Defining relations of the special unitary group over a~quadratic extension of an ordered Euclidean field

Zh. S. Satarov

Sbornik. Mathematics, Tome 54 (1986) no. 2, pp. 415-419

Résumé

Let $k$ be an ordered Euclidean field (i.e., an ordered field in which the group of nonzero squares coincides with the group of positive elements) and $K$ its quadratic extension. Further, let $\overline\xi$ denote the image of the element $\xi$ under the nontrivial automorphism of the extension $K/k$. We consider the special unitary group $SU(n, K)$ of degree $n\geqslant2$ over the field $K$, i.e., the subgroup of matrices $a$ of the general linear group $GL(n, K)$ for which $aa^*=e$ and $\det a=1$, where $^*$ denotes taking conjugate-transpose, i.e., $(a^*)_{ij}=\overline a_{ji}$. Defining relations in a certain natural system of generators are found for the group $SU(n,\, K)$, $n\geqslant2$. Bibliography: 8 titles.