This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Introduction to lie algebras and representation theory

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

LetH be a Hopf algebra over a field with bijective antipode,A a rightH-comodule algebra,B the subalgebra ofH-coinvariant elements and can:A ⊗
 B
 A →A ⊗H the canonical map. ThenA is a faithfully flat (as left or rightB-module) Hopf Galois extension iffA is coflat asH-comodule and can is surjective (Theorem I). This generalizes results on affine quotients of affine schemes by Oberst and Cline, Parshall and Scott to the case of non-commutative algebras. The dual of Theorem I holds and generalizes results of Gabriel on quotients of formal schemes to the case of non-cocommutative coalgebras (Theorem II). Furthermore, in the dual situation, a normal basis theorem is proved (Theorem III) generalizing results of Oberst-Schneider, Radford and Takeuchi.

Principal homogeneous spaces for arbitrary Hopf algebras

Basic Algebraic Geometry

We determine the isomorphic classes of 6 or 8 dimensional semisimple Hopf algebrasA over an algebraically closed field such that (dimA)1≠0.

Semisimple Hopf algebras of dimension 6, 8

Self-Dual Hopf Algebras of Dimension p3 Obtained by Extension

Let f, g be finite-dimensional Lie algebras over a field of characteristic zero. Regard f and g∗, the dual Lie coalgebra of g, as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair (f, g∗) of Lie bialgebras is given, which has structure maps ⇀,ρ. Then it induces a matched pair (U f, Ug◦,⇀′, ρ′) of Hopf algebras, where U f is the universal envelope of f and Ug◦ is the Hopf dual of Ug. We show that the group Opext(U f, Ug◦) of cleft Hopf algebra extensions associated with (U f, Ug◦,⇀′, ρ′) is naturally isomorphic to the group Opext(f, g∗) of Lie bialgebra extensions associated with (f, g∗,⇀, ρ). An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If g = [g, g], there follows a bijection between the set Ext(U f, Ug◦) of all cleft Hopf algebra extensions of U f by Ug◦ and the set Ext(f, g∗) of all Lie bialgebra extensions of f by g∗.

/pdf/extensions-of-hopf-algebras-and-lie-bialgebras-2vrp5ogjyu.pdf

Akira Masuoka

Papers

Semisimple Hopf algebras of dimension 6, 8

Self-Dual Hopf Algebras of Dimension p3 Obtained by Extension

Extensions of Hopf Algebras and Lie Bialgebras

Abelian and non-abelian second cohomologies of quantized enveloping algebras

Semisimple hopf algebras of dimension 2p