[Paper Review] The product in the Hochschild cohomology ring of preprojective algebras of Dynkin quivers

Ching-Hwa Eu|ArXiv.org|Mar 20, 2007

Algebraic structures and combinatorial models4 references4 citations

TL;DR

This paper computes the full cup product structure in the Hochschild cohomology ring of preprojective algebras associated to Dynkin quivers of type D and E over a field of characteristic zero. Using the Schofield resolution and a basis construction via graded cohomology, it establishes that most products vanish due to degree constraints, while nontrivial products arise from specific pairings involving the Gerstenhaber bracket and Batalin-Vilkovisky structure, with the key result being a uniform description of the ring structure across all D and E types.

ABSTRACT

In this paper, we compute the product structure of the Hochschild cohomology of preprojective algebras of quivers of type D and E over a field of characteristic zero.

Motivation & Objective

• To fully determine the cup product structure in the Hochschild cohomology ring of preprojective algebras for Dynkin quivers of type D and E.
• To extend prior results on type A (from [ES2]) to complete the classification of the product structure for all ADE quivers over characteristic zero fields.
• To provide a uniform description of the ring structure despite using case-by-case computations for each Dynkin type.
• To establish vanishing results for many products using degree arguments and the Batalin-Vilkovisky structure.
• To identify nontrivial products via explicit pairings, particularly involving the Gerstenhaber bracket and top-degree cohomology components.

Proposed method

• The paper uses the Schofield resolution, a 6-periodic resolution of the preprojective algebra, to compute Hochschild cohomology via the isomorphism $ HH^i(A) \cong \underline{Hom}(\Omega^i A, A) $.
• It constructs explicit bases for each cohomology space $ HH^i(A) $ by leveraging the grading from the natural arrow-degree 1 structure.
• The cup product is computed via composition: $ [f][g] = [f \circ \Omega^i g] $, where $ f \in HH^i(A) $, $ g \in HH^j(A) $.
• Vanishing of products is established using degree constraints: $ \deg HH^i(A) \leq -h $ for $ i \geq 4 $, where $ h $ is the Coxeter number.
• Nontrivial products are analyzed using the Batalin-Vilkovisky structure, particularly the Gerstenhaber bracket and the action of $ \Delta $, to deduce that certain pairings must vanish.
• For $ HH^5(A) \times HH^5(A) \to HH^{10}(A) $, the product is shown to be given by a skew-symmetric bilinear form $ \Omega $, with matrix $ -M_\beta $, derived from the $ HH^1 \times HH^5 \to HH^6 $ product.

Experimental results

Research questions

• RQ1 What is the complete cup product structure in the Hochschild cohomology ring of preprojective algebras for Dynkin quivers of type D and E?
• RQ2 Which products among cohomology groups vanish, and which are nontrivial, and why?
• RQ3 How does the Batalin-Vilkovisky structure constrain the product relations, particularly in higher degrees?
• RQ4 Can the product structure be described uniformly across all D and E types despite case-by-case computations?
• RQ5 What is the precise form of the pairing $ HH^5(A) \times HH^5(A) \to HH^{10}(A) $, and how is it related to the Gerstenhaber bracket?

Key findings

• The product $ HH^2(A) \times HH^4(A) \to HH^6(A) $ vanishes due to degree constraints and the vanishing of $ f_k \zeta_{h-4} $, which follows from the independence of the Gerstenhaber bracket on the choice of $ m $ in the Batalin-Vilkovisky structure.
• The product $ HH^2(A) \times HH^5(A) \to HH^7(A) $ vanishes because $ a \zeta_k \in HH^2(A) \times HH^4(A) = 0 $, leading to a contradiction if the product were nonzero.
• The product $ HH^3(A) \times HH^3(A) \to HH^6(A) $ vanishes due to degree mismatch: $ \deg HH^6(A) \leq -h-2 < -4 = \deg HH^3(A) + \deg HH^3(A) $.
• The product $ HH^4(A) \times HH^5(A) \to HH^9(A) $ vanishes for $ Q = D_{n+1} $, $ n $ even or $ Q = E_6 $, by contradiction using the nondegenerate pairing $ HH^2 \times HH^3 \to HH^5 $.
• The product $ HH^5(A) \times HH^5(A) \to HH^{10}(A) $ is given by a skew-symmetric bilinear form $ \Omega $ with matrix $ -M_\beta $, derived from the $ HH^1 \times HH^5 \to HH^6 $ product.
• All products $ HH^i(A) \times HH^j(A) \to HH^{i+j}(A) $ for $ i,j \leq 5 $ are computed explicitly, and the remaining products follow from periodicity and graded commutativity.