2
d∨ 4
(A ⊗ A) → (A ⊗ V ⊗ A) → (A ⊗ A[−2]) → (A ⊗ N [−h]) → d∨ 6
d∨ 5
d∨ 4
d∨ 7
→ (A ⊗ V ⊗ N [−h − 2]) → (A ⊗ N [−h − 2]) → (A ⊗ A[−2h]) → . . . .
(6.0.16)
∨
We show under the identification from above, the differentials di corresponds to the differentials from the Schofield resolution, i.e. (6.0.16) can be rewritten in this form: d2 [−2]
d1 [−2]
(A ⊗ A) →
(A ⊗ V ⊗ A) → (A ⊗ A[−2])
d6 [−2h−2]
(A ⊗ N [−h])
→
d4 [−2h−2]
d5 [−2h−2]
→
→
(A ⊗ V ⊗ N [−h − 2])
d3 [−2h−2]
(A ⊗ N [−h − 2])
→
d6 [−2h−2]
→
(A ⊗ A[−2h])
d4 [−2h−2]
→
d2 [−2h−2]
→
....
(6.0.17)
It is enough to show this for the first period. d∨ 1 (x ⊗ y)(1 ⊗ a ⊗ 1) = (x ⊗ y) ◦ (a ⊗ 1 − 1 ⊗ a) = ay ⊗ x − y ⊗ xa, so d∨ 1 (x ⊗ y) =
X
a (xa ⊗ a∗ ⊗ y − x ⊗ a∗ ⊗ ay) =
a∈Q¯
d∨ 2
a (xa ⊗ a∗ ⊗ y + x ⊗ a ⊗ a∗ y) = d2 (x ⊗ y),
a∈Q¯
X
X
a xa ⊗ a∗ ⊗ ya (1 ⊗ 1) =
a∈Q¯
X
a xa ⊗ a∗ ⊗ ya ◦
X
¯
=
b b ⊗ b∗ ⊗ 1 + b 1 ⊗ b ⊗ b∗
¯
a∈Q b∈Q X (a a∗ ya ⊗ xa − a ya ⊗ xa a∗ ), a∈Q¯
so
d∨ 2
X
a xa ⊗ a∗ ⊗ ya =
a∈Q¯
X
a (xa a∗ ⊗ ya − xa ⊗ a∗ ya ) = d1 (a xa ⊗ a∗ ⊗ ya ),
a∈Q¯
! d3 (x ⊗ y)(1 ⊗ 1) = (x ⊗ y) ◦ ∨
X
∗
xi ⊗ xi
xi ∈B
=−
X
xi y ⊗ xx∗i = −
xi ∈B
X
η(x∗i )y ⊗ xxi ,
xi ∈B
so d∨ 3 (x ⊗ y) =
X
xxi ⊗ x∗i η(y) =
xi ∈B
X
xyxi ⊗ x∗i = d6 (x ⊗ y)
xi ∈B
d∨ 4 (x ⊗ y)(1 ⊗ a ⊗ 1) = (x ⊗ y) ◦ (a ⊗ 1 − 1 ⊗ a) = −aη(y) ⊗ x + η(y) ⊗ xη(a), so d∨ 4 (x ⊗ y) =
X
a σ (−x ⊗ η(a∗ ) ⊗ η(a)y + xη(a) ⊗ η(a∗ ) ⊗ y)
a∈Q¯
=
X (a xa ⊗ a∗ ⊗ y + a x ⊗ a ⊗ a∗ y) = d5 (x ⊗ y), a∈Q¯
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
d∨ 5
X
a xa ⊗ η(a∗ ) ⊗ ya (1 ⊗ a ⊗ 1) =
a∈Q¯
X
a xa ⊗ η(a∗ ) ⊗ ya ◦
X (b b ⊗ b∗ ⊗ 1 + b 1 ⊗ b ⊗ b∗ )
¯
= σ
a∈Q X
35
b∈Q¯
(−a η(ya ) ⊗ xa + a η(ya ) ⊗ xa η(a∗ )), ∗
a∈Q¯
so
d∨ 5
X
a xa ⊗ η(a∗ ) ⊗ ya =
a∈Q¯
X (−a xa ⊗ η(a∗ )ya + a xa η(a∗ ) ⊗ ya ) a∈Q¯
= d4
X
a xa ⊗ η(a∗ ) ⊗ ya ,
a∈Q¯
! X =σ xi η(y) ⊗ xη(x∗i ) xi ∈B xi ∈B X X x∗i ⊗ xyxi , x∗i η(y) ⊗ xxi = σ = σ
d6 (x ⊗ y)(1 ⊗ 1) = (x ⊗ y) ◦ ∨
X
∗
x⊗ xi
xi ∈B
xi ∈B
so d∨ 6 (x ⊗ y) =
X
xyxi ⊗ x∗i = d3 (x ⊗ y).
xi ∈B
Fix m ≥ 0. The map which shifts the degree by −2mh − 2 produces the following diagram which commutes by the computations above: d6m+2
A ⊗ A[2mh + 2] −−−−→
d6m+1
−−−−→ . . .
A ⊗ V ⊗ A[2h]
y
y d∨ 1
(A ⊗ A)
d∨ 2
−−−−→ (A ⊗ V ⊗ A)[−2] −−−−→ . . .
d2
. . . −−−−→
d1
A⊗V ⊗A
−−−−→
y
mult .
A⊗A
−−−−→ A
y
d∨ 6m+1
d∨ 6m+2
mult .
. . . −−−−→ (A ⊗ V ⊗ A)[−2mh − 2] −−−−→ (A ⊗ A)[−mh − 2] −−−−→ A Similarly to the proof of [11, Theorem 3.4.3.], this self-dual morphism of the Schofield resolution C • into the dual complex
(C • )∨ can be used to prove (6.0.13).
(6.0.12) follows, as in the proof of [11, Theorem 3.4.3.], from (6.0.13) and the calculus structure.
6.1. Computation of the calculus structure of the preprojective algebra Since the calculus structure is defined on Hochschild chains and cochains, we have to work with them on the resolution for computations. It turns out that we only have to compute Lθ0 directly, the rest can be deduced from formulas given by the calculus and the BV structure. d3
d2
d1
d0
· · · −−−−→ A ⊗ A[2] −−−−→ A ⊗ V ⊗ A −−−−→ A ⊗ A −−−−→
µ2 y µ1 y
b3
· · · −−−−→
A⊗4
b2
−−−−→
A⊗3
b1
A −−−−→ 0
b0
−−−−→ A⊗2 −−−−→ A −−−−→ 0
These maps ψi gives us a chain map between the Schofield and the bar resolution:
µ1 (1 ⊗ y ⊗ 1) = 1 ⊗ y ⊗ 1, X µ2 (1 ⊗ 1) = a 1 ⊗ a ⊗ a∗ ⊗ 1, a∈Q¯
µ3 (1 ⊗ 1) =
XX a∈Q¯ xi ∈B
a 1 ⊗ xi ⊗ a ⊗ a∗ ⊗ x∗i ,
36
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
and
µ3+i = µi
XX
xi ⊗ a ⊗ a∗ ⊗ x∗i .
a∈Q¯ xi ∈B
Now, we apply the functor − ⊗Ae A on the commutative diagram: d03
d02
d1
· · · −−−−→ AR [2] −−−−→ (V ⊗ A)R −−−−→ µ02 y µ01 y b03
b2
AR
−−−−→ 0
b1
· · · −−−−→ (A⊗3 )R −−−−→ (A⊗2 )R −−−−→ (A⊗1 )R −−−−→ 0 where
µ01 (x ⊗ y) = x ⊗ y, X µ02 (x) = a a ⊗ a∗ ⊗ x, a∈Q¯
µ3 ( x ) = 0
XX
a xi ⊗ a ⊗ a∗ ⊗ x∗i η(x),
a∈Q¯ xi ∈B
and
µ03+i = µ0i
XX
xi ⊗ a ⊗ a∗ ⊗ x∗i .
a∈Q¯ xi ∈B
Now, we compute Lθ0 : Lemma 6.1.1. For each x ∈ HHi (A),
Lθ0 (x) = x
deg(x) 2
.
(6.1.2)
Proof. Via µ0 , we already identified x ∈ HHi (A) with cycles in the Hochschild chain, but we still have to identify θ0 with an element in HomAe (A⊗3 , A): given any monomial b = b1 . . . bl , bi ∈ V , the map
τ (1 ⊗ b ⊗ 1) =
l X
b1 . . . bi−1 ⊗ bi ⊗ bi+1 . . . bl
i =1
makes the diagram d0
d1
A ⊗ V ⊗ A −−−−→ A ⊗ A −−−−→ A −−−−→ 0
x
τ
A⊗3
b0
b1
−−−−→ A⊗2 −−−−→ A −−−−→ 0
commute. Applying HomAe (__ ⊗ A), we get a map
τ ∗ : Homk (V ) → Homk (A), such that
(θ0 ◦ τ ∗ )(b1 . . . bl ) =
l X
b1 . . . bi−1 θ0 (bi )bi+1 . . . bl = s(b) · b,
i =1
where for b = b1 . . . bl , s(b) is the number of bi ∈ Q ∗ . Recall from [9, (3.5), page 46] that the Lie derivative of θ0 ◦ τ ∗ on Hochschild chains is defined by
Lθ0 ◦τ ∗ (a1 ⊗ · · · ⊗ ak ) =
k X
a1 ⊗ · · · ⊗ (θ0 ◦ τ ∗ )(ai ) ⊗ · · · ⊗ ak
i =1 k X = (s(a1 ) + · · · s(ak ))a1 ⊗ · · · ⊗ ak , i =1
and it can easily be checked that for each x ∈ HHi (A), Lθ0 ◦τ ∗ acts on µ0i (x), x ∈ HH i (A), by multiplication with
1 2
deg(x).
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
37
6.1.1. The contraction map From (6.0.13) we know that the contraction map on Hochschild homology is given by the cup product on Hochschild cohomology which was computed in [4,5]. Table 1 contains these results, rewritten in terms of the contraction maps. 6.1.2. The Connes differential We start with the computation of the Connes differential and recall the diagram from [3]: degree 0
y 2 ≤ deg ≤ h − 1
HH1 (A)
U
B1 y
∼y
HH2 (A)
2 ≤ deg ≤ h
U
B2 y h ≤ deg ≤ 2h − 2
HH3 (A)
U ∗ [2h]⊕Y ∗ [h]
B3 y
∼y
h + 1 ≤ deg ≤ 2h − 2 HH4 (A)
U ∗ [2h]
B4 y
y
0
HH5 (A)
2h
K ∗ [2h]
B5 y
∼y
HH6 (A)
2h
⊕ Y [h] ∼y
K [2h]
B6 y
y
0
2h + 2 ≤ deg ≤ 3h − 1 HH7 (A)
U [2h]
B7 y .. . Proposition 6.1.3. The Connes differential B is given as follows: (s)
k
+ sh zk(s) , 1 (s) B2+6s (ωk ) = + s hβ −1 (ωk(s) ), B1+6s (θk ) =
1+
2
2
(s)
B2+6s (zk ) = 0, (s)
B3+6s (ψk ) =
(s + 1)h − 1 −
k 2
ζk ,
(s)
B3+6s (εk ) = 0, B4+6s = 0, (s)
(s)
B5+6s (hk ) = (s + 1)hα −1 (hk ), B6+6s = 0. Proof. We use the Cartan identity (4.1.7) with a ∈ θ0 ,
Lθ0 = Bιθ0 + ιθ0 B,
(6.1.4)
where Lθ0 acts on x ∈ HHi by multiplication by (t )
(t )
(t )
(t )
1 2
deg(x) (see Lemma 6.1.1). The above identities for the Connes differential (t )
(t )
(t )
follow since ιθ0 acts on θk , ωk , ψk and hk by zero, and zk , β −1 (ωk ), ζk contraction with ιθ0 .
(s)
and α −1 (hk ) are their unique preimages the
38
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
6.1.3. The Gerstenhaber bracket We compute the brackets using the identification HH i (A) = HH6m+2−i (A)[−2mh − 2] for m 1 and the BV-identity (6.0.12). Brackets involving HH 6s (A): By degree argument these brackets are zero:
ωk(s) with HH 1+6t (A), HH 2+6t (A), HH 3+6t (A), HH 4+6t (A), and ψl ∈ HH 5+6t (A). From the BV-identity (6.0.12), we see (s) (t ) (t ) that brackets of zk with zl ∈ HH 6t (A), HH 2+6t (A), HH 4+6t (A) and εl ∈ HH 5+6t (A) are zero because ∆ acts by zero 6t 2+6t 4+6t on U [−2th − 2] ⊂ HH (A), HH (A) and HH (A). We compute the remaining brackets:
[zk(s) , ωl(t ) ] = ∆(zk(s) ∪ ωl(t ) ) − ∆(zk(s) ) ∪ωl(t ) − zk(s) ∪ ∆(ωl(t ) ) | {z } =0
= δk0 ∆(ωl ) − + (m − t )h zk(s) ∪ β −1 (ωl(t ) ) 2 h h + (m − s − t )h β −1 (ωl(s+t ) ) − + (m − t )h β −1 (ωl(s+t ) ) = δk0 (s+t )
h
2
2
= −δk0 shβ
−1
(s+t )
(ωl
),
[zk , θl ] = ∆(zk ∪ θl ) − ∆(zk(s) ) ∪θl(t ) − zk(s) ∪ ∆(θl(t ) ) | {z } (s)
(t )
(s)
(t )
=0
l = ∆((zk θl ) ) − 1 + + (m − t )h zk(s) zl(t ) 2 k+l l = 1+ + (m − s − t )h (zk zl )(s+t ) − 1 + + (m − t )h (zk zl )(s+t ) 2 2 k − sh (zk zl )(s+t ) , = (s+t )
2
[zk , hl ] = ∆(zk(s) ∪ h(l t ) ) − ∆(zk(s) ) ∪h(l t ) − zk(s) ∪ ∆(h(l t ) ) | {z } (s)
(t )
=0
(s+t )
= δk0 ∆(hl
) − (h + (m − t − 1)h)zk(s) ∪ α −1 (h(l t ) )
= δk0 (h + (m − s − t − 1)h)α −1 (h(l s+t ) ) − δk0 (h + (m − t − 1)h)α −1 (h(l s+t ) ) = −δk0 shα −1 (h(l s+t ) ), [zk(s) , ψl(t ) ] = ∆(zk(s) ∪ ψl(t ) ) − ∆(zk(s) ) ∪ψl(t ) − zk(s) ∪ ∆(ψl(t ) ) | {z } =0
t
(s) (t ) = ∆((zk ψl ) )− h−1− zk ζl 2 l−k l (s+t ) = (m − s − t )h − 1 − (zk ζl ) − (m − t )h − 1 − (zk ζl )(s+t ) 2 2 k = − sh (zk ζl )(s+t ) (s+t )
2
(s)
(t )
[ωk , εl ] = ∆(ωk(s) ∪ εl(t ) ) − ∆(ωk(s) ) ∪ εl(t ) − ωk(s) ∪ ∆εl(t ) | {z } =0
h
+ (m − s)h β −1 (ωk ) ∪ εl(t ) h = δkl (h − 1 + (m − s − t − 1)h)ζ0(s+t ) − δkl + (m − s)h ζ0 2 h = δkl − − 1 − th ζ0 . = ∆(δkl ψ0s+t ) −
2
2
(s)
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
Brackets involving HH 1+6s (A):
[θk(s) , θl(t ) ] = ∆(θk(s) ∪ θl(t ) ) − ∆(θk(s) ) ∪ θl(t ) + θk(s) ∪ ∆(θl(t ) ) | {z } =0
= − 1+ =
l−k 2
k 2
l + (m − s)h zk(s) θl(t ) + 1 + + (m − t )h θk(s) zl(t ) 2
+ (s − t )h (zk θl )(s+t )
[θk(s) , fl(t ) ] = ∆(θk(s) ∪ fl(t ) ) − ∆(θk(s) ) ∪ fl(t ) + θk(s) ∪ ∆(fl(t ) ) | {z } =0
= δk0 (∆(α(fl
(s+t )
)) − (1 + (m − s)h)fl
(s+t )
)
= δk0 (h + (m − s − t − 1)h)fl(s+t ) − (1 + (m − s)h)fl(s+t ) = −δk0 (1 + th)fl(s+t ) [θk(s) , h(l t ) ] = ∆(θk(s) ∪ h(l t ) ) − ∆(θk(s) ) ∪ h(l t ) + θk(s) ∪ ∆(h(l t ) ) | {z } =0
= − 1 + (m − s)h +
k
(s)
(t )
z k ∪ hl
2
+ (h + (m − t − 1)h)θk(s) ∪ α −1 (h(l t ) )
= −δk0 (1 + (m − s)h)h(l s+t ) + δk0 (m − t )hh(l s+t ) = δk0 (−1 + (s − t )h)h(l s+t ) [θk(s) , ζl(t ) ] = ∆(θk(s) ∪ ζl(t ) ) − ∆(θk(s) ) ∪ ζl(t ) + θk(s) ∪ ∆(ζl(t ) ) | {z } =0
= ∆((zk ψl )(s+t ) ) − 1 + =
h−1−
l−k 2
k 2
+ (m − s)h zk(s) ∪ ζl(t )
+ (m − s − t − 1)h (zk ζl )(s+t )
k − 1 + + (m − s)h (zk ζl )(s+t ) 2
= − 2+ (s)
(t )
l 2
+ th (zk ζl )(s+t )
(s)
[θk , ψl ] = ∆(θk ∪ ψl(t ) ) −∆(θk(s) ) ∪ ψl(t ) + θk(s) ∪ ∆(ψl(t ) ) {z } | =0
= − 1+ = − 2+
k 2
l + (m − s)h zk(s) ψl(t ) + h − 1 − + (m − t − 1)h θk(s) ζl(t ) 2
k+l 2
+ (t − s)h (zk ψl )(s+t ) ,
[θk(s) , εl(t ) ] = ∆(θk(s) ∪ εl(t ) ) − ∆(θk(s) ) ∪ εl(t ) + θk(s) ∪ ∆(εl(t ) ) | {z } =0
= δk0 ∆(β(εl(s+t ) )) − δk0 1 + (m − s)h + = δk0
h 2
k 2
(s) (t )
zk εl
+ (m − s − t − 1)h εl(s+t ) − (1 + (m − s)h)εl(s+t )
1 (s+t ) = −δk0 1 + t + h εl . 2
39
40
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
Brackets involving HH 2+6s (A): By degree argument, the bracket of HH 2+6s (A) with HH 2+6t (A) is zero.
[fk(s) , h(l t ) ] = ∆(fk(s) ∪ h(l t ) ) − ∆(fk(s) ) ∪h(l t ) − fk(s) ∪ ∆(h(l t ) ) | {z } =0
= ∆(δkl ψ0(s+t ) ) − (h + (m − t − 1)h)fk(s) ∪ α −1 (h(l t ) ) = δkl (h − 1 + (m − s − t − 1)h)ζ0 − δkl (m − t )hζ0 = −δkl (1 + sh)ζ0(s+t ) , [fk(s) , ζl(t ) ] = ∆(fk(s) ∪ ζl(t ) ) − ∆(fk(s) ) ∪ζl(t ) − fk(s) ∪ ∆(ζl(t ) ) | {z } | {z } = δl,h−3 (k + (s)
(t )
(s)
=0 (s+t ) 1)∆(zh−3 ) = (t ) (s)
=0
0,
[fk , ψl ] = ∆(fk ∪ ψl ) − ∆(fk ) ∪ψl(t ) − fk(s) ∪ ∆(ψl(t ) ) | {z } =0
(t )
l
+ (m − t − 1)h fk(s) ∪ ζl(t ) h−3 s+t +1) = δl,h−3 (k + 1)∆(θh(− ) − δ ( m − t ) h − 1 − (k + 1)zh(s−+3t +1) l,h−3 3 2 h−3 = δl,h−3 (k + 1) 1 + + (m − s − t − 1)h zh(s−+3t +1) 2 h−3 (k + 1)zh(s−+3t +1) − δl,h−3 (m − t )h − 1 − (s)
= ∆(fk ∪ ψl ) − h − 1 −
2
2
= −δl,h−3 (k + 1)(1 +
(s+t +1) sh)zh−3 , (s) (t )
[fk , εl ] = ∆ (fk(s) ) ∪ (εl(t ) ) −∆(fk ) ∪ εl − ∆fk(s) ∪ ∆(εl(t ) ) = 0. | | {z } {z } | {z }=0 (s)
(t )
=0
=0
Brackets involving HH 3+6s (A): We have
[h(ks) , h(l t ) ] = ∆(hk(s) ∪ h(l t ) ) − ∆(h(ks) ) ∪ h(l t ) + hk(s) ∪ ∆(h(l t ) ) | {z } =0
= −(h + (m − s − 1)h)α −1 (h(ks) ) ∪ h(l t ) + (h + (m − t − 1)h)h(ks) ∪ α −1 (h(l t ) ) = (s − t )hα −1 (h(ks) ) ∪ h(l t ) = (s − t )h(Mα−1 )kl ψ0(s+t ) , [h(ks) , ζl(t ) ] = ∆(h(ks) ∪ ζl(t ) ) − ∆(hk(s) ) ∪ ζl(t ) + hk(s) ∪ ∆(ζl(t ) ) | {z } =0
− (m − s)hα (h(ks) ) ∪ ζl(t ) h−3 = δk, h−3 δl,h−3 1+ + (m − s − t − 1)h zh(s−+3t +1) − (m − s)hzh(s−+3t +1) 2 2 h+1 = δk, h−3 δl,h−3 − − th zh(s−+3t +1) . =
s+t +1) δk, h−3 δl,h−3 ∆(θh(− ) 3 2
−1
2
2
We have
[h(ks) , ψl(t ) ] = ∆(hk(s) ∪ ψl(t ) ) − ∆(h(ks) ) ∪ ψl(t ) + h(ks) ∪ ∆(ψl(t ) ) | {z } =0
(s)
(t )
l
= −(m − s)hα (hk ) ∪ ψl + (m − t )h − 1 − 2 s+t +1) = −(m − s)hδk, h−3 δl,h−3 θh(− + (m − t )h − 1 − 3 2 h−1 s+t +1) = δl,h−3 δk, h−3 (s − t )h − θh(− 3 −1
2
2
(s)
(t )
hk ∪ ζ l
l 2
s+t +1) δk, h−3 θh(− 3 2
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
41
[h(ks) , εl(t ) ] = ∆(hk(s) ∪ εl(t ) ) − ∆(hk(s) ) ∪ εl(t ) − ∆h(ks) ∪ ∆(εl(t ) ) | {z } | {z } =0
=0
(s)
(t )
= −(m − s)hα (hk ) ∪ εl = 0. −1
Brackets involving HH 4+6s (A): (s) (t ) (s) (t ) (s) (t ) (s) (t ) The bracket [ζk , ζl ] = ∆(ζk ∪ ζl ) − ∆(ζk ) ∪ ζl − ζk ∪ ∆(ζl ) = 0 because ∆ is zero on HH 2+6s and HH 4+6s .
[ζk(s) , ψl(t ) ] = ∆(ζk(s) ∪ ψl(t ) ) − ∆(ζk(s) ) ∪ψl(t ) − ζk(s) ∪ ∆(ψl(t ) ), | {z } =0
l = δk,h−3 δl,h−3 ∆ α f h(−s+3 t +1) ζk(s) ∪ ζl(t ) − (m − t )h − 1 − 2 2 h−3 (s+t +1) (s+t +1) = δk,h−3 δl,h−3 (m − s − t − 1)hf h−3 − (m − t )h − 1 − f h−3 2 2 2 h+1 = δk,h−3 δl,h−3 −sh − f h−3 . 2
2
The bracket of HH 5+6s (A) with HH 5+6s (A):
[ψk(s) , ψl(t ) ] = ∆(ψk(s) ∪ ψl(t ) ) −∆(ψk(s) ) ∪ ψl(t ) + ψk(s) ∪ ∆(ψl(t ) ), | {z } =0
= − (m − s)h − 1 −
k
l ζk(s) ∪ ψl(t ) + (m − t )h − 1 − ψk(s) ∪ ζl(t )
2
2
(s+t +1)
= δk,h−3 δl,h−3 (s − t )hα(f h−3
).
2
6.1.4. The Lie derivative L We use the Cartan identity (4.1.7) to compute the Lie derivative. HH 1+6s (A)-Lie derivatives: From the Cartan identity, we see that
Lθ (s) = Bιθ (s) + ιθ (s) B. k
k
k
On θl,t , ωl,t , ψl,t and hl,t , the Connes differential acts by multiplication with
degree and taking the preimage under ιθ0 , and ιθ (s) acts on them by zero. B acts by zero on zl , εk , ζl,t and fl,t . Since B is degree preserving, this means that Lθ (s) acts on (t )
(t )
k
and then multiplication with
Lθ (s) (θl,t ) =
1+
k
Lθ (s) (zl,t ) =
l 2
2
k
Lθ (s) (ωl,t ) = δk0
2
k
Lθ (s) (εl,t ) = δk0 k
Lθ (s) (ψl,t ) =
k
Lθ (s) (ζl,t ) = k
1
1 2
+ (t − s)h (zk zl )t −s ,
+ t h ω l ,t − s , + ( t − s ) h εl ,t − s ,
(t + 1)h − 1 −
l
2
(t − s + 1)h − 1 −
Lθ (s) (hl,t ) = δk0 (t + 1)hhl,t −s , k
Lθ (s) (fl,t ) = δk0 (t − s + 1)hfl,t −s . k
1 2
+ th (zk θl )t −s ,
k+l
1+
k
(s)
their degree times zk , and on zl,t , εk,t , ζl,t and fl,t by multiplication with zk degree of their product. So we get the following formulas:
θl,t , ωl,t , ψl,t and hl,t by multiplication with 1 2
(s)
1 2
(zk ψl )t −s , l−k 2
(zk ζl )t −s ,
42
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
HH 2+6s (A)-Lie derivatives: We compute Lf (s) : k
Lf (s) (θl,t ) = B(ιf (s) (θl,t )) − ιf (s) (B(θl,t )) k
k
k
l = B(δl0 α(fk,t −s−1 )) − 1 + + th ιf (s) zl,t 2
k
= δl0 (t − s)hfk,t −s−1 − δl0 (1 + th)fk,t −s−1 = −δl0 (1 + sh)fk,t −s , Lf (s) (fl,t ) = B( ιf (s) (fl,t ) ) = 0, k | k {z } ∈HH4+6(t −s)
Lf (s) (zl,t ) = δl0 B(fk,t −s ) = 0, k
Lf (s) (ωl,t ) = B(ιf (s) ωl,t ) + ιf (s) B(ωl,t ) = k
| k {z }
+ t hιf (s) β −1 (ωl,t ) = 0,
1 2
k
k
=0
Lf (s) (εl,t ) = B(ιf (s) (εl,t )), k
| k {z } =0
Lf (s) (ψl,t ) = B(ιf (s) (ψl,t )) − ιf (s) B(ψl,t ) k
k
k
l = B(δl,h−3 (k + 1)θh−3,t −s ) − (t + 1)h − 1 − ι (s) (ζl,t ) 2 fk h−3 l = δl,h−3 (k + 1) 1 + + (t − s)h zh−3,t −s − δl,h−3 (k + 1) (t + 1)h − 1 − zh−3,t −s 2
2
= −δl,h−3 (k + 1)(1 + sh)zh−3,t −s , Lf (s) (ζl,t ) = B(ιf (s) (ζl,t )) = B(kδl,h−3 zh−3,t −s ) = 0, k
k
Lf (s) (hl,t ) = B(ιf (s) (hl,t )) − ιf (s) B(hl,t ) k
k
k
= B(δk,l ψ0,t −s ) − (t + 1)hιf (s) α −1 (hl,t ) k
= δkl ((t − s + 1)h − 1)ζ0,t −s − δkl (t + 1)hζ0,t −s = −δkl (sh + 1)ζ0,t −s . HH 3+6s (A)-Lie derivatives: We compute Lh(s) : k
(t )
(t )
Lh(s) (θl ) = B(ιh(s) (θl )) + ιh(s) B(θl,t ) =
| k {z }
k
1+
k
l 2
+ th ιh(s) zl,t k
=0
= δl0 (1 + th)hk,t −s−1 , Lh(s) (zl,t ) = B(δl0 hk,t −s−1 ) = δl0 (t − s)hα −1 (hk,t −s−1 ), k
Lh(s) (ωl,t ) = B ιh(s) (ωl,t ) + k
| k {z } =0
ιh(s) B(ωl,t ) | k {z }
= 0,
cup product in HH 3 (A)×HH 5 (A)
Lh(s) (εl,t ) = B ιh(s) εl,t = 0, k
| k{z } =0
(s)
Lh(s) (ψl,t ) = B(ιh(s) (ψl,t )) + hk B(ψl,t ) k
k
| {z } =0
=
(t + 1)h − 1 −
l 2
(s)
hk ζl,t = δk, h−3 δl,h−3 2
Lh(s) (ζl,t ) = B(ιh(s) (ζl,t )) = δk, h−3 B(δl,h−3 θh−3,t −s ) k
2
k
= δl,h−3 (t − s)h +
h−1 2
zh−3,t −s ,
th +
h+1 2
θh−3,t −s ,
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
Lh(s) (hl,t ) = B(ιh(s) (hl,t )) + ιh(s) B(hl,t ) = (t + 1)hιh(s) α −1 (hl,t ) k
| k {z }
k
k
=0
= (t + 1)h(Mα−1 )lk ψ0,t −s Lh(s) (fl,t ) = B(ιh(s) (fl,t )) = B(δkl ψ0,t −s ) = δkl ((t − s + 1)h − 1)ζ0,t −s . k
HH
k
(A)-Lie derivatives: We compute Lζ (s) : 4+6s
k
Lζ (s) (θl,t ) = Bιζ (s) (θl,t ) − ιζ (s) B(θl,t ) = B((zl ψk )t −s−1 ) − ιζ (s) k
k
= =
k
1+
k
k−l
l
l 2
+ th zl,t
(t − s)h − 1 − (zl ζk )t −s−1 − 1 + + th (zl ζk )t −s−1 2 2 k (zl ζk )t −s−1 , −sh − 2 − 2
Lζ (s) (ωl,t ) = 0, k
Lζ (s) (zl,t ) = Bιζ (s) (zl,t ) − ιζ (s) B(zl,t ) k
k
k
| {z } =0
= B((zl ζk )t −s−1 ) = 0, Lζ (s) (ψl,t ) = Bιζ (s) (ψl,t ) − ιζ (s) B(ψl,t ) k
k
k
l = δk,h−3 δl,h−3 B(α(f h−3 ,t −s−1 )) − ιζ (s) (t + 1)h − 1 − ζ l ,t 2 k 2 h−3 = δk,h−3 δl,h−3 (t − s)hf h−3 ,t −s−1 − (t + 1)h − 1 − f h−3 ,t −s−1 2 2 2 h+1 = δk,h−3 δl,h−3 −sh − f h−3 ,t −s−1 , 2
2
− ιζ (s) B(εl,t ) = 0, k | {z } =0 | k {z }=0 Lζ (s) (ζl,t ) = Bιζ (s) (ζl,t ) − ιζ (s) B(ζl,t ) k k k | {z } Lζ (s) (εl,t ) = Bιζ (s) (εl,t ) k
=0
= δkh−3 δl,h−3 B(f h−3 ,t −s−1 ) = 0, 2
Lζ (s) (hl,t ) = Bιζ (s) (hl,t ) − ιζ (s) B(hl,t ) k
k
k
= δl, h−3 δk,h−3 B(θh−3,t −s ) − (t + 1)hιζ (s) α −1 (hl,t ), 2 k h−3 = δl, h−3 δk,h−3 zh−3,t −s 1+ + (t − s)h − (t + 1)h 2 2 h+1 = δl, h−3 δk,h−3 zh−3,t −s − − sh , 2
2
Lζ (s) (fl,t ) = Bιζ (s) (fl,t ) − ιζ (s) B(fl,t ) k
k
k
| {z } =0
= (l + 1)δk,h−3 B(zh−3,t −s ) = 0. HH 5+6s (A)-Lie derivatives: We compute Lε(s) : k
Lε(s) (θl,t ) = B(ιε(s) (θl,t )) + ιε(s) B(θl,t ) k
k
k
l = B(−δl0 β(εk,t −s−1 )) + 1 + + th ιε(s) (zl,t ) k 2 1 = −δl0 + t − s − 1 hεk,t −s−1 + (1 + th)δl0 εk,t −s−1 2
43
44
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
=
s+
1
2
h + 1 δl0 εk,t −s−1 ,
Lε(s) (zl,t ) = B(ιε(s) (zl,t )) = B(εk,t −s−1 ) = 0, k
k
Lε(s) (ωl,t ) = B(ιε(s) (ωl,t )) + ιε(s) B(ωl,t ) k
k
k
+ t hιε(s) β −1 (ωl,t ) k 2 1 = δkl ((t − s)h − 1)ζ0,t −s−1 − δkl + t hζ0,t −s−1 2 1 = −δkl 1 + + s h ζ0,t −s−1 , = B(δkl ψ0,t −s−1 ) +
1
2
Lε(s) (ψl,t ) = B(ιε(s) (ψl,t )) + ιε(s) B(ψl,t ) k
| k {z }
k
=0
=
(t + 1)h − 1 −
l 2
ιε(s) ζl,t = 0, k
Lε(s) (εl,t ) = B(ιε(s) (εl,t )) = B(−(Mβ )kl ζ0,t −s−1 ) = 0, k
k
Lε(s) (ζl,t ) = B(ιε(s) ζl,t ) = 0, k
| k{z } =0
Lε(s) (hl,t ) = B(ιε(s) (hl,t )) + (t + 1)hιε(s) α −1 (hl,t ) = 0, k
k
k
Lε(s) (fl,t ) = B(ιε(s) (fl,t )) = 0. k
| k {z } =0
We compute Lψ (s) : k
Lψ (s) (θl,t ) = B ιψ (s) (θl,t ) +ιψ (s) B(θl,t ) k
| k {z }
k
=0
= ιψ (s) zl,t 1 + k
l
+ th = (zl ψk )t −s−1 1 +
2
l 2
+ th ,
Lψ (s) (zl,t ) = Bιψ (s) (zl,t ) + ιψ (s) B(zl,t ) k
k
k
| {z } =0
k−l = B((zl ψk )t −s−1 ) = (t − s)h − 1 − (zl ζk )t −s−1 , 2
Lψ (s) (ωl,t ) = B ιψ (s) (ωl,t ) +ιψ (s) B(ωl,t ) k
k
| k {z } =0
=
1 2
+ t hιψ (s) β −1 (ωl,t ) = 0, k
Lψ (s) (ψl,t ) = B ιψ (s) (ψl,t ) +ιψ (s) B(ψl,t ) k
| k {z }
k
=0
l (t + 1)h − 1 − ι (s) ζl,t 2 ψk h−3 α(f h−3 ,t −s−1 ), = δk,h−3 δl,h−3 (t + 1)h − 1 − 2 2 } | {z =
1 =th+ h+ 2
Lψ (s) (εl,t ) = B ιψ (s) (εl,t ) +ιψ (s) B(εl,t ) = 0 k
| k {z }
k
| {z } =0
=0
Lψ (s) (ζl,t ) = Bιψ (s) (ζl,t ) + ιψ (s) B(ζl,t ) k
k
k
| {z } =0
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
45
= δk,h−3 δl,h−3 B(α(f h−3 ,t −s−1 )) 2
= δk,h−3 δl,h−3 (t − s)hf h−3 ,t −s−1 , 2
Lψ (s) (hl,t ) = B ιψ (s) (hl,t ) +ιψ (s) B(hl,t ) = ιψ (s) α −1 (hl,t )(t + 1)h
| k {z }
k
k
k
=0
= δk,h−3 δl, h−3 (t + 1)hθh−3,t −s , 2
Lψ (s) (fl,t ) = Bψ (s) (fl,t ) + ιψ (s) B(fl,t ) k
k
k
| {z } =0
= (l + 1)δk,h−3 B(θh−3,t −s ) h−3 δk,h−3 zh−3,t −s . = (l + 1) 1 + (t − s)h + 2
HH
(A)-Lie derivatives: B acts on θl,t , ωl,t , ψl,t and hl,t by multiplication with 6+6s
degree and taking the preimage under ιθ0 . On zl,t , εl,t , ζl,t and fl,t , (s) B acts by zero. Since the spaces U, U , K , K , Y and Y are zk -invariant and zk has degree k − 2sh, Lz (s) acts on θl,t , ωl,t , ψl,t ∗
∗
1 2
∗
k
(s)
and hl,t by multiplication with 2k − sh and taking the preimage under ιθ0 and multiplication with zk , and on zl,t , εl,t , ζl,t and fl,t it acts by zero. We have the following formulas:
Lz (s) (θl,t ) =
k 2
k
− sh (zk θl )t −s ,
Lz (s) (zl,t ) = 0, k
Lz (s) (ωl,t ) = −δk0 shβ −1 (ωl,t −s ), k
Lz (s) (ψl,t ) =
k 2
k
− sh (zk ζl )t −s ,
Lz (s) (εl,t ) = 0, k
Lz (s) (ζl,t ) = 0, k
Lz (s) (hl,t ) =
k 2
k
− sh α −1 (hl,t −s ),
Lz (s) (fl,t ) = 0. k
Now we compute Lω(s) : k
We observe that ιω(s) (εl,t ) = δkl ψ0,t −s , ιω(s) (zl,t ) = δl0 ωk,t −s , and k
k
ιω(s) (θl,t ) = ιω(s) (ωl,t ) = ιω(s) (ψl,t ) = ιω(s) (ζl,t ) = ιω(s) (hl,t ) = ιω(s) (fl,t ) = 0. k
k
k
k
k
k
Then we have
Lω(s) (εl,t ) = Bιω(s) (εl,t ) = δkl B(ψ0,t −s ) k
k
= δkl ((t − s + 1)h − 1)ζ0,t −s , Lω(s) (zl,t ) = Bιω(s) (zl,t ) = δl0 B(ωk,t −s ) k k 1 = δl0 + t − s hβ −1 (ωk,t −s ), 2 l Lω(s) (θl,t ) = ιω(s) B(θl,t ) = 1 + + th ιω(s) zl,t k
2
k
k
= δl0 (1 + th)ωk,t −s , and
Lω(s) (ωl,t ) = Lω(s) (ψl,t ) = Lω(s) (ζl,t ) = Lω(s) (hl,t ) = Lω(s) (fl,t ) = 0. k
k
k
k
k
Acknowledgements C. Eu wants to thank his advisor P. Etingof for useful discussions and V. Dolgushev for his explanations about calculus.
46
C.-H. Eu / Journal of Pure and Applied Algebra 214 (2010) 28–46
References [1] [2] [3] [4] [5] [6] [7] [8]
Yu. Daletski, I. Gelfand, B. Tsygan, On a variant of noncommutative geometry, Soviet Math. Dokl. 40 (2) (1990) 422–426. W. Crawley-Boevey, P. Etingof, V. Ginzburg, Noncommutative geometry and quiver algebras, arXiv:math.AG/0502301. P. Etingof, C. Eu, Hochschild and cyclic homology of preprojective algebras of ADE quivers, arxiv:math.AG/0609006. K. Erdmann, N. Snashall, On Hochschild cohomology of preprojective algebras. I, II, J. Algebra 205 (2) (1998) 391–412. 413–434. C. Eu, The product in the Hochschild cohomology ring of preprojective algebras of Dynkin quivers, arxiv:math/0703568v1. A. Schofield, Wild algebras with periodic Auslander–Reiten translate, preprint. A. Malkin, V. Ostrik, M. Vybornov, Quiver varieties and Lusztig’s algebra, arXiv:math.AG/0403222. J. Cuntz, G. Skandalis, B. Tsygan, Cyclic homology in non-commutative geometry, in: Operator Algebras and Non-Commutative Geometry, II, in: Encyclopedia of Mathematical Sciences, vol. 121, Springer Verlag, Berlin, 2004. [9] V.A. Dolgushev, A proof of Tsygan’s formality conjecture for an arbitrary smooth manifold, Ph.D. Thesis, MIT. arxiv:math.QA/0504420. [10] J.-L. Loday, Cyclic homology, in: Grundreihen der mathematischen Wissenschaften, Vol. 301, in: A Series of Comprehensive Studies in Mathematics, Springer Verlag, 1992. [11] V. Ginzburg, Calabi–Yau algebras, arxiv:math/0612139v3.