32 research outputs found
On the Quillen determinant
We explain the bundle structures of the {\it Determinant line bundle} and the
{\it Quillen determinant line bundle} considered on the connected component of
the space of Fredholm operators including the identity operator in an intrinsic
way. Then we show that these two are isomorphic and that they are non-trivial
line bundles and trivial on some subspaces. Also we remark a relation of the
{\it Quillen determinant line bundle} and the {\it Maslov line bundle}.Comment: 11 pages, to appear in the Journal of Geometry and Physic
Free nilpotent and -type Lie algebras. Combinatorial and orthogonal designs
The aim of our paper is to construct pseudo -type algebras from the
covering free nilpotent two-step Lie algebra as the quotient algebra by an
ideal. We propose an explicit algorithm of construction of such an ideal by
making use of a non-degenerate scalar product. Moreover, as a bypass result, we
recover the existence of a rational structure on pseudo -type algebras,
which implies the existence of lattices on the corresponding pseudo -type
Lie groups. Our approach substantially uses combinatorics and reveals the
interplay of pseudo -type algebras with combinatorial and orthogonal
designs. One of the key tools is the family of Hurwitz-Radon orthogonal
matrices
Automorphism groups of pseudo H-type algebras
In the present paper we determine the group of automorphisms of pseudo H-type Lie algebras, that are two step nilpotent Lie algebras closely related to the Clifford algebras Cl(Rr,s).publishedVersio