The finite dual of commutative-by-finite Hopf algebras
16T05
Rings and Algebras (math.RA)
Mathematics - Quantum Algebra
FOS: Mathematics
Quantum Algebra (math.QA)
Mathematics - Rings and Algebras
0101 mathematics
01 natural sciences
DOI:
10.1017/s0017089522000052
Publication Date:
2022-03-21T00:07:37Z
AUTHORS (3)
ABSTRACT
AbstractThe finite dual
$H^{\circ}$
of an affine commutative-by-finite Hopf algebra H is studied. Such a Hopf algebra H is an extension of an affine commutative Hopf algebra A by a finite dimensional Hopf algebra
$\overline{H}$
. The main theorem gives natural conditions under which
$H^{\circ}$
decomposes as a crossed or smash product of
$\overline{H}^{\ast}$
by the finite dual
$A^{\circ}$
of A. This decomposition is then further analysed using the Cartier–Gabriel–Kostant theorem to obtain component Hopf subalgebras of
$H^{\circ}$
mapping onto the classical components of
$A^{\circ}$
. The detailed consequences for a number of families of examples are then studied.
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