In this article -that has also the intention to survey some known results in the theory of compact quantum groups using methods different from the standard and with a strong algebraic flavor- we consider compact o-coalgebras and Hopf algebras. In the case of a o-Hopf algebra we present a proof of the characterization of the compactness in terms of the existence of a positive definite integral, and use our methods to give an elementary proof of the uniqueness - up to conjugation by an automorphism of Hopf algebras- of the compact involution appearing in [4]. We study the basic properties of the positive square root of the antipode square that is a Hopf algebra automorphism that we call the positive antipode. We use it -as well as the unitary antipode and the Nakayama automorphism- in order to enhance our understanding of the antipode itself.
We develop an algebraic theory of colored, semigrouplike-flavored and pathlike co-, bi- and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in combinatorics, topology, number theory and physics. In particular, we can precisely give conditions for the invertibility of characters that is needed for renormalization in the formulation of Connes and Kreimer. These are met in the relevant examples. In order to construct antipodes, we discuss formal localization constructions and quantum deformations. These allow to define and explain the ap…
For an algebraic compact quantum group $H$ we establish a bijection between the set of right coideal $*$-subalgebras $A\to H$ and that of left module quotient $*$-coalgebras $H\to C$. It turns out that the inclusion $A\to H$ always splits as a map of right $A$-modules and right $H$-comodules, and the resulting expectation $E:H\to A$ is positive (and lifts to a positive map on the full $C^*$ completion on $H$) if and only if $A$ is invariant under the squared antipode of $H$. The proof proceeds by Tannaka-reconstructing the coalgebra $C$ corresponding to $A\to H$ by means of a fiber functor fro…
We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same cod…
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the uni…
Any simplicial Hopf algebra involves $2n$ different projections between the Hopf algebras $H_n,H_{n-1}$ for each $n \geq 1$. The word projection, here meaning a tuple $\partial \colon H_{n} \to H_{n-1}$ and $i \colon H_{n-1} \to H_{n}$ of Hopf algebra morphisms, such that $\partial \, i = \mathrm{id}$. Given a Hopf algebra projection $(\partial \colon I \to H,i)$ in a braided monoidal category $\mathfrak{C}$, one can obtain a new Hopf algebra structure living in the category of Yetter-Drinfeld modules over $H$, due to Radford's theorem. The underlying set of this Hopf algebra is obtained by an…
In our previous publications we have introduced analogs of partial derivatives on the algebras U(gl(N)). In the present paper we compare two methods of introducing these analogs: via the so-called quantum doubles and by means of a coalgebraic structure. In the case N=2 we extend the quantum partial derivatives from U(u(2)) (the compact form of the algebra U(gl(2))) on a bigger algebra, constructed in two steps. First, we define the derivatives on a central extension of this algebra, then we prolongate them on some elements of the corresponding skew-field by using the Cayley-Hamilton identities…
We prove a number of structural and representation-theoretic results on linearly reductive quantum groups, i.e. objects dual to that of cosemisimple Hopf algebras: (a) a closed normal quantum subgroup is automatically linearly reductive if its squared antipode leaves invariant each simple subcoalgebra of the underlying Hopf algebra; (b) for a normal embedding $\mathbb{H}\trianglelefteq \mathbb{G}$ there is a Clifford-style correspondence between two equivalence relations on irreducible $\mathbb{G}$- and, respectively, $\mathbb{H}$-representations; and (c) given an embedding $\mathbb{H}\le \mat…
Towards the Baum-Connes' Analytical Assembly Map for the Actions of Discrete Quantum Groups
给定一个离散量子群的作用（在Van Daele，Kustermans和Effros-Ruan的意义下）${\cal A}$在$C^*$ -代数${\cal C}$上，满足类似于经典离散群$Γ$在某个空间上的真$Γ$ -紧作用的正则性假设，我们能够构造出规范映射$μ^i_r$（\$μ_i $ respectively) ($ i=0，1 $) from the $ { \cal A} $-equivariant K-homology groups $ KK_i^{ \cal A}（{ \cal C}，C|）$ to the K-theory groups $ K_i（\hat{{\cal A}_r}）$ ($ K_i（\hat{\cal A}）$ respectively), where $ \hat{{\cal A}_r} $ and $ \hat{\cal A} $ stand for the quantum analogues of the reduced and full group $ C^*$-代数（c.f. [11]，[6]）。我们遵循经典Baum-Connes映射的构造步骤，尽管在量子群的上下文中，必须认真考虑不变权重的非平凡模性质（以及相关的事实，即对极的平方不是单位的），这使得猜测和证明相关希尔伯特模结构的正确定义有点棘手。
