Multi-Braided Creatures & Quantum Groups

by Micho Durdevich

Table of Contents

Blueprint Category
Elementary Consequences
Canonical Twist & Beyond
Alternative Formulation


In 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 no points at all.

However, quantum spaces corresponding to standard quantum groups are always inherently inhomogeneous as there is always one classical point--corresponding to the "neutral element" represented by the counit map (indeed, taking the classical Gelfand-Neimark theory into accout, it it is natural to assume that "points" of a quantum space correspond to characters of the associated *-algebra, and in the standard theory of Hopf algebras the counit is always a character).

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 the set of points of the initial quantum group. This "classical part" of a compact quantum group is a standard compact group, in a natural manner.

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 stabilizations. In particular, a reasonable theory of quantum groups should be closed under tensoring with matrix algebras. Unfortunately, the standard theory is not closed under stabilizations.

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.
All these structures 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 diagrammatic category which can be viewed as a blueprint for all quantum groups. This is a monoidal category M generated by a single object (so that all objects are labeled by natural numbers) and two morphisms of type 1|2 and 2|1, that play the role of abstract coproduct and product. Because of its universal nature, calculations performed in M are automatically valid in any concrete realization. In other words "a true quantum group" in a monoidal category C is interpretable as a covariant functor from the blueprint category M with values in C.

We shall then consider some elementary consequences of the introduced axioms. All considerations will be performed in terms of diagrams-pictures representing morphisms in our category, and their algebraic interrelations. In particular, we shall introduce left and right "transfer operators" and prove the existence of analogs of the co-unit and the antipode. It is important to stress that in our formulation we do not require the existence of any "base field" object (no 0-object in our category). Comparing with the standard theory of Hopf algebras, this means that we would rather consider the composition of the counit and the unit, instead of dealing with them separately. This unified morphism will be called the co-unit.

After presenting the basic diagrammatic setup, we shall discuss a very interesting problematics of a possibility to play a flipping game: to introduce certain morphisms that allow us to transpose objects and morphisms, moving them from one position in a diagram to another.

In particular, as we shall see, there is always a canonical flip-over morphism (acting in object 2), which expresses the "multiplicativity property" of of the coproduct moprhism. If this flip-over morphism satisfies some additional conditions (invertibility and two nice octagonal diagrams) then it is possible introduce an infinite family of twist morphisms (indexed by pairs of independent integers) that allow us to express twisting properties of the product and the coproduct (and hence, of all other morphisms) in a concise and very elegant way. The initial flip-over morphism corresponds to the index (1|1). Furthermore, all these twist morphisms are mutually maximally compatible, in a braided sense. In particular, they all satisfy the braid equation. This justifies the name for our structures: Multi-Braided Quantum Groups.

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 theorems  in our approach. Our multi-braided formalism is much more general than the conceptual framework of braided categories, which can be undestood as a singular case of the theory when all twisting operators coincide, and the co-unit is recognized as a composite object.

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.

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