Yangians of Lie (super)algebras

Abstract

This thesis is concerned with extending some well-known results about the Yangians Y (gl N) and Y (slN) to the case of super-Yangians. First we produce a new presentation of the Yangian Y (gl m|n), using the Gauss decomposition of a matrix with non-commuting entries. Then, by writing the quantum Berezinian in terms of generators from the new presentation we prove that its coefficients generate the centre Zm|n of Y (gl m|n). We show that the Yangian Y (slm|n) is isomorphic to a subalgebra of the Yangian Y (gl m|n), and in particular if m ̸ = n, then Y (gl m|n) ∼ = Zm|n ⊗ Y (slm|n). Finally, we show that a Yangian Y (psl n|n) associated with the projective special linear Lie superalgebra may be obtained from Y (sln|n) by quotienting out the ideal generated by the coefficients of the quantum Berezinian. iii ivAcknowledgements I gratefully acknowledge the help of my supervisor Alex Molev, who provided the original plan for this thesis project and has been helpful and supportive throughout