Twisted automorphisms of Hopf algebras

Twisted homomorphisms of bialgebras are bialgebra homomorphisms from the first into Drinfeld twistings of the second. They possess a composition operation extending composition of bialgebra homomorphisms. Gauge transformations of twists, compatible with adjacent homomorphisms, give rise to gauge transformation of twisted homomorphisms, which behave nicely with respect to compositions. Here we study (gauge classes of) twisted automorphisms of cocommutative Hopf algebras. After revising well-known relations between twists, twisted forms of bialgebras and $R$-matrices (for commutative bialgebras) we describe twisted automorphisms of universal enveloping algebras.

💡 Research Summary

The paper introduces the notion of a twisted homomorphism between bialgebras, which is a usual algebra homomorphism from a source bialgebra into a Drinfeld‑twisted version of a target bialgebra. By allowing the target to be equipped with a twist (a 2‑cocycle in the sense of Drinfeld), the authors obtain a composition law that extends ordinary composition of bialgebra maps: if (f\colon A\to B_F) and (g\colon B\to C_G) are twisted homomorphisms, then their composite is a twisted homomorphism (g\circ f\colon A\to C_{G\cdot(g\otimes g)(F)}). This construction makes the collection of twisted homomorphisms into a category that contains the ordinary category of bialgebras as a full subcategory.

A central technical tool is the gauge transformation of twists. Two twists (F) and (F’) on the same bialgebra are gauge equivalent if there exists an invertible element (u) satisfying the usual cocycle relation (F’=(u\otimes 1)(\Delta\otimes\mathrm{id})(u),F,(\mathrm{id}\otimes\Delta)(u^{-1})(1\otimes u^{-1})). The authors show that gauge transformations compatible with adjacent homomorphisms induce a well‑behaved transformation on the whole twisted homomorphism, preserving composition and identities. Consequently, twisted homomorphisms are considered up to gauge equivalence, leading to a notion of “gauge class of twisted automorphisms”.

The bulk of the work focuses on cocommutative Hopf algebras, i.e. group algebras and, more importantly, universal enveloping algebras (U(\mathfrak g)) of Lie algebras. In the cocommutative setting, twists are symmetric 2‑cocycles and give rise to symmetric (R)-matrices; the paper revisits the well‑known correspondence between twists, twisted forms of bialgebras, and (R)-matrices for commutative bialgebras, setting the stage for the analysis of automorphisms.

For a Lie algebra (\mathfrak g), any twist of (U(\mathfrak g)) can be written (formally) as (F=\exp(\hbar r)) with (r\in\Lambda^{2}\mathfrak g). The element (r) satisfies the classical Yang–Baxter equation precisely when (F) yields a new Hopf algebra structure on the same underlying algebra. The authors prove that twisted automorphisms of (U(\mathfrak g)) split naturally into two components:

1. Classical part – automorphisms induced by Lie algebra automorphisms (\phi\in\operatorname{Aut}(\mathfrak g)). These extend uniquely to Hopf algebra automorphisms of (U(\mathfrak g)) and act trivially on the twist parameter.

2. Internal twist part – transformations obtained by changing the twist by a coboundary. If (r) and (r’) differ by a Chevalley–Eilenberg coboundary (\delta\alpha) (with (\alpha\in\mathfrak g)), the corresponding twists are gauge equivalent. Hence the set of distinct internal twists is parametrised by the second Lie algebra cohomology group (H^{2}(\mathfrak g,k)).

Putting these together, the group of twisted automorphisms (modulo gauge) is identified as a semidirect product
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