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)]

摘要

We detail the automatic construction of R matrices corresponding to (thetensor products of) the (0|\alpha) families of highest-weight representationsof the quantum superalgebras U_q[gl(m|n)]. These representations areirreducible, contain a free complex parameter \alpha, and are 2^{mn}dimensional. Our R matrices are actually (sparse) rank 4 tensors, containing atotal of 2^{4mn} components, each of which is in general an algebraicexpression in the two complex variables q and \alpha. Although the constructions are straightforward, we describe them in fullhere, to fill a perceived gap in the literature. As the algorithms aregenerally impracticable for manual calculation, we have implemented the entireprocess in Mathematica; illustrating our results with U_q[gl(3|1)].