Twisted bialgebroids versus bialgebroids from a Drinfeld twist

TL;DR

This paper compares two methods of constructing bialgebroids using Drinfeld twists, showing they produce equivalent results, which aids in describing quantum deformed phase spaces and spacetime noncommutativity in quantum gravity.

Contribution

It demonstrates the equivalence of twisting a bialgebroid and constructing a bialgebroid from a twisted bialgebra using normalized cocycle twists.

Findings

Both techniques yield the same bialgebroid structure.

The approach helps justify deformed spacetime coordinates.

Facilitates better description of quantum phase spaces.

Abstract

Bialgebroids (resp. 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 underlying module algebras (=quantum spaces). Smash product construction combines these two into the new algebra which, in fact, does not depend on the twist. However, we can turn it into bialgebroid in the twist dependent way. Alternatively, one can use Drinfeld twist techniques in a category of bialgebroids. We show that both techniques indicated in the title: twisting of a bialgebroid or constructing a bialgebroid from the twisted bialgebra give rise to the same result in the case of normalized cocycle twist. This can be useful for 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.