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Let $R=k[[u,v]]$ be a power series ring over algebraically closed field of characteristic zero. The quaternionic $R$-algebra is $A=R\langle x,y\rangle/I$, where $I=(x^2-a, y^2-b, xy+yx-2c)$ and $a,b,c\in R$. I am interested in existence of such algebras for general $a,b,c$ and an explicit matrix representation for them. It is possible to construct some specific examples as the rings of invariants (see for example here, page 74) but what is the situation in general?

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    $\begingroup$ What do you mean by existence? They do exist :-) You want them to satisfy some condition which is not written? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 1, 2012 at 18:56
  • $\begingroup$ The definition you wrote is one that guarantees existence. If I'm not mistaken, you can make a matrix representation by taking $1,x,y,xy$ as a basis over the quotient field of $R$. $\endgroup$
    – S. Carnahan ♦
    Commented Sep 2, 2012 at 9:08

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