The centralizer construction and Yangian-type algebras

Abstract

Let d be a positive integer. The Yangian Yd=Y(gl(d,C)) of the general linear Lie algebra gl(d,C) has countably many generators and quadratic-linear defining relations, which can be packed into a single matrix relation using the Yang matrix — the famous RTT presentation. Alternatively, Yd can be built from certain centralizer subalgebras of the universal enveloping algebras U(gl(N,C)), with the use of a limit transition as N→∞. This approach is called the centralizer construction.The paper shows that a generalization of the centralizer construction leads to a new family {Yd,L:L=1,2,3,…} of Yangian-type algebras (the Yangian Yd being the first term of this family). For the new algebras, the RTT presentation seems to be missing. Nevertheless, the algebras Yd,L share a number of properties of the Yangian Yd, including the existence of defining quadratic-linear commutation relations.