On Hopf Algebras and Their Generalizations

Abstract

We survey Hopf algebras and their generalizations. In particular, we compare and contrast three well-studied generalizations (quasi-Hopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (Hopf monads and hopfish algebras). Each of these notions was originally introduced for a specific purpose within a particular context; our discussion favors applicability to the theory of dynamical quantum groups. Throughout the note, we provide several definitions and examples in order to make this exposition accessible to readers with differing backgrounds.

Key Words:

• Hopf algebra
• Hopf algebroid
• Hopfish algebra
• Hopf monad
• Quantum group
• Quasi-Hopf algebra
• Weak Hopf algebra

2000 Mathematics Subject Classification:

• Primary 16W30
• Secondary 17B37, 18D10, 20G42, 81R50

ACKNOWLEDGMENTS

The author would like to thank Marcelo Aguiar, Gabriella Böhm, Pavel Etingof, Ignacio Lopez Franco, Ludmil Hadjiivanov, Mitja Mastnak, Susan Montgomery, Frederic Patras, Fernando Souza, Ivan Todorov, Alexis Virelizier, Alan Weinstein, Milen Yakimov, and the anonymous referee, for helpful comments and useful questions during various stages of the work that led to this article.

Notes

¹In the introduction to Sweedler (1969), Sweedler remarks that this notation was developed during years of joint work with Robert Heyneman. However, it is much more common to see references to the Sweedler notation than to the Heyneman–Sweedler notation. We follow the tradition here.

²We are intentionally being vague, in order to include several classes of algebras here; if, for instance, G is an (affine) algebraic group, we could be thinking of the coordinate ring of G; if G is a Poisson group, we could be looking at the Poisson algebra C ^∞(G).

³Without additional conditions on G and F _(G), the range of the most natural comultiplication map defined as dual to the multiplication will not be contained in F _(G)⊗ F _(G), hence the need for this embedding.

⁴This may remind some readers the philosophy of noncommutative geometry a la Connes. A similar approach to quantum groups would involve viewing them as symmetry objects of some quantum space; see Manin (1988).

⁵More precisely, a quasitriangular Hopf algebra is a Hopf algebra H together with an invertible element  ∊ H ⊗ H such that:

1. Δ ^op (h) = Δ(h)⁻¹ for all h ∊ H;

2. (Δ ⊗ id)() = ₁₃₂₃ and (id ⊗ Δ)() = ₁₃₁₂.

A coquasitriangular Hopf algebra is defined in a dual manner, see Majid (1995) and Schauenburg (1992b).

⁶We presented the commutative diagram version of the counit axiom earlier; in closed form it reads:

⁷We note here that our definitions in this section are not quite as general as the original ones in Drinfeld (1989). Drinfeld's original definitions involved two invertible elements l and r, or equivalently a left unit constraint l and a right unit constraint r, satisfying the so-called Triangle Axiom. However, Drinfeld showed, also in Drinfeld (1989), that he could always reduce his quasi-Hopf algebras into quasi-bialgebras where r = l = 1. Therefore, we will not be worried much about our more restrictive definitions.

⁸Obviously the phrase “nice enough” needs to be explained for this vague sentence to become a precise (and correct) mathematical statement. For instance, for a Tannaka–Krein type construction (cf. Section 8), we need a fiber functor from our category into the category of vector spaces. We do not attempt this here.

⁹In this context, some authors prefer to use terms like “gauge transformation” or “skrooching” in place of the term “twist.”

¹⁰Note that the answer to the following question is not automatically clear and still unknown to the author: Given a monoidal category ℳ which is the representation category of a quasi-Hopf algebra, how nice is the strict monoidal category tensor equivalent to ℳ? For instance, it is known that ℳ′ will not always be the representation category of a Hopf algebra. (This comment is due to G.B., P.E., and the anonymous referee).

¹¹We should point out that there is an alternative use in the literature for the term weak Hopf algebra. Some authors use the term to refer to a bialgebra B with a weak antipode S in Hom(B, B), i.e., id∗S∗id = id and S∗id∗S = S, where ∗ is the convolution product; see for instance Li (1998) and Li and Duplij (2002). Wisbauer (2001) briefly discusses how the two concepts are related.

¹²We define only the finite-dimensional version here. In this context, it seems to be standard to restrict to finite dimensions because it is much easier to dualize the notion in that case. For a more detailed discussion of why finite-dimensionality is in general preferred, we refer the reader to Böhm et al. (1999).

¹³Defining the multiplication as a partial function allows us to avoid giving a precise domain for it. However, when it is relevant, the domain of m is called the set of composable pairs. Clearly, it is a subset of G × G.

¹⁴Even though the category of L-bimodules is not necessarily symmetric, it is still possible to define an isomorphism T: B ⊗ _L B → B ⊗ _{L ^op} B given by the flip.

¹⁵We require that the antipode S be a bijection for the sake of simplicity. For a study of Hopf algebroids with antipodes which are not necessarily bijective, see Böhm (2005b).

¹⁶We should note here that some of the above information in our definition is redundant. In fact one can start with a left bialgebroid ℋ _L  = (H, L, s _L , t _L , γ _L , π _L ) and an anti-isomorphism S of the total ring H satisfying certain conditions, and from here can reconstruct a right bialgebroid ℋ _R using the same total ring such that the triple (ℋ _L , ℋ _R , S) is a Hopf algebroid. See Böhm (2005a) for more details. An even more streamlined set of defining axioms is proposed in Böhm and Brzeziński (2006).

¹⁷It may be interesting to note here that duality for bialgebroids (over a noncommutative base) is much simpler, see Kadison and Szlachányi (2003).

¹⁸We note here that there is at least one other category-theoretic Hopf-like object studied in the recent years. We do not focus on the relevant constructions here; we simply refer the reader to Day et al. (2003) and Day and Street (2004), where the ideas are developed in great detail. Some connections to the picture from Section 6 may be found in Böhm and Szlachányi (2004) and Böhm (To appear).

Communicated by M. Cohen.