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Universal tangle invariant and commutants of quantum algebras

H C Lee

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We construct a universal tangle invariant on a quantum algebra. We show that the invariant maps tangle to commutants of the algebra; every (1,1)-tangle is mapped to a Casimir operator of the algebra; the eigenvalue of the Casimir operator in an irreducible representation of the algebra is a link polynomial for the closure of the tangle. This result is applied to a discussion of the Alexander - Conway polynomial and quantum holonomy in Chern - Simons theory in three dimensions.


PACS

02.30.Tb Operator theory

02.10.De Algebraic structures and number theory

02.10.Ud Linear algebra

03.65.Fd Algebraic methods

MSC

08A40 Operations, polynomials, primal algebras

15A18 Eigenvalues, singular values, and eigenvectors

16W30 Coalgebras, bialgebras, Hopf algebras (See also 16S40, 57T05); rings, modules, etc. on which these act

81R15 Operator algebra methods (See also 46Lxx, 81T05)

Subjects

Mathematical physics

Quantum information and quantum mechanics

Dates

Issue 2 (21 January 1996)

Received 25 April 1995



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