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Bialgebroids (resp. Hopf algebroids) are bialgebras (Hopf algebras) overnoncommutative rings. Drinfeld twist techniques are particularly useful in the(deformation) quantization of Lie algebras as well as underlying modulealgebras (=quantum spaces). Smash product construction combines these two intothe new algebra which, in fact, does not depend on the twist. However, we canturn it into bialgebroid in the twist dependent way. Alternatively, one can useDrinfeld twist techniques in a category of bialgebroids. We show that bothtechniques indicated in the title: twisting of a bialgebroid or constructing abialgebroid from the twisted bialgebra give rise to the same result in the caseof normalized cocycle twist. This can be useful for better description of aquantum deformed phase space. We argue that within this bialgebroid frameworkone can justify the use of deformed coordinates (i.e. spacetimenoncommutativity) which are frequently postulated in order to explain quantumgravity effects.

Twisted bialgebroids versus bialgebroids from a Drinfeld twist