The centralizer construction and Yangian-type algebras

Abstract

Let $d$ be a positive integer. The Yangian $Y_d=Y(\mathfrak{gl}(d,\mathbb C))$ of the general linear Lie algebra $\mathfrak{gl}(d,\mathbb C)$ 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, $Y_d$ can be built from certain centralizer subalgebras of the universal enveloping algebras $U(\mathfrak{gl}(N,\mathbb C))$, with the use of a limit transition as $N\to\infty$. This approach is called the \emph{centralizer construction}. The paper shows that a generalization of the centralizer construction leads to a new family $\{Y_{d,L}: L=1,2,3,\dots\}$ of Yangian-type algebras (the Yangian $Y_d$ being the first term of this family). For the new algebras, the RTT presentation seems to be missing, but a number of properties of the Yangian $Y_d$ persist. In particular, $Y_{d,L}$ possesses a system of quadratic-linear defining relations. The algebras $Y_{d,L}$ ($d=1,2,3,\dots$) provide a kind of quantization for a special double Poisson bracket (in the sense of Van den Bergh) on the free associative algebra with $L$ generators.