Premium
On the Group of Symplectic Matrices Over a Free Associative Algebra
Author(s) -
Cohn P. M.,
Gerritzen L.
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1017/s0024610700001939
Subject(s) - mathematics , symplectic group , symplectic representation , symplectic geometry , symplectic vector space , pure mathematics , free algebra , symplectic matrix , vector space , bilinear form , polynomial ring , algebra over a field , division algebra , general linear group , moment map , cellular algebra , discrete mathematics , filtered algebra , algebra representation , polynomial , symmetric group , mathematical analysis
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 GL n 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].