Elsevier

Journal of Algebra

Volume 586, 15 November 2021, Pages 607-642
Journal of Algebra

Skew braces as remnants of co-quasitriangular Hopf algebras in SupLat

https://doi.org/10.1016/j.jalgebra.2021.07.006Get rights and content

Abstract

Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, SupLat, of complete lattices and join-preserving morphisms. We connect the two methods by showing that any Hopf algebra, H in SupLat, has a corresponding group, R(H), which we call its remnant and a co-quasitriangular structure on H induces a YBE solution on R(H), which is compatible with its group structure. Conversely, any group with a compatible YBE solution can be realised in this way. Additionally, it is well-known that any such group has an induced secondary group structure, making it a skew left brace. By realising the group as the remnant of a co-quasitriangular Hopf algebra, H, this secondary group structure appears as the projection of the transmutation of H. Finally, for any YBE solution, we obtain a SupLat-FRT Hopf algebra in SupLat, whose remnant recovers the universal skew brace of the solution.

Introduction

Originally appearing in statistical mechanics [35], the Yang-Baxter equation and its solutions play a fundamental role in the theory of quantum groups [7], [8], [14], [15], braided categories and knot theory [16]. The equation is best known in its linear form, but in [9] Drinfeld proposed the open problem of classifying solutions of the Yang-Baxter equation over sets. A set X together with a map r:X×XX×X is called a set-theoretical solution to the Yang-Baxter equation (YBE) ifr12r23r12=r23r12r23 holds, where rij:X×X×XX×X×X are the applications of r to the i and j-th components of X3 i.e. r23=id×r. These objects have garnered large interest due their interactions with combinatorics [12], ring theory [32], [33] and their applications to knot theory [4]. More recently, the work of Rump on involutive YBE solutions [29], [28] inspired Guarnieri and Vendramin to develop of the theory of skew braces, which are sets with two compatible group structures [13]. In particular, any set-theoretical YBE solution has a corresponding universal skew brace, allowing us to classify set-theoretical YBE solutions, by first classifying such algebraic structures.

Now, when we look back at linear YBE solutions on vector spaces, there is a well established relation between these solutions and (co-)quasitriangular Hopf algebras. (Co-)quasitriangular Hopf algebras and bialgebras provide solutions of YBE via their (co-)representation theory. Conversely, the Fadeev-Reshetikhin-Takhtajan (FRT) construction produces such a bialgebra from any linear YBE solution [11]. From a categorical point of view, the FRT bialgebra and its co-quasitriangular structure can also be obtained by Tannaka-Krein reconstruction, from the braided category which the YBE solution generates. If the YBE solution is bi-invertible, it generates a rigid braided category and the same formalism can be applied to obtain a Hopf algebra [30]. Hence, it is natural to ask whether skew braces can be viewed as Hopf algebras in a suitable category related to sets. In this work, we show that the correct category to consider is that of complete lattices and join preserving morphisms, SupLat, and construct skew braces from co-quasitriangular Hopf algebras in this category and vice-versa.

A suitable categorical framework to study set-theoretical YBE solutions should allow us to (a) apply the usual categorical Hopf algebra techniques to obtain new skew braces, (b) use a similar Tannaka-Krein reconstruction and recover the universal skew brace and (c) explain the nature of the two products on a skew brace and their interaction, which has been subject to several studies already. Studying Hopf algebras in the category of sets itself fails for our purposes because of two key reasons:

  • (A)

    Hopf algebras in the category of sets and functions, Set, are groups and one can easily check that any (co-)quasitriangular structure on a group must be trivial. Hence, we can not obtain YBE solutions by looking at (co-)modules over a group in Set.

  • (B)

    The key ingredient for obtaining a Hopf algebra by Tannaka-Krein reconstruction requires the underlying object of the YBE solution to be dualizable, while the only dualizable object in Set is the set of one element.

The first naive solution is to look at the category of sets and relations, Rel, where every set has itself as a dual, making the category rigid. However, Rel is not cocomplete and the colimit needed for Tannaka-Krein reconstruction, (5), will not exist. The second naive solution is to move into the cocompletion of Rel, namely [Relop,Set], via the Yoneda embedding. But this category is rather large and the Hopf algebras constructed will not be very intuitive. Instead, we remedy these issues by embedding the category of sets into the category of complete lattices and join-preserving morphisms, SupLat, via the power-set functor: In particular, all dualizable objects in SupLat are of the form P(X), for a set X, Lemma 2.4, and YBE solutions on them correspond to set-theoretical YBE solutions. The other major benefit of working in SupLat, is that we can formulate a deeper connection between co-quasitriangular Hopf algebras in this category and groups with braiding operators, which are groups with a compatible YBE solutions on their underlying sets, see (8) and (9).

Our results can be summarised as follows: Given a Hopf algebra structure on a complete lattice H in SupLat, we can form a new Hopf algebra by quotienting out the “kernel” of the counit, Lemma 3.5. The counit of this new Hopf algebra will send all non-trivial elements to 1P(1) and in Lemma 3.6, we show that this condition is equivalent to the Hopf algebra being the “group algebra”, see Example (3.2), of a group. Hence, this process provides a corresponding group for every Hopf algebra in SupLat, which we call its remnant and denote by R(H).

It is well-known that the multiplication of a co-quasitriangular Hopf algebra is braided-commutative with respect to a naturally induced braiding, (19), on the Hopf algebra. Hence, we demonstrate that given a co-quasitriangular structure on H, the induced braiding of H restricts to a braiding operator on its remnant, Theorem 4.2. Additionally, any group with a braiding operator possesses a secondary group structure on the same set which makes it a skew brace. On the other hand, Majid has shown that any co-quasitriangular Hopf algebra has an induced secondary multiplication and an antipode which provide it with a braided Hopf algebra structure, called its transmutation, in its category of comodules [24]. A corollary of our work is that the secondary group structure on the remnant agrees with the projection of the transmuted multiplication of H, Theorem 4.3. Finally, in Section 5.2, we show that any skew brace can be recovered as the remnant of a co-quasitriangular Hopf algebra in SupLat.

In Section 5.1, we use Tannaka-Krein duality in SupLat to construct a unique co-quasitriangular Hopf algebra for any set-theoretical YBE-solution. We call this construction the SupLat-FRT Hopf algebra of the solution. We then recover the universal skew brace of the solution as the remnant of this Hopf algebra, Theorem 5.1. For this construction, we require the YBE solution to be dualisable in SupLat, or equivalently non-degenerate, see Section 2.1. Hence, throughout our work a set-theoretical YBE solution will refer to a non-degenerate solution. In Section 5.3, we compare our SupLat-FRT algebras and the linear FRT (Hopf) bialgebras coming from the linearisation of set-theoretical YBE solutions.

Similar ideas were discussed in [19], [20], where Hopf algebras in Rel are shown to correspond to groups with unique factorisations, G=G+G, and quasitriangular structures on them are classified. In SupLat, this work translates into the classification of Hopf algebra structures on free lattices i.e. P(X) for a set X. In Section 4.1, we review these results and discuss how parts of their proofs can be eased by our work. In [19], the authors observe that a quasitriangular structure on these Hopf algebras induces a YBE solution on G. In Theorem 4.7, we show that this result is a particular instance of our work. Moreover, in [18] the authors describe the properties of the universal skew brace of a set-theoretical solution by taking inspiration from [19], [20], but do not directly connect the works. By providing the correct categorical setting here, we are able to present a single machinery which captures both constructions.

Appendix A provides additional details on a natural bijection, l:XX, which is induced on any non-degenerate set-theoretical YBE solution (X,r). This bijection appears when we view (X,r) as a dualizable object in Rel and again in the transmutation of the SupLat-FRT Hopf algebras.

Outline of Paper. The first aspect of this paper is to describe co-quasitriangular Hopf algebras in SupLat and the induced skew brace structure on their remnant groups. In Sections 2.2 and 2.3, we briefly recall the basics of skew braces and the category SupLat, respectively and then construct the remnant of an arbitrary Hopf algebra in SupLat in Section 3. The reader should note that in Section 3, we present the structure of Hopf algebras in SupLat as definitions, so that previous knowledge of Hopf algebras is not essential for reading the article. The main results relating co-quasitriangular Hopf algebras and skew braces appear in Section 4. Our second main result is the SupLat-FRT construction in Section 5, where we associate a co-quasitriangular Hopf algebra in SupLat for any set-theoretical YBE solution and recover the universal skew brace of the solution. While we briefly recall the necessary background on Tannaka-Krein reconstruction in Section 2.1, we are forced to assume familiarity with the notion of Hopf algebras in symmetric monoidal categories in these sections and refer to Chapter 9 of [26] for additional details. Finally, we provide an overview of future research directions in Section 6.

Acknowledgments. The author would like to thank Shahn Majid, for many helpful discussions on the topic, as well as the referee for fruitful suggestions on the content and organisation of the article and pointing out several relevant references.

Section snippets

Preliminaries

In this section, we recall the necessary background on the construction of Hopf algebras from YBE solutions in a general symmetric monoidal category, the theory of skew braces and the category SupLat.

Notation. All Hopf algebras considered in this work, will have invertible antipodes, and as noted later, all YBE solutions considered are assumed to be non-degenerate. We will freely use either m and . to denote the multiplication operation, unless otherwise stated. We will denote the elements of

Hopf algebras in SupLat

In this section, we introduce the remnant group of a Hopf algebra in SupLat and show that any co-quasitriangular Hopf algebra gives rise to a skew brace.

As pointed out earlier, for a lattice L, the lattice LL can be viewed as a quotient of P(L×L). Hence, any element of can be written as iI{(li,li)} for some elements li,liL. Therefore, we will use the shorter notation iI(li,li), instead of iI{(li,li)}, unless otherwise stated.

Co-quasitriangular Hopf algebras and skew braces

In this section, we show that the remnant of a co-quasitriangular Hopf algebra has an induced braiding operator. Consequently, we show that the secondary product ⋆ on a such a group agrees with the restriction of the transmuted product of the Hopf algebra to its remnant.

Before we prove our results, we review the definitions of co-quasitriangular structures on Hopf algebras in SupLat and present Majid's transmutation theory in this case. Recall from the last section, that we are simply

SupLat-FRT reconstruction

In this section, we use the Tannaka-Krein formalism formulated in Section 2.1, first to recover the universal group of a set-theoretical YBE solution, Theorem 5.1. Secondly, to obtain a co-quasitriangular Hopf algebra for every group with a braiding operator, whose remnant recovers the group and the operator, Theorem 5.2. In Section 5.3 we compare our SupLat-FRT Hopf algebras with the linear FRT bialgebras and their Hopf extension coming from set-theoretical YBE solutions.

Outlook on future work

Having established the correct setting to use Hopf algebraic techniques to study YBE set-theoretical solutions, there are several directions in which we can further develop this theory.

On one hand, this setting allows us to produce new skew braces from old ones, using the machinery of Hopf algebras:

  • (I)

    Given a co-quasitriangular Hopf algebra (CQHA), one can discuss Drinfeld co-twists on this structure, which produce new CQHA structures on the same lattice. Furthermore, co-twists on the SupLat-FRT

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