Multiplicative preprojective algebras of Dynkin quivers

TL;DR

This paper investigates when multiplicative preprojective algebras of Dynkin quivers are isomorphic to additive ones, establishing conditions based on invertibility of bad primes and exploring their Hochschild homology and deformation properties.

Contribution

It provides explicit isomorphisms over localized integers, identifies obstructions via Hochschild homology, and compares multiplicative and additive preprojective algebras in various types.

Findings

Isomorphism holds when bad primes are invertible in the base ring.

Differences in Hochschild homology indicate non-isomorphism when bad primes are not invertible.

Zeroth Hochschild homology relates to unobstructed deformations of Ginzburg dg-algebras.

Abstract

For a commutative ring $R$ and an ADE Dynkin quiver $Q$, we prove that the multiplicative preprojective algebra of Crawley-Boevey and Shaw, with parameter $q=1$, is isomorphic to the (additive) preprojective algebra as $R$-algebras if and only if the bad primes for $Q$ (2 in type D, 2 and 3 for $Q = E_{6}$, $E_{7}$ and 2, 3 and 5 for $Q= E_{8}$) are invertible in $R$. We construct an explicit isomorphism over $\mathbb{Z}[1/2]$ in type D, over $\mathbb{Z}[1/2, 1/3]$ for $Q = E_{6}$, $E_{7}$ and over $\mathbb{Z}[1/2, 1/3, 1/5]$ for $Q=E_{8}$. Conversely, if some bad prime is not invertible in $R$, we show that the additive and multiplicative preprojective algebras differ in zeroth Hochschild homology, and hence are not isomorphic. In fact, one only needs the vanishing of certain classes in zeroth Hochschild homology of the multiplicative preprojective algebra, utilizing a rigidification argument for isomorphisms that may be of independent interest. In the setting of Ginzburg dg-algebras, our obstructions are new in type E and give a more elementary proof of the negative result of Etg\"{u}--Lekili arXiv:1502.07922 in type D. Moreover, the zeroth Hochschild homology of the multiplicative preprojective algebra can be interpreted as the space of unobstructed deformations of the multiplicative Ginzburg dg-algebra by Van den Bergh duality. Finally, we observe that the multiplicative preprojective algebra is not symmetric Frobenius if $Q \neq A_1$, a departure from the additive preprojective algebra in characteristic 2 for $Q = D_{2n}$, $n \geq 2$ and $Q =E_{7}$, $E_{8}$.