The magic bimodule J

A tilted algebra B has global dimension at most 2.
(This means: Ext3 vanishes, but Ext2 usually does not vanish.)

Consider J = Ext2(DB,B).
(B = BB is the left regular representation, whereas DB is the dual of the right regular representation).

We consider the semi-direct extension C(B) of B by J.
This algebra C(B) is called the cluster tilted algebra corresponding to B.

These cluster tilted algebras have been introduced by Buan-Marsh-Reiten
(in a different, but equivalent way);
the description given here is due to Zhu and Assem-Bruestle-Schiffler.


In order to construct the semi-direct extension C(B) of B by J, one first considers the following matrix ring
and takes as C(B) the subring of all elements of the form
with b in B and x in J.