[Paper Review] On the first Hochschild cohomology group of a cluster-tilted algebra

Ibrahim Assem, María Julia Redondo+1 more|arXiv (Cornell University)|Apr 7, 2012

Algebraic structures and combinatorial models21 references4 citations

TL;DR

This paper establishes a short exact sequence for the first Hochschild cohomology group of a cluster-tilted algebra $ B $, showing $ 0 \to k^{n_{B,C}} \to \mathrm{HH}^1(B) \to \mathrm{HH}^1(C) \to 0 $, where $ C $ is the associated tilted algebra and $ n_{B,C} $ counts equivalence classes of arrows not in $ C $. For representation-finite cluster-tilted algebras, $ \dim_k \mathrm{HH}^1(B) $ equals the number of chordless cycles minus the number of inner arrows in the quiver of $ B $, providing a combinatorial computation method.

ABSTRACT

Given a cluster-tilted algebra B, we study its first Hochschild cohomology group HH^1(B) with coefficients in the B-B-bimodule B. If C is a tilted algebra such that B is the relation extension of C, then we show that if C is constrained, or else if B is tame, then HH^1(B) is isomorphic, as a k-vector space, to the direct sum of HH^1(C) with k^{n\_{B,C}}, where n\_{B,C} is an invariant linking the bound quivers of B and C. In the representation-finite case, HH^1(B) can be read off simply by looking at the quiver of B.

Motivation & Objective

• To relate the first Hochschild cohomology of a cluster-tilted algebra $ B $ to that of its associated tilted algebra $ C $, especially in the tame and representation-finite cases.
• To resolve Skowroński’s question on whether vanishing $ \mathrm{HH}^1(B) $ implies simple connectivity for cluster-tilted algebras.
• To provide a combinatorial formula for $ \dim_k \mathrm{HH}^1(B) $ in the representation-finite case using quiver structure.
• To define and compute the invariant $ n_{B,C} $, which measures the difference in relations between $ B $ and $ C $, and show its invariance in the representation-finite setting.

Proposed method

• Define an equivalence relation on arrows in the quiver of $ B $ not present in the quiver of $ C $, with the number of equivalence classes denoted $ n_{B,C} $.
• Use the relation-extension construction to relate $ B $ and $ C $, leveraging known results on cluster-tilted algebras as extensions of tilted algebras.
• Apply the short exact sequence $ 0 \to k^{n_{B,C}} \to \mathrm{HH}^1(B) \to \mathrm{HH}^1(C) \to 0 $, proven for tame algebras.
• Characterize inner arrows as those lying on two chordless cycles, and use this to define the combinatorial invariant $ n_B $ in the representation-finite case.
• Use Euler’s formula on planar quivers to derive the formula $ n_B = 1 + \text{outer arrows} - n $ for connected quivers.
• Prove that $ \mathrm{HH}^1(B) = 0 $ if and only if $ B $ is hereditary and its quiver is a tree, confirming Skowroński’s conjecture for this class.

Experimental results

Research questions

• RQ1 Does the short exact sequence $ 0 \to k^{n_{B,C}} \to \mathrm{HH}^1(B) \to \mathrm{HH}^1(C) \to 0 $ hold for tame cluster-tilted algebras?
• RQ2 Is the vanishing of $ \mathrm{HH}^1(B) $ equivalent to $ B $ being simply connected (i.e., hereditary with quiver a tree)?
• RQ3 Can $ \dim_k \mathrm{HH}^1(B) $ be computed purely from the quiver structure in the representation-finite case?
• RQ4 What is the combinatorial meaning of $ n_{B,C} $, and does it remain invariant under different choices of $ C $?
• RQ5 How does deleting a vertex affect the Hochschild cohomology dimension, and can this be captured via a degree function?

Key findings

• For any tame cluster-tilted algebra $ B $, there is a short exact sequence $ 0 \to k^{n_{B,C}} \to \mathrm{HH}^1(B) \to \mathrm{HH}^1(C) \to 0 $, where $ n_{B,C} $ is the number of equivalence classes of arrows not in the quiver of $ C $.
• The first Hochschild cohomology group $ \mathrm{HH}^1(B) $ vanishes if and only if $ B $ is hereditary and its quiver is a tree, confirming Skowroński’s conjecture for cluster-tilted algebras.
• In the representation-finite case, $ \dim_k \mathrm{HH}^1(B) = n_B = \text{number of chordless cycles} - \text{number of inner arrows} $ in the quiver of $ B $.
• For a connected quiver $ \tilde{Q} $, $ n_B = 1 + \text{number of outer arrows} - n $, where $ n $ is the number of vertices.
• The Hochschild degree of a vertex $ x $, defined as $ \deg_{\mathrm{HH}}(x) = n_B - n_{B/Be_xB} $, equals the number of chordless cycles through $ x $ minus the number of inner arrows on those cycles.
• In the example of a type $ \mathbb{E}_8 $ cluster-tilted algebra with 4 chordless cycles and 2 inner arrows, $ \mathrm{HH}^1(B) = k^2 $, confirmed by both the cycle-minus-inner-arrow formula and the outer-arrow formula.