Bialgebra actions, twists, and universal deformation formulas

Abstract

We introduce the concept of a twisting element based on a bialgebra and show how it can be used to twist a large class of algebras, coalgebras and certain subcategories of their respective module and comodule categories. We prove that this subcategory of modules over the original algebra is equivalent to the corresponding category of modules over the twisted algebra. The relation between twisting elements and universal deformation formulas is also given, along with new formulas which are associated to enveloping algebras of non-abelian Lie algebras.

Noncommutative Instantons from Twisted Conformal Symmetries

“…Given a right C ∞ (S 4 θ )-module E, its dual module is defined by 21) and is naturally a left C ∞ (S 4 θ )-module. In the case that E is also a left…”

“…Twisting of algebras and coalgebras has been known for some time [18,19,21]. The twists relevant for toric noncommutative manifolds are associated to the Cartan subalgebra of a Lie algebra and were already introduced in [31].…”

Twists of quantum groups and noncommutative field theory

“…One can check that with the deformed coproduct (15) these CR are invariant under the action of M µν [3]. One has to mention that the interpretation of the Weyl -Moyal product using abelian twist has been known for some time in the deformation quantization (see [17,18] and references therein).…”

Universal Deformation Formulas

“…In this section we briefly recall some constructions of [6,7]. Let B = (B, · , ∆, 1, ǫ) be an unital and counital associative and coassociative bialgebra over the ground field k, with the multiplication · : B ⊗ B → B, comultiplication ∆ : B → B ⊗ B, unit 1 ∈ B and counit ǫ : B → k. It is a standard fact that the tensor product 2 The following definition, as well as Theorem 2.2, can be found in [7].…”

“…We start this section by recalling some notions of [7] adapted to deformations over an arbitrary local complete Noetherian ring R = (R, I) with the maximal ideal I and residue field k = R/I. For terminology and basic definitions related to deformations over R we refer e.g.…”