DIAMETER OF COMMUTING GRAPHS OF SYMPLECTIC ALGEBRAS

Let $F$ be an algebraically closed field of characteristic $0$ and let $\operatorname{sp}(2l,F)$ be the rank $l$ symplectic algebra of all $2l\times 2l$ matrices $x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$ over $F$, where $A^{t}$ is the transpose of $A$ and $B,C$ are symmetric matrices of order $l$. The commuting graph $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ of $\operatorname{sp}(2l,F)$ is a graph whose vertex set consists of all nonzero elements in $\operatorname{sp}(2l,F)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$. We prove that the diameter of $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ is $4$ when $l>2$.

TWISTED YANGIANS AND FOLDED ${\mathcal W}$ -ALGEBRAS

We show that the truncation of twisted Yangians are isomorphic to finite [Formula: see text] -algebras based on orthogonal or symplectic algebras. This isomorphism allows us to classify all the finite-dimensional irreducible representations of the quoted [Formula: see text] -algebras. We also give an R matrix for these [Formula: see text] -algebras, and determine their center.