Multi-Braided Creatures & Quantum Groups

Table of Contents

Introduction
Blueprint Category
Elementary Consequences
Canonical Twist & Beyond
Alternative Formulation

Variations & Concluding Remarks

We can obtain the same theory by using a slightly different set of axioms, going somewhat backwards. As in our main formulation, we can start from a monoidal category over natural numbers, with generating morphisms
Alternative Generating Morphisms
and assume
  • Associativity & Coassociativity;
  • Convolution Property;
  • Standard Jumping Property of Co-unit;
  • Strange Jumping Property:

    Jumping Property

In this formulation we do not have to postulate the cancellation property. It follows as a simple theorem, from the jumping property of the co-unit.

Indeed, the jumping property allows us to introduce the twist operator as in the previous section. Using the twisting operator, and performing some elementary transformations, it is easy to see that the co-unit acts trivially on the left and on the right, on any morphism in our category. This immediately implies the cancellation law.

It is easy to see that the circlular 1|1 morphisms naturally give rise to our previously introduced elliptic 2|2 morphisms

Defining Elliptic Morphisms

so that we have

Composition Law

and in particular

Closing the Circle

which closes the circle, establishing the equivalence with the first formulation, presented in Section 2. In conclusion, we can define multi-braided quantum groups as certain "images" of this game:

DEFINITION: Let C be an arbitrary monoidal category. A multi-braided quantum group (multi-braided Hopf algebra) in C is any covariant functor from M2 to C.

It is worth mentioning that octagonal conditions we introduced in the previous section are equivalent to the co/associativity of the canonical co/product maps, that we can naturally define on the composite object 2.

The formulation we presented essentially relies on the fact that both product and coproduct are co/associative morphisms. This assumption is especially important in considerations involving twist operators, and various jumping properties.

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