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I wonder if there is a name for:

1) Algebras which are Morita equivalent to their centers, or

2) dg-algebras which are derived Morita equivalent to their Hochshild cohomology?

For instance, (finite-dimensional?) semi-simple Frobenius algebras satisfy (1), as often do the smash products of Hopf algebras with commutative algebras. Item (1) also seems closely related to the defining property of an Azumaya algebra. I am less familiar with examples of (2).

The request for a name is just to help me search for examples, classification tools, and intuition in the literature, as (1) and (2) as I wrote them contain too many buzzwords for me to search effectively, it seems.

Anyone who can offer such insight directly will of course be greatly appreciated!

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If I were forced to neologise, «Morita-commutative» :) –  Mariano Suárez-Alvarez May 26 '12 at 23:32
Over an algebraically-closed field, the only finite dimensional examples of (1) are the semi-simple algebras, so I guess over a general field one gets only the separable ones, no? –  Mariano Suárez-Alvarez May 27 '12 at 0:41
@Mariano: Surely not! There are plenty of non-semisimple finite dimensional commutative algebras. Anything with a nilpotent element, for instance. –  Evan Jenkins May 27 '12 at 0:56
Actually, remove the "for instance"; these are precisely the non-semisimple finite dimensional commutative algebras. –  Evan Jenkins May 27 '12 at 1:02
@Theo: I believe that the Hochschild cohomology of k[x] / x^2 is one-dimensional in every positive degree when char k is not 2, so it would seem that Hochschild cohomology can definitely see non-semisimplicity. Also, your objection doesn't address the main question: even if the Hochschild cohomology (resp. center) can't detect a difference between two dg-algebras (resp. algebras), that has no bearing on whether either is derived Morita equivalent (resp. Morita equivalent) to its Hochschild cohomology (resp. center). –  Evan Jenkins May 27 '12 at 6:16

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