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Noncommutative differential geometry, and the matrix representations of generalised algebras

Research output: Contribution to journalJournal article

<mark>Journal publication date</mark>05/1998
<mark>Journal</mark>Journal of Geometry and Physics
Issue number3-4
Number of pages18
Pages (from-to)227-244
<mark>Original language</mark>English


The underlying algebra for a noncommutative geometry is taken to be a matrix algebra, and the set of derivatives the adjoint of a subset of traceless matrices. This is sufficient to calculate the dual I-forms, and show that the space of 1-forms is a free module over the algebra of matrices. The concept of a generalised algebra is defined and it is shown that this is required in order for the space of 2-forms to exist. The exterior derivative is generalised for higher-order forms and these ale also shown to be: free modules over the matrix algebra. Examples of mappings that preserve the differential structure are given. Also given are four examples of matrix generalised algebras. and the corresponding noncommutative geometries. including the cases where the generalised algebra corresponds to a representation of a Lie algebra or a q-deformed algebra.