1,353 research outputs found
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
Bialgebra deformations and algebras of trees
Let A denote a bialgebra over a field k and let A sub t = A((t)) denote the ring of formal power series with coefficients in A. Assume that A is also isomorphic to a free, associative algebra over k. A simple construction is given which makes A sub t a bialgebra deformation of A. In typical applications, A sub t is neither commutative nor cocommutative. In the terminology of Drinfeld, (1987), A sub t is a quantum group. This construction yields quantum groups associated with families of trees
Oriented Quantum Algebras and Invariants of Knots and Links
In GT/0006019 oriented quantum algebras were motivated and introduced in a
natural categorical setting. Invariants of knots and links can be computed from
oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for
Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras
from a more algebraic perspective, and develop a more detailed theory for them
and their associated invariants.Comment: LAteX document, 45 pages, 17 figure
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