When is a Cleft Extension H-Azumaya?

Abstract

We show which H op -cleft extensions k# ' H op of k for a dual quasi-triangular Hopf algebra (H, r) are H-Azumaya. The result is given in terms of bijectivity of a map ' : H ! H* defined in terms of the universal r-form r and the 2-cocycle ', generalizing a well-known result for the commutative and co-commutative case. We illustrate the Theorem with an explicit computation for the Hopf algebras of type E(n). (2000): 16W30, 16H05, 16K50. Mathematics Subject Classifications

Extending lazy 2-cocycles on Hopf algebras and lifting projective representations afforded by them

“…Let us note that (2.14) and (2.15) appear also in [8], in a slightly different form, and they actually hold for any left 2-cocycle, not necessarily lazy. …”

“…Let σ : H ⊗H → k be a normalized and convolution invertible left 2-cocycle. It is well-known (see [17], [8]) that the following formulae hold:…”

“…Let us note that (2.14) and (2.15) appear also in [8], in a slightly different form, and they actually hold for any left 2-cocycle, not necessarily lazy. Proposition 2.6 Let σ be a normalized and convolution invertible left 2-cocycle on H. Then we have, for all h ∈ H:…”