We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele in [A. Van Daele, Multiplier Hopf algebras, {\em Trans. Amer. Math. Soc.}, {\bf 342}(2), (1994) 917--932.] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly non-unital, idempotent, non-degenerate, $k$-projective) algebra over a commutative ring $k$ is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of $k$-modules, into a diagram of strict monoidal forgetful functors.
Deep Dive into Multiplier Hopf and bi-algebras.
We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele in [A. Van Daele, Multiplier Hopf algebras, {\em Trans. Amer. Math. Soc.}, {\bf 342}(2), (1994) 917–932.] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly non-unital, idempotent, non-degenerate, $k$-projective) algebra over a commutative ring $k$ is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of $k$-modules, into a diagram of strict monoidal forgetful functors.
A Hopf algebra over a commutative ring k is defined as a k-bialgebra, equipped with an antipode map. A k-bialgebra can be understood as a coalgebra (or comonoid) in the monoidal category of k-algebras; an antipode is the inverse of the identity map in the convolution algebra of k-endomorphisms of the k-bialgebra. From the module theoretic point of view, a kbialgebra can also be understood as a k-algebra that turns its category of left (or, equivalently, right) modules into a monoidal category, such that the forgetful functor to the category of k-modules is a strict monoidal functor.
During the last decades, many generalizations of and variations on the definition of a Hopf algebra have emerged in the literature, such as quasi-Hopf algebras [5], weak Hopf algebras [1], Hopf algebroids [2] and Hopf group (co)algebras [3]. For most of these notions, the above categorical and module theoretic interpretations remain valid in a certain form, and in some cases, this was exactly the motivation to introduce such a new Hopf-type algebraic structure.
Multiplier Hopf algebras were introduced by Van Daele in [10], motivated by the theory of (discrete) quantum groups. The initial data of a multiplier Hopf algebra are a non-unital algebra A, a so-called comultiplication map ∆ : A → M(A ⊗ A), where M(A ⊗ A) is the multiplier algebra of A ⊗ A, which is the “largest” unital algebra containing A ⊗ A as a twosided ideal, and certain bijective endomorphisms on A ⊗ A. It is well-known that the dual of a Hopf algebra is itself a Hopf algebra only if the original Hopf algebra is finitely generated and projective over its base ring. A very nice feature of the theory of multiplier Hopf algebras is that it lifts this duality to the infinite dimensional case. In particular, the (reduced) dual of a co-Frobenius Hopf algebra is a multiplier Hopf algebra, rather than a usual Hopf algebra.
From the module theoretic point of view, some aspects in the definition of a multiplier Hopf algebra remain however not completely clear. For example, a multiplier Hopf algebra is introduced without defining first an appropriate notion of a “multiplier bialgebra”. In particular, from the definition of a multiplier Hopf algebra a counit can be constructed, rather than being given as part of the initial data. Furthermore, a categorical characterization of multiplier Hopf algebras as in the classical and more general cases as mentioned before seems to be missing or at least unclear at the moment. In this paper we try to shed some light on this situation.
Our paper is organized as follows. In the first Section, we recall some notions related to non-unital algebras and non-unital modules. We repeat the construction of the multiplier algebra of a non-degenerate (idempotent) algebra, and show how this notion is related to extensions of non-unital algebras. We show how non-degenerate idempotent k-projective algebras constitute a monoidal category.
In the second Section we then introduce the notion of a multiplier bialgebra as a coalgebra in this monoidal category of non-degenerate idempotent k-projective algebras. We show that a non-degenerate idempotent k-projective algebra is a multiplier bialgebra if and only if the category of its extensions, as well as the categories of its left and right modules are monoidal and fit, together with the category of modules over the commutative base ring, into a diagram of strict monoidal forgetful functors (see Theorem 2.9). The main difference from the unital case is that monoidality of the category of left modules is not equivalent to monoidality of the category of right modules.
In the last Section, we recall the definition of a multiplier Hopf algebra by Van Daele. We show that a multiplier Hopf algebra is always a multiplier bialgebra and we give an interpretation of the antipode as a type of convolution inverse. We conclude the paper by providing a categorical way to introduce the notions of module algebra and comodule algebra over a multiplier bialgebra, which in the multiplier Hopf algebra case were studied in [12].
Notation. Throughout, let k be a commutative ring. All modules are over k and linear means k-linear. Unadorned tensor products are supposed to be over k. M k denotes the category of k-modules. By an algebra we mean a k-module A equipped with an associative klinear map µ A : A ⊗ A → A; it is not assumed to possess a unit. An algebra map between two algebras is a k-linear map that preserves the multiplication. A unital algebra is an algebra A with unit element 1 A . An algebra map between two unital algebras that maps the one unit element to the other, will be called a unital algebra map. A right A-module M is a k-module equipped with a k-linear map µ M,A : M ⊗ A → M such that the associativity condition
The k-module of right A-linear (resp. left B-linear, (B, A)-bilinear) maps between two right A-modules (resp. left B-modules, (B, A)bimodules) M and N is denoted by Hom A (M, N) (resp. B Hom(M, N),
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