Twisted bialgebroids versus bialgebroids from a Drinfeld twist

Abstract

Bialgebroids (respectively Hopf algebroids) are bialgebras (Hopf algebras) over noncommutative rings. Drinfeld twist techniques are particularly useful in the (deformation) quantization of Lie algebras as well as the underlying module algebras (=quantum spaces). A smash product construction combines both of them into the new algebra which, in fact, does not depend on the twist. However, we can turn it into a bialgebroid in a twist-dependent way. Alternatively, one can use Drinfeld twist techniques in a category of bialgebroids. We show that both the techniques indicated in the title—the twisting of a bialgebroid or constructing a bialgebroid from the twisted bialgebra—give rise to the same result in the case of a normalized cocycle twist. This can be useful for the better description of a quantum deformed phase space. We argue that within this bialgebroid framework one can justify the use of deformed coordinates (i.e. spacetime noncommutativity), which are frequently postulated in order to explain quantum gravity effects.