Classification of Bicovariant Differential Calculi on the Quantum Groups S L q (n+1) and S p q (2n)
Abstract
For transcendental values of
q
all bicovariant first order differential calculi on the coordinate Hopf algebras of the quantum groups
S
L
q
(n+1)
and
S
p
q
(2n)
are classified. It is shown that the irreducible bicovariant first order calculi are determined by an irreducible corepresentation of the quantum group and a complex number
ζ
such that
ζ
n+1
=1
for
S
L
q
(n+1)
and
ζ
2
=1
for
S
p
q
(2n)
. Any bicovariant calculus is inner and its quantum Lie algebra is generated by a central element. The main technical ingredient is a result of the Hopf algebra
R(
G
q
)
0
for arbitrary simple Lie algebras.