© 2001 by London Mathematical Society
© The London Mathematical Society
On the Group of Symplectic Matrices Over a Free Associative Algebra
University College London Gower Street, London WC1E 6BT
Ruhr-Universität Bochum, Universitätsstrasse 150 D-44780 Bochum, Germany
Received 12 January 2000. Revision received 11 July 2000.
Symplectic groups are well known as the groups of isometries of a vector space with a non-singular bilinear alternating form. These notions can be extended by replacing the vector space by a module over a ring R, but if R is non-commutative, it will also have to have an involution. We shall here be concerned with symplectic groups over free associative algebras (with a suitably defined involution). It is known that the general linear group GLn over the free algebra is generated by the set of all elementary and diagonal matrices (see [1, Proposition 2.8.2, p. 124]). Our object here is to prove that the symplectic group over the free algebra is generated by the set of all elementary symplectic matrices. For the lowest order this result was obtained in [4]; the general case is rather more involved. It makes use of the notion of transduction (see [1, 2.4, p. 105]). When there is only a single variable over a field, the free algebra reduces to the polynomial ring and the weak algorithm becomes the familiar division algorithm. In that case the result has been proved in [3, Anhang 5].