Multiplicative preprojective algebras of Dynkin quivers |
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Affiliation: | 1. Sistiana Mare 56, 34011 Trieste, Italy;2. Departamento de Matemáticas, Universidad de La Serena, Cisternas 1200, 1700000 La Serena, Chile;3. IMUV, Universidad de Valparaíso, Gran Bretaña 1111, 2340000 Valparaíso, Chile;1. Institute of Mathematics, Czech Academy of Sciences, ?itná 25, 115 67, Prague, Czech Republic;2. Tallinn University of Technology, Akadeemia tee 21b, 12618, Tallinn, Estonia |
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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 , 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 , and 2, 3 and 5 for – are invertible in R. We construct an explicit isomorphism over in type D, over for , and over for . 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 Etgü–Lekili [5, Theorem 13] in type D. Moreover, the zeroth Hochschild homology of the multiplicative preprojective algebra, computed in Section 4, 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 , a departure from the additive preprojective algebra in characteristic 2 for , and , . |
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Keywords: | Preprojective algebra Hochschild homology Ginzburg dg algebra Frobenius structure |
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