COSET SPACES FOR QUANTUM GROUPS
Consideration of symmetries often simplifies
problems in physics and geometry. Quantum groups
are analogues of groups, and they
can describe a novel kind of symmetry.
We view them as objects of noncommutative geometry.
They may act on algebras of noncommutative or quantum
observables.
We propose a theory of coset spaces
for quantum groups in the language of coactions of
Hopf algebras and analyse an example which should
be thought as a quantum group analogue of the flag
variety.
In the classical case, the flag variety is the coset space
of the special linear group modulo its Borel subgroup of
lower triangular matrices.
We introduce and study a notion of localized coinvariants;
the quantum group coset space is viewed as
a system of algebras of localized coinvariants,
equipped with a quantum version of the locally trivial
principal bundle where the total space is described
by the quantum special linear group and
the base space is described by the
system of algebras of localized coinvariants.
We use quasideterminants,
the commutation relations between the quantum minors
and the noncommutative Gauss decomposition to formulate
and prove the main results.
We apply our axiomatization of quantum group
fibre bundles to obtain a generalization
of a concept of Perelomov coherent states to the Hopf
algebra setting and obtain the corresponding
resolution of unity formula.