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Register a new userZeroth Hochschild homology of preprojective algebras over the integers
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Authors
Travis Schedler
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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 $text{Sym } HH_0(Pi)$ for all quivers, the BV algebra structure on Hochschild cohomology, including how the Lie algebra structure $HH_0(Pi_Q)$ naturally arises from it, and the cyclic homology groups of $Pi_Q$.
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We give an interpretation of the $(q,t)$-deformed Cartan matrices of finite type and their inverses in terms of bigraded modules over the generalized preprojective algebras of Langlands dual type in the sense of Geiss-Leclerc-Schr{o}er~[Invent.~math.~{bf{209}} (2017)]. As an application, we compute the first extension groups between the generic kernels introduced by Hernandez-Leclerc~[J.~Eur.~Math.~Soc.~{bf 18} (2016)], and propose a conjecture that their dimensions coincide with the pole orders of the normalized $R$-matrices between the corresponding Kirillov-Reshetikhin modules.
We determine higher topological Hochschild homology of rings of integers in number fields with coefficients in suitable residue fields. We use the iterative description of higher THH for this and Postnikov arguments that allow us to reduce the necessary computations to calculations in homological algebra, starting from the results of Bokstedt and Lindenstrauss-Madsen on (ordinary) topological Hochschild homology.
Geiss-Leclerc-Schroer [Invent. Math. 209 (2017)] has introduced a notion of generalized preprojective algebra associated with a generalized Cartan matrix and its symmetrizer. This class of algebra realizes a crystal structure on the set of maximal dimensional irreducible components of the nilpotent variety [Selecta Math. (N.S.) 24 (2018)]. For general finite types, we give stratifications of these components via partial orders of torsion classes in module categories of generalized preprojective algebras in terms of Weyl groups. In addition, we realize Mirkovic-Vilonen polytopes from generic modules of these components, and give a identification as crystals between the set of Mirkovic-Vilonen polytopes and the set of maximal dimensional irreducible components except for type $mathsf{G}_2$. This generalizes results of Baumann-Kamnitzer [Represent. Theory 16 (2012)] and Baumann-Kamnitzer-Tingley [Publ. Math. Inst. Hautes Etudes Sci. 120 (2014)].
It is shown that except in three cases conjugacy classes of classical Weyl groups $W(B_{n})$ and $W(D_{n})$ are of type ${rm D}$. This proves that Nichols algebras of irreducible Yetter-Drinfeld modules over the classical Weyl groups $mathbb W_{n}$ (i.e. $H_{n}rtimes mathbb{S}_{n}$) are infinite dimensional, except the class of type $(2, 3),(1^{2}, 3)$ in $mathbb S_{5}$, and $(1^{n-2}, 2)$ in $mathbb S_{n}$ for $n >5$.
Beliakova-Putyra-Wehrli studied various kinds of traces, in relation to annular Khovanov homology. In particular, to a graded algebra and a graded bimodule over it, they associate a quantum Hochschild homology of the algebra with coefficients in the bimodule, and use this to obtain a deformation of the annular Khovanov homology of a link. A spectral refinement of the resulting invariant was recently given by Akhmechet-Krushkal-Willis. In this short note we observe that quantum Hochschild homology is a composition of two familiar operations, and give a short proof that it gives an invariant of annular links, in some generality. Much of this is implicit in Beliakova-Putyra-Wehrlis work.
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