Mathematics – Representation Theory
Scientific paper
2007-04-24
Mathematics
Representation Theory
103 pages, 2 figures
Scientific paper
We determine the $\Z$-module structure and explicit bases for the preprojective algebra $\Pi$ and all of its Hochschild (co)homology, for any non-Dynkin quiver. This answers (and generalizes) a conjecture of Hesselholt and Rains, producing new $p$-torsion elements in degrees $2 p^\ell, \ell \geq 1$. We relate these elements by $p$-th power maps and interpret them in terms of the kernel of Verschiebung maps from noncommutative Witt theory. We also define a Lie bialgebra structure on $HH_0(\Pi)$ (from the necklace Lie bialgebra), relate it to Goldman/Turaev's Lie bialgebra of loops, compute it for extended Dynkin quivers, and compute the Poisson center of $\Sym HH_0(\Pi)$ for all quivers. We then compute the BV algebra structure on Hochschild cohomology, show that the Lie algebra structure $HH_0(\Pi_Q)$ naturally arises from it, and compute all cyclic homology groups of $\Pi_Q$. In the process, we define and study related algebraic structures: a ``noncommutative BV structure'' generalizing the necklace Lie bialgebra, and ``free-product'' deformations of $\Pi_Q$, which yield all ordinary deformations as quotients.
Schedler Travis
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