Oriented Quantum Algebras and Coalgebras, Invariants of Oriented 1-1 Tangles, Knots and Links

In this paper we study oriented quantum coalgebras which are structures closely related to oriented quantum algebras. We study the relationship between oriented quantum coalgebras and oriented quantum algebras and the relationship between oriented quantum coalgebras and quantum coalgebras. We show that there are regular isotopy invariants of oriented 1-1 tangles and of oriented knots and links associated to oriented and twist oriented quantum coalgebras respectively. There are many parallels between the theory of oriented quantum coalgebras and the theory of quantum coalgebra

Cohomology and deformation of module-algebras

An algebraic deformation theory of module-algebras over a bialgebra is constructed. The cases of module-coalgebras, comodule-algebras, and comodule-coalgebras are also considered.Comment: To appear in J. Pure Applied Algebra. 14 page

Monoidal Hom-Hopf algebras

Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Hom-algebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras and Lie algebras.Comment: 25 pages; extended version: compared to the version that appeared in Comm. Algebra, the Section Preliminary Results and Remarks 5.1 and 6.1 have been adde

Lie monads and dualities

We study dualities between Lie algebras and Lie coalgebras, and their respective (co)representations. To allow a study of dualities in an infinite-dimensional setting, we introduce the notions of Lie monads and Lie comonads, as special cases of YB-Lie algebras and YB-Lie coalgebras in additive monoidal categories. We show that (strong) dualities between Lie algebras and Lie coalgebras are closely related to (iso)morphisms between associated Lie monads and Lie comonads. In the case of a duality between two Hopf algebras -in the sense of Takeuchi- we recover a duality between a Lie algebra and a Lie coalgebra -in the sense defined in this note- by computing the primitive and the indecomposables elements, respectively.Comment: 27 pages, v2: some examples added and minor change

Bases as Coalgebras

The free algebra adjunction, between the category of algebras of a monad and the underlying category, induces a comonad on the category of algebras. The coalgebras of this comonad are the topic of study in this paper (following earlier work). It is illustrated how such coalgebras-on-algebras can be understood as bases, decomposing each element x into primitives elements from which x can be reconstructed via the operations of the algebra. This holds in particular for the free vector space monad, but also for other monads, like powerset or distribution. For instance, continuous dcpos or stably continuous frames, where each element is the join of the elements way below it, can be described as such coalgebras. Further, it is shown how these coalgebras-on-algebras give rise to a comonoid structure for copy and delete, and thus to diagonalisation of endomaps like in linear algebra