Research Projects

A significant aspect of the developments in Serre’s problem on projective modules was the study of classical groups and its generalisations. The name ‘classical groups’ was first used by H. Weyl in his book “The Classical Groups Their Invariants and Representations”, to collectively represent the general linear groups, the symplectic groups, the orthogonal groups, and the unitary groups. The study of classical groups is important because of its significance in many vital areas of physical sciences. In this project I will study a generalisation of the classical orthogonal group called the Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal group. In 1968, A. Roy defined a set of transformations which generalises the classical Eichler-Siegel-Dickson transformations over fields to commutative rings. These elementary orthogonal transformations are defined on quadratic spaces with a hyperbolic summand over a commutative ring with unity in which 2 is an invertible element. These are called Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal transformations. The group generated by such transformations is called the DSER elementary orthogonal group. If we choose a particular quadratic form, then the DSER group will reduce to the classical orthogonal group. Thus the DSER group is a generalisation for the orthogonal group, and we can improve many classical results in the literature by obtaining similar results for the more general DSER group. This project aims to prove two important results on the DSER elementary orthogonal group. We intend to prove the nilpotency of the quotient group of the orthogonal group of a quadratic space with a hyperbolic summand by the DSER elementary orthogonal group. We would also like to prove the equivalence of the Local-Global principle and normality for DSER elementary orthogonal groups. Normality of the elementary subgroup of the general linear group is a major result in algebraic K-theory. It was proved in 1977 by A.A. Suslin. Later similar results were proved for other classical groups also. The orthogonal and symplectic group analogues were proved by A.A. Suslin and V.I. Kopeiko. The normality results for DSER elementary orthogonal group were studied by A.A. Ambily and R.A. Rao, and they obtained the normality of DSER elementary orthogonal group in the corresponding orthogonal group. After having the normality, a natural question to ask is whether the quotient group SK₁ is nilpotent. This project aims to address this question. Quillen’s Local-Global principle is a significant result established for proving Serre’s problem on projective modules. In 2005, R. Basu, R.A. Rao and R. Khanna proved the equivalence of the normality of elementary subgroups and the Local-Global principle for classical groups such as general linear, symplectic and orthogonal groups of matrices of order n. Here we try to obtain the equivalence of the Local-Global principle and normality for DSER elementary orthogonal groups.