Some Bialgebroids Constructed by Kadison and Connes–Moscovici are Isomorphic

Abstract

We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-Rsmash product over a Hopf algebra is isomorphic to a diagonal crossed product.

Pre-torsors and equivalences

“…In the first one a × B -Hopf algebra corresponding to a torsor coming from a cleft extension by a Hopf algebroid is computed. This turns out to be a generalisation of bialgebroids studied in [11] and [17] (and shown to be mutually isomorphic in [23]). In the second appendix we describe differential structures and differentiable bimodules associated to unital faithfully flat pre-torsors.…”

On anchored Lie algebras and the Connes-Moscovici's bialgebroid construction

“…and [5,Theorem 4.1]. (f) Connes-Moscovici's bialgebroid (see [7,31] and [4,Example 3.4.6]). Let H be a Hopf algebra (in fact, a bialgebra would suffice) and let A be an H-module algebra.…”

“…Following the foregoing and, in particular, in view of the results in [31], two observations were made, that triggered the present investigation: (i)that whenever a Lie algebra L acts by derivations on an associative algebra A (for the sake of simplicity, let us call it an A-anchored Lie algebra), then A becomes naturally an U k (L)-module algebra and (ii)that 2010 Mathematics Subject Classification. Primary: 16B50; 16S10; 16S30; 16T15; 16W25; 18A40.…”

“…This procedure has been introduced independently, and under different forms, by Connes and Moscovici [7] in their study of the index theory of transversely elliptic operators, and by Kadison [19] in connection with his work on (pseudo-)Galois extensions. Later, Panaite and Van Oystaeyen proved in [31] that the two constructions were in fact equivalent (isomorphic as A-bialgebroids) and that, as algebras, they were particular instances of the L-R-smash product introduced in [30]. Nevertheless, by following [4], we will refer to the bialgebroid A ⊙ H ⊙ A as the Connes-Moscovici's bialgebroid.…”

On the Semiprime Smash Product Question

“…It has been show in [30] (see also [15,22,47]) that A ⊗ A op ⊗ H becomes an associative unital algebra with this multiplication, which I denote by A e ⊲⊳ H, where A e = A ⊗ A op stands for the enveloping algebra of A. Note that if H is cocommutative, then A e becomes a left H-module algebra and A e ⊲⊳ H = A e #H is the ordinary smash product.…”