![]() The DifferentialGeometry software now contains the commands for the complete analysis of the structure theory for any semi-simple Lie algebra (defined perhaps as a matrix algebra, a Lie algebra of symmetries of a differential equation, the isometries of a metric, the automorphisms of a Lie algebra, etc). Twenty-four new commands have been added to the LieAlgebra package for working with simple (and semi-simple) Lie algebras. Structure Theory of Semi-simple Lie Algebras The sub-package LieAlgebras:-LieAlgebraRepresentations has been deprecated and the 11 commands in that package have been integrated directly into the LieAlgebras package. The sub-package LieAlgebras:-LieAlgebraCohomology has been deprecated and the 3 commands in that package have been integrated directly into the LieAlgebras package. Integration of the sub-packages LieAlgebraCohomology and LieAlgebraRepresentations This procedure uses a simple algorithm which does not require eigenvalue computation and works well with the Maple assuming feature.įor General Relativity students and researchers new to Maple, see Overview of General Relativity in DifferentialGeometry. The signature of a metric can now be computed with QuadraticFormSignature. Line congruences play a distinguished role in the study of space-times and the properties of such congruences can now be easily calculated with the command CongruenceProperties. The commands RainichConditions and RainichElectromagneticField can be used to determine if a given space-time is an electro-vac space-time and to find the electromagnetic field for that space-time. The Tensor sub-package contains 4 new commands for analyzing the geometric properties of space-times. The Library sub-package contains a new command MetricSearch which provides a simple method for searching the DifferentialGeometry database of solutions to the Einstein equations. For an introduction to this new functionality in DifferentialGeometry, see Working with Abstract Forms. These capabilities extend those available in the difforms package. The capabilities of other commands in the DifferentialGeometry package, notably DGinfo, ExteriorDerivative, Hook and LieDerivative, have been extended to work with abstract differential forms. Structure equations for the abstract differential forms can be included in the new calling sequences for DGsetup. The functionality of the command DGsetup has been extended to create a new computational environment for working with abstract differential forms, that is, differential forms which are defined without reference to any underlying system of coordinates. Updates to Differential Geometry in Maple 16ĭifferentialGeometry for Maple 16, with 28 new commands, contains important new functionality for working with abstractly defined differential forms, Lie algebras and general relativity.
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