Proceedings of the 44th Symposium on Ring Theory and ...

Remark 6. For i = 1, . . . , 4, let Λ i = KΓ i /I i be <strong>the</strong> cluster-tilted algebra <strong>of</strong> type D n
corresponding to Γ i . Then we see from [3, 24] that
(1) Λ 1 is <strong>the</strong> path algebra <strong>of</strong> a Dynkin quiver <strong>of</strong> type D m .
(2) Λ 2 is <strong>of</strong> type D 4 , and I 2 = 〈a 1 a 2 , b 1 b 2 , a 2 a 0 , b 2 b 0 , a 0 a 1 − b 0 b 1 〉. We immediately see
that Λ 2 is a special biserial algebra <strong>of</strong> [22], but not a self-injective algebra.
(3) Λ 3 is <strong>of</strong> type D m , and I 3 = 〈p | p is a path <strong>of</strong> length m − 1〉. Hence Λ 3 is a
(m−1, 1)-stacked monomial algebra, and is also a self-injective Nakayama algebra.
(4) Λ 4 is <strong>of</strong> type D 2m , and it follows by [3, Lemma 4.5] that Λ 4 is derived equivalent
to <strong>the</strong> (2m − 1, 1)-stacked monomial algebra Λ ′ = KQ ′ /I ′ , where Q ′ is <strong>the</strong> cyclic
quiver with 2m vertices
1 2
• • ❄ ❄❄❄❄
2m 8
•
8888 • 3
.
7 • ❄ ❄❄❄❄ • 4
• 8
•
8888
6 5
and I ′ is generated by all paths <strong>of</strong> length 2m − 1. Note that Λ ′ is a self-injective
Nakayama algebra, and moreover is a cluster-tilted algebra <strong>of</strong> type D 2m ([21, 24]).
In [19], Happel described <strong>the</strong> Hochschild cohomology for path algebras. Using this result
and [18, Theorem 3.4], we have <strong>the</strong> following proposition:
Proposition 7. For <strong>the</strong> algebras Λ 1 , Λ 3 and Λ 4 above, we have
HH ∗ (Λ 1 ) ≃ HH ∗ (Λ 1 )/N Λ1 ≃ K
HH ∗ (Λ 3 )/N Λ3 ≃ HH ∗ (Λ 4 )/N Λ4 ≃ K[x].
Finally we describe <strong>the</strong> Hochschild cohomology ring modulo nilpotence <strong>of</strong> <strong>the</strong> algebra
A k := Γ 2 /J k , where k ≥ 0 and J k is <strong>the</strong> ideal generated by <strong>the</strong> following elements:
(a 1 a 2 a 0 ) k a 1 a 2 , b 1 b 2 , (a 2 a 0 a 1 ) k a 2 a 0 , b 2 b 0 , (a 0 a 1 a 2 ) k a 0 a 1 − b 0 b 1 .
If k = 0, <strong>the</strong>n J 0 = I 2 , and so A 0 = Γ 2 /J 0 coincides with <strong>the</strong> algebra Λ 2 . Note that, for
all k ≥ 0, A k is a special biserial algebra and not a self-injective algebra.
Now <strong>the</strong> dimensions <strong>of</strong> <strong>the</strong> Hochschild cohomology groups <strong>of</strong> A k are given as follows:
Theorem 8 ([14]). For k ≥ 0 and i ≥ 0 we have
⎧
k + 1 if i ≡ 0 (mod 6)
k + 1 if i ≡ 1 (mod 6)
k if i ≡ 2 (mod 6)
⎪⎨
dim K HH i k + 1 if i ≡ 3 (mod 6) and char K | 3k + 2
(A k ) =
k if i ≡ 3 (mod 6) and char K ∤ 3k + 2
k + 1 if i ≡ 4 (mod 6) and char K | 3k + 2
k if i ≡ 4 (mod 6) and char K ∤ 3k + 2
⎪⎩
k if i ≡ 5 (mod 6).
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