A Brief Introduction to Quantum Geometry

Quantum Groups and Quantum Bundles

In accordance with our basic dictionary, it is natural to assume that the group structure on a quantum space G is described by a *-homomorhism such that the diagram

is commutative, and such that

In such a way we arrive to compact quantum groups. As a very important special class of compact quantum groups, let us mention matrix groups. These structures are specified by a C*-algebra A, together with a *-homomorphism and a matrix such that the *-algebra generated by the entries is everywhere dense in the C*-algebra A, and such that

As a basic example of a compact quantum group, let us mention a quantum version of the SU(2) group [W3]. By definition the corresponding C*-algebra A is generated by elements and , and relations

where . The coproduct is specified by the above matrix rule

It is easy to see that the defining relations for A are actually equivalent to the unitarity of the above matrix.

Quantum groups provide a conceptual framework for generalizing the classical concept of symmetry. Indeed, in classical geometry, symmetries of a quantum space X are interpretable as automotphisms of the associated algebra A. This is straghtforwardly generalizable to the quantum level--we can define symmetries of a quantum space as automorphisms of the associated non-commutative algebra A. So symmetries always form a subgroup of the automorphism group Aut(A). Another way of incorporating the idea of symmetry is to generalize the concept of the group action rather than the one of the individual symmetries. In such a way we arrive to the concept of an action of a quantum group on a quantum space.

A fundamental class of quantum spaces possessing a built-in quantum group symmetry is given by quantum principal bundles. These objects are quantum counterparts of classical principal bundles. Quantum groups play the role of structure groups and general quantum spaces play the role of the base manifolds--the main geometrical idea is the same as in the classical theory:

If the quantum principal bundle and the structure group are represented by C*-algebras B and A respectively, then the right action of G on P is represented by a *-homomorphism , such that the diagram

is commutative. The classical condition that the structure group is acting freely on the bundle is expressed as the density condition

The above diagram corresponds to the requirement that the structure group is really "acting" on the bundle.

The base manifold M is described by the F-fixed point subalgebra of B. Geometrically, this means that the functions on the base are just the functions on the bundle, constant along the action orbits.

As a quantum object, the structure group G is not understandable as a collection of elements. Therefore the action F is not reducible to a collection of single symmetry transformations. In other words the action of the whole quantum group is considered as a quantum symmetry (in the commutative case, F contains the information about all possible transformations of P induced by the elements of G).