Zeroth Hochschild homology of preprojective algebras over the integers

2007/04/24 by Travis Schedler, Schedler, Travis
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT

paper · pdf · doi:10.48550/arxiv.0704.3278

69 pages, 2 figures; final pre-publication version; many corrections and improvements throughout. Note though the first version has additional results (for instance, it computes the higher Hochschild (co)homology and its structures)

arxiv created 2016/05/28 · arxiv updated 2016/05/31

Abstract

We determine the Z-module structure of the preprojective algebra and its zeroth Hochschild homology, for any non-Dynkin quiver (and hence the structure working over any base commutative ring, of any characteristic). This answers (and generalizes) a conjecture of Hesselholt and Rains, producing new p-torsion classes in degrees 2p^l, l >= 1, We relate these classes by p-th power maps and interpret them in terms of the kernel of Verschiebung maps from noncommutative Witt theory. An important tool is a generalization of the Diamond Lemma to modules over commutative rings, which we give in the appendix. In the previous version, additional results are included, such as: the Poisson center of Sym HH₀(Π) for all quivers, the BV algebra structure on Hochschild cohomology, including how the Lie algebra structure HH₀(Π_Q) naturally arises from it, and the cyclic homology groups of Π_Q.

Related

• Hochschild and cyclic (co)homology of preprojective algebras of quivers of type T
• Hochschild and cyclic homology of preprojective algebras of ADE quivers
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• The product in the Hochschild cohomology ring of preprojective algebras of Dynkin quivers
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