A PBW commutator lemma for U_q[gl(m|n)]

We present and prove in detail a Poincare--Birkhoff--Witt commutator lemma for the quantum superalgebra U_q[gl(m|n)].Comment: 16 pages, no figure

Automatic Evaluation of the Links-Gould Invariant for all Prime Knots of up to 10 Crossings

This paper describes a method for the automatic evaluation of the Links-Gould two-variable polynomial link invariant (LG) for any link, given only a braid presentation. This method is currently feasible for the evaluation of LG for links for which we have a braid presentation of string index at most 5. Data are presented for the invariant, for all prime knots of up to 10 crossings and various other links. LG distinguishes between these links, and also detects the chirality of those that are chiral. In this sense, it is more sensitive than the well-known two-variable HOMFLY and Kauffman polynomials. When applied to examples which defeat the HOMFLY invariant, interestingly, LG `almost' fails. The automatic method is in fact applicable to the evaluation of any such state sum invariant for which an appropriate R matrix and cap and cup matrices have been determined.Comment: 28 pages, 6 figures. Minor corrections and references added since version

Automatic Construction of Explicit R Matrices for the One-Parameter Families of Irreducible Typical Highest Weight (0|\alpha) Representations of U_q[gl(m|n)]

We detail the automatic construction of R matrices corresponding to (the tensor products of) the (0|\alpha) families of highest-weight representations of the quantum superalgebras U_q[gl(m|n)]. These representations are irreducible, contain a free complex parameter \alpha, and are 2^{mn} dimensional. Our R matrices are actually (sparse) rank 4 tensors, containing a total of 2^{4mn} components, each of which is in general an algebraic expression in the two complex variables q and \alpha. Although the constructions are straightforward, we describe them in full here, to fill a perceived gap in the literature. As the algorithms are generally impracticable for manual calculation, we have implemented the entire process in Mathematica; illustrating our results with U_q[gl(3|1)].Comment: 65 pages, 6 tables. David De Wit: <http://www.kurims.kyoto-u.ac.jp/~ddw