Tensors on free modules of finite rank
|Reported by:||egourgoulhon||Owned by:|
|Component:||linear algebra||Keywords:||free module, tensor, tensor product|
|Authors:||Eric Gourgoulhon, Michal Bejger||Reviewers:|
|Report Upstream:||N/A||Work issues:|
Description (last modified by egourgoulhon)
This ticket implements:
- tensor products of the type M\otimes ...\otimes M \otimes M* \otimes...\otimes M* where M is a free module of finite rank over a commutative ring R and M* is its dual (k factors of M and l factors of M*, say)
- the elements of the above tensor products, considered as tensors of type (k,l) on M, i.e. multilinear forms (M*)k \times Ml --> R, thanks to the canonical isomorphism (M*)* = M (which holds since M is a free module of finite rank)
- the following tensor operations:
- operations inherent to the module structure (addition, multiplication by a ring element)
- tensor product of two tensors
- tensor contraction
- symmetry / antisymmetry handling (on subset of the tensor arguments or on all arguments)
- exterior product of alternating forms
No distinguished basis is assumed on the free module M; on the contrary many bases can be introduced. Each tensor has then various representations, via its components in the various bases.
Motivation and context
The ticket has been motivated by tensors on smooth manifolds over R, within the
SageManifolds project. In this context, tensors on free modules appear at two levels:
- tensors on tangent spaces:
- commutative ring R: real field R
- free module M: tangent vector space at a given manifold's point
- tensor fields on a manifold:
- commutative ring R: the algebra Coo(U) of smooth functions U--> R, where U is a parallelizable open set of the manifold
- free module M: the set X(U) of smooth vector fields on U (since U is parallelizable, this is a free module; its rank is the manifold's dimension)
Apart from the numerous doctests in the code, some pieces of documentation are
- the tutorial worksheet posted here (a pdf version is here)
- the "tensors on free modules" reference manual (a pdf version is here); it can also be generated via the command sage -docbuild tensors_free_module html
See also this page.
- Although developed in the context of SageManifolds (ticket:14865), the ticket is self-contained and does not depend on other parts of SageManifolds. It this respect, it can be viewed as some attempt to include a first subset of SageManifolds in Sage, with a moderate size: the ticket comprises 9634 lines of Python code (most of them being doctests), while at present (version 0.5) SageManifolds contains 33272 lines of code.
- The ticket follows Sage's Parent/Element scheme and the (new) category framework. In particular, the ticket's free module class (FiniteRankFreeModule) passes the module TestSuite.
- It turned out to be necessary to develop a new class to implement free modules of finite rank. Indeed, the category of free modules does not exist yet in Sage: only those of generic modules (Modules) or free modules with a distinguished basis (ModulesWithBasis) are available. Now, the tangent space at a given point of a manifold is a vector space without any distinguished basis (in other words, while the tangent space is isomorphic to Rn, there is no canonical isomorphism, each isomorphism relying on the choice of some coordinate chart). The new class, FiniteRankFreeModule, does not rely on any distinguished basis. It inherits directly from Parent, with the category set to Modules(). In particular, it does not inherit from sage.modules.module.FreeModule_generic since the latter does not conform to the new coercion model and seems to assume a distinguished basis (cf. its method basis()).
Change History (17)
comment:3 Changed 6 months ago by git
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comment:8 Changed 4 months ago by git
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