On symmetric, smooth and Calabi–Yau algebras

One possible definition for a Calabi–Yau algebra is a symmetric smooth PI algebra. Our main purpose here is to prove some necessary and sufficient criteria for verifying the (local) symmetric property, in smooth PI algebras. Many known smooth PI algebras are shown to have this property. In particular quantum enveloping algebras of complex semi-simple Lie algebras, in the root of unity case and the enveloping algebra of sl(n) with (p,n)=1, in characteristic p, are typical examples. A surprising result is that the inj.dimT, is finite, where T is the trace ring of m, n×n generic matrices over a field of zero characteristic.