Defining relations of a classical unitary group over the ring of dual numbers (Q1909853)

We find defining relations for classical unitary groups \(U(n,D)\), \(n\geq 2\), over the ring of dual numbers \(D\). This result gives us as a corollary, presentations (via generators and relations) of special unitary groups \(SU(n,D)\) and projective unitary groups \(PU(n,D)\), \(PSU(n,D)\). We note that the result obtained here can be carried over without essential modifications to unitary groups \(U(n,K)\), \(SU(n,K)\), \(PU(n,K)\), \(PSU(n,K)\), \(n\geq 2\), over arbitrary (in general non-commutative) local rings \(K\), for which the anti-automorphism of second order \(^-\) satisfies the requirements: (1) \(N(x)=N(\overline{x})\) \(\forall x\in K\); (2) there is a complete system of centrally stable residues of the ring \(K\) (i.e., residues \(x\in\text{cent }K\), \(\overline{x}=x\)) with respect to the equivalence \(x\sim y\Leftrightarrow N(x)=N(y)\); (3) there is fulfilled the inclusion \(N(K^*)+N(K)\subseteq N(K^*)\) (here \(N(x)=x \overline{x}\) is the norm of the element \(x\)). Thus, the solution technique of this article represents a rather essential generalization of earlier results.