[Paper Review] Hochschild and cyclic homology of central extensions of preprojective algebras of ADE quivers

Ching-Hwa Eu|ArXiv.org|Jun 16, 2006

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper computes the Hochschild and cyclic homology of centrally extended preprojective algebras of ADE quivers using a 4-periodic projective resolution, establishing periodicity in homology and cohomology. It identifies the universal deformation of the algebra via an isomorphism between the deformation space and the second Hochschild cohomology group, showing the deformation theory is unobstructed.

ABSTRACT

Let A be the central extension of the preprojective algebra of an ADE quiver introduced by P. Etingof and E. Rains in math/0503393. The paper math/0606403 computes the structure of the zeroth Hochschild (co)homology of A. We generalize the results of math/0606403 by calculating the additive structure of all the Hochschild homology and cohomology groups of A and the cyclic homology of A, and to describe the universal deformation of A. Namely, we show that the (co)homology is periodic with period 4, and compute the first four (co)homology groups in each case.

Motivation & Objective

• To compute the additive structure of Hochschild homology and cohomology, and cyclic homology, of centrally extended preprojective algebras of ADE quivers.
• To establish periodicity in the Hochschild (co)homology and cyclic homology of these algebras, with period 4.
• To determine the universal deformation of the centrally extended preprojective algebra by analyzing HH²(A).
• To generalize prior results on centers and trace spaces (from [ELR]) to full (co)homological structures.
• To provide a framework for computing product structures in (co)homology, which is left for future work.

Proposed method

• Constructs a 4-periodic projective resolution of the centrally extended preprojective algebra A using graded bimodules over the path algebra.
• Uses the periodic resolution to compute the first four Hochschild homology and cohomology groups, establishing periodicity.
• Applies the Hilbert series formalism to compute generating functions for homology and cohomology, leveraging results from [ELR] and combinatorial identities from [RS].
• Defines a chain map f₁ and f₂ between resolutions and proves commutativity to relate differentials in the resolution.
• Uses the Hom functor over A^e to dualize the resolution and compute Hochschild cohomology via the induced maps f₁* and f₂*.
• Establishes an isomorphism between the deformation space E' and HH²(A) by showing that the map θ induces an isomorphism from E' to A/([A,A] + μZ), leading to the universal deformation.

Experimental results

Research questions

• RQ1 What is the additive structure of Hochschild homology and cohomology for centrally extended preprojective algebras of ADE quivers?
• RQ2 Does the Hochschild (co)homology of these algebras exhibit periodicity, and if so, with what period?
• RQ3 How can the universal deformation of the centrally extended preprojective algebra be explicitly described?
• RQ4 What is the relationship between the deformation space and the second Hochschild cohomology group HH²(A)?
• RQ5 Can the Hilbert series of the homology and cohomology groups be computed using known results and combinatorial identities?

Key findings

• The Hochschild homology and cohomology of the centrally extended preprojective algebra A are periodic with period 4.
• The first four Hochschild homology and cohomology groups are computed explicitly, and their Hilbert series are determined using results from [ELR] and combinatorial identities.
• The cyclic homology of A is computed, and its Hilbert series are derived from the same data.
• The deformation space E' is identified as a complement to ker(θ), and the map θ induces an isomorphism from E' to A/([A,A] + μZ), which is isomorphic to HH²(A).
• The subdeformation A' defined by A' = P[z][[t₁,…,tₛ]] / (∑[a,a*] = μz + ∑tᵢwᵢ) is the universal deformation of A.
• The deformation theory of A is unobstructed, as HH²(A) classifies all formal deformations and the isomorphism with E' ensures universality.