Hopf Monoids in Varieties
نویسنده
چکیده
We show that entropic varieties provide the canonical setting for a generaliziation of Hopf algebra theory. In particular we show that all naturally occurring functors in this context have the expected adjoints, when generalized to this level. In particular, universal measuring comonoids exist over every entropic variety, as do generalized group algebras, and these carry a canonical Hopf structure. This is done partly by specializing more general results from category theory, and partly by generalizing classical and recent results about Hopf algebras over modules. As a byproduct a list of questions concerning properties of the monoidal structure of an entropic variety is produced, which may be of interest more generally. MSC 2010: Primary 08B99, Secondary 16T05
منابع مشابه
Hopf Monoids in semi-additive Varieties
We study Hopf monoids in entropic semi-additive varieties (equivalently, entropic Jónsson-Tarski varieties and categories of semimodules over a commutative semiring, respectively) with an emphasis on adjunctions related to the enveloping monoid functor and the primitive element functor. These investigations are based on the concept of the abelian core of a semi-additive variety variety and its ...
متن کاملEntropic Hopf algebras
The concept of a Hopf algebra originated in topology. Classically, Hopf algebras are defined on the basis of unital modules over commutative, unital rings. The intention of the present work is to study Hopf algebra formalism (§1.2) from a universal-algebraic point of view, within the context of entropic varieties. In an entropic variety, the operations of each algebra are homomorphisms, and ten...
متن کاملWeak Hopf Monoids in Braided Monoidal Categories
We develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R ⊗ R.
متن کاملWeak Hopf Algebras and Singular Solutions of Quantum Yang-baxter Equation
We investigate a generalization of Hopf algebra slq (2) by weakening the invertibility of the generator K, i.e. exchanging its invertibility KK = 1 to the regularity KKK = K. This leads to a weak Hopf algebra wslq (2) and a J-weak Hopf algebra vslq (2) which are studied in detail. It is shown that the monoids of group-like elements of wslq (2) and vslq (2) are regular monoids, which supports th...
متن کاملHopf monoids from class functionson unitriangular matrices
We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal’s category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatori...
متن کامل