论文标题
中央式结构和扬吉型代数
The centralizer construction and Yangian-type algebras
论文作者
论文摘要
让$ d $成为一个积极的整数。 Yangian $ y_d = y（\ mathfrak {gl}（d，\ m asthbb c））$的$ line lie lie代数$ \ mathfrak {gl} {gl}（d，d，\ mathbb c）$可以计算许多生成器和Quadratic-linear关系，并且可以将其包装到单个矩阵中，并将其包装到单个矩阵中 - 扬声 - 另外，可以从通用包络代数$ u（\ mathfrak {gl}（n，\ mathbb c））$的某些centralizer subalgebras构建$ y_d $，并将限制过渡用作$ n \ to \ infty $。此方法称为\ emph {Centralizer construction}。 该论文表明，centralizer构造的概括导致一个新的家庭$ \ {y_ {d，l}：l = 1,2,3，\ dots \} $ yangian-type代数（Yangian $ y_d $是这个家庭的第一个任期）。对于新的代数，RTT演示文稿似乎丢失了，但是Yangian $ y_d $的许多属性仍然存在。特别是，$ y_ {d，l} $具有二次线性定义关系系统。 代数$ y_ {d，l} $（$ d = 1,2,3，\ dots $）在免费的$ l $发电机上的自由关联代数上为特殊的双泊松支架（在van den bergh的意义上）提供了一种量化。
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.
