DIAMETER OF COMMUTING GRAPHS OF SYMPLECTIC ALGEBRAS. Issue 3 (4th June 2019)

• Record Type: Journal Article
• Authors: GENG, XIANYA; FAN, LITING; MA, XIAOBIN
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 100:Issue 3(2019)
• Page Start: 419
• Page End: 427
• Publication Date: 2019-06-04
• Subjects: primary 05C50, -- secondary 15A27, -- 15A30; commuting graphs, -- diameter, -- Lie algebras; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972719000583
• Languages: English
• ISSNs: 0004-9727

Abstract

Abstract : 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 $l>2$ .