Representations Theory and Doubles of Yangians of Classical Lie Superalgebras

Abstract

Some basic results of the theory of Yangians of Lie superalgebras are described. The Yangian of a classical Lie superalgebra is described as a result of quantization of the Lie superbialgebra of polynomial loops. Two systems of generators and defining relations are introduced, and their equivalence is proved. The PBW theorem for the Yangians of classical Lie superalgebras is formulated and proved. All irreducible finite-dimensional representations of the Yangians of Lie superalgebras of type A(m,n) are described in terms of Drinfel’d polynomials. A notion of the double of a Yangian and a formula for the universal R-matrix for the double of a Yangian are discussed for the Yangian of a Lie superalgebra of type A(m,n).