
Differential Geometry


 
For Maple 17, the DifferentialGeometry package contains over 40 new commands or commands with enhanced functionality or improved efficiency.
Updates to the DifferentialGeometry package:
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DGsetup has a new keyword argument for declaring certain coordinates to be complex.

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The new command DGconjugate calculates the complex conjugate of a vector, tensor or differential form, quaternion or octonion.

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The new commands DGRe and DGIm calculate the real and imaginary parts of a vector, tensor, differential form, quaternion or octonion.

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The new commands DGsolve, DGNullSpace and DGImageSpace provide useful utilities for solving equations whose unknowns are tensors or differential forms.

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The new command `&algmult` computes noncommutative multiplication in general algebras.

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A new geometric type, "multivector", has been created. This significantly improves computational efficiency when working with contravariant skewsymmetric tensors. The commands convert, Hook, RaiseLowerIndices, and LieDerivative have been extended to accept multivector arguments.

ExteriorDifferentialSystems
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This is a new package for the study of differential systems.

Library
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Over 100 new metrics have been added to the database of solutions to the Einstein equations.

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A command line version of the MetricSearch maplet is now available.

LieAlgebras
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The Lie algebra package is being extended to work with more general algebras. The new AlgebraData and AlgebraLibraryData commands creates the multiplication structures for quaternions, octonions, Jordan algebras and Clifford algebras.The commands AlgebraNorm and AlgebraInverse calculate the norms and inverses of quaternions and octonions.

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The command LieAlgebraData has been rewritten to give massive improvements in efficiency for calculating the structure equations for matrix algebras. The command also has a new calling sequence for working with graded Lie algebras.

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The functionality of the SimpleLieAlgebraData command has been extended to include the exceptional Lie algebras ${g}_{2}$ and ${f}_{4}$.

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The new commandChevalleyBasis calculates the Chevalley basis of a semisimple Lie algebra.

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The Derivations command has been rewritten to give significant improvements in efficiency. The command can now also compute the derivations of a general algebra.

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The Codifferential command has been added to calculate the codifferential of a multivector in a Lie algebra.

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The SymbolAlgebra command computes the graded nilpotent Lie algebra associated to any distribution of vector fields.

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The TanakaProlongation command calculates the Tanaka prolongation of any graded nilpotent Lie algebra. This command gives a powerful new tool for finding differential systems with maximal symmetry algebras.

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The new command Rank1Elements calculates the rank 1 matrices in the span of given list of matrices.

Tensor
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The index structure for connections has been changed to conform with the most common usage.

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The new command FormInnerProduct provides for fast evaluation of the inner products of differential forms.

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A powerful new command has been developed for calculating the infinitesimal holonomy of an affine connection.

Example 1:

Early in the theoretical development of simple Lie algebras, it was conjectured by Killing that there exists a certain small number of exceptional Lie algebras not belonging to the families of simple Lie algebras, which were wellknown at the time. It was an important problem to explicitly construct these exceptional algebras. In 1908, E. Cartan proved that the smallest of these exceptional Lie algebras, the 14dimensional Lie algebra denoted by ${g}_{2}$, is the algebra of derivations of the 8dimensional algebra of octonions $\mathrm{\𝕆}$. With Maple 17, it is easy to verify Cartan's computations.
First, use the new AlgebraLibraryData command to retrieve the structure equations for the octonions:
>

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

>

$\mathrm{StrEq}:=\mathrm{AlgebraLibraryData}\left(''Octonions''\,\mathrm{Oct}\right)$

${\mathrm{StrEq}}{:=}\left[{{\mathrm{e1}}}^{{2}}{\=}{\mathrm{e1}}{\,}{\mathrm{e1}}{\.}{\mathrm{e2}}{\=}{\mathrm{e2}}{\,}\right]$ 


