## Multi-Braided Creatures & Quantum Groups## by Micho Durdevich |

Introduction |

## IntroductionIn this lecture we are going to present basic ideas of a new diagrammatic formalism for (generalized) quantum groups. The theory goes far beyond the standard Hopf/braided-Hopf algebras and includes many important examples of quantum spaces equipped with a group-like structure, naturally emerging from non-commutative geometric considerations.As a highlighting motivating idea, we can mention a principal
feature distinguishing quantum spaces from classical spaces:
Quantum spaces can not be viewed as collections of points equipped with
an appropriate additional geometrical structure (as explained in detail,
in my introductory lecture on quantum geometry). In
most cases, true quantum spaces will have However, quantum spaces corresponding to standard quantum groups are always
This fact can be understood as a consequence of a too strong set of axioms imposed within the standard formulation. In particular, any group-like quantum object based on a simple algebra is not includable within the standard approach. Such things are clearly not acceptable from both conceptual and aestetic point of views--quantum analogs of groups should retain a similar kind of internal homogeneity and isotropy, as their classical counterparts. The mentioned inhomogeneity of standard quantum groups becomes transparent if
we consider the theory of locally trivial quantum principal bundles over classical
smooth manifolds, possessing compact quantum structure groups.
Surprisingly at a first sight, the classification problem of such structures reduces to
the classification of classical principal bundles, over the same base space, the
structure group of which is given by A geometrical explanation of the bundle reduction phenomenon is simple: Quantum groups are always interpretable as spaces consisting of two parts--classical and quantum (completely "pointless" and described by the ideal over all commutators). Now, if we try to constuct a locally trivial quantum principal bundle over a classical space, we have to introduce a system of "transition funcions" that would glue the fibers over the intersections of locally-trivializing regions. These transition functions preserve the structure of the fiber, and as a consequence they map the classical part into the classical part and the quantum part into the quantum part. In addition, the transition functions are covariant, with respect to the quantum group structure. It follows therefore that they are completely determined by their reductions on the corresponding classical part of the quantum structure group. The reduced transition functions give us the corresponding classical cocycle. Another geometrical indication that standard Hopf/braided-Hopf algebras are insufficient to
address the diversity of group-like quantum spaces comes from K-theory and cyclic
cohomology. If two C*-algebras are strongly Morita equivalent, they have
the same K-groups and cyclic cohomology groups. At the geometrical level, it means
that the underlying quantum spaces are very similar. If fact, the formulation of A Connes
implicitly assumes that strongly Morita-equivalent C*-algebras define exactly the same
geometrical objects.
From this point of view, it would be natural to expect that a proper geometrical formulation
of quantum groups is closed under Let us list some interesting examples of quantum objects that are naturally includable in the present formalsm. - General braided Clifford algebras (including standard Clifford and Weyl algebras);
- Quantum tori (both irrational and rational);
- Matrix algebras over arbitrary quantum groups. In other words, the theory is closed under tensoring with matrix algebras, and compact operators;
- Various algebras emerging from quantum and classical
*combinatorics*(as Hecke algebras and their approriate extensions, for example); - Gauge bundles associated to quantum principal bundles;
- Structures appearing when deforming the co-product/product or both, of a given quantum group (in other words, the theory is closed under the appropriate deformations);
- Cuntz algebras :)
- Various "pointless" objects naturally appearing in quantizing a classical group;
- Subquantum extensions of quantum observable algebras.
can not be viewed as standard Hopf algebras (including
Hopf algebras in braided categories).
The paper is organized in the following way: Our exposition begins by introducing the main We shall then consider some elementary consequences of the introduced
axioms. All considerations will be performed in terms of After presenting the basic diagrammatic setup, we shall discuss a very interesting
problematics of a possibility to play a In particular, as we shall see, there is always a canonical flip-over morphism (acting
in object An important feature of our theory is that, in contrary to the standard braided category formulation,
we do not postulate any braid equation, nor any a priori given type of a twisting propery.
All such properties come out as Finally, in the last section an alternative diagrammatic axiomatics is presented, and some concluding remarks are made. This lecture follows the formalism I developed some time ago, substantially generalizing it further to include various additional structures. [Next Segment]: Blueprint Category of Diagrams |