Quantum supergroups and link polynomials. (English) Zbl 0797.17008
Summary: A general method for constructing invariants for quantum supergroups is applied to obtain a closed formula for link polynomials. For type I quantum supergroups, a realization of the braid group and corresponding link polynomial is determined for each irreducible representation of the quantum supergroup in a certain class. Although these realizations are not matrix representations in the usual sense, link polynomials are defined which are generalizations of those previously obtained from quantum groups. To illustrate the theory, link polynomials corresponding to the defining representations of the quantum supergroups \(\text{U}_ q[\text{gl}(m| n)]\), \(\text{U}_ q[\text{C}(m+ 1)]\) are determined explicitly.
MSC:
17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
20F36 | Braid groups; Artin groups |
57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |