⟨Leibniz-Grassmann-Clifford-Hestenes⟩ differential geometric algebra / multivector simplicial complex
The Grassmann.jl package provides tools for doing computations based on multi-linear algebra, differential geometry, and spin groups using the extended tensor algebra known as Leibniz-Grassmann-Clifford-Hestenes geometric algebra. Combinatorial products included are ∧, ∨, ⋅, *, ⋆, ', ~, d, ∂ (which are the exterior, regressive, inner, and geometric products; along with the Hodge star, adjoint, reversal, differential and boundary operators). The kernelized operations are built up from composite sparse tensor products and Hodge duality, with high dimensional support for up to 62 indices using staged caching and precompilation. Code generation enables concise yet highly extensible definitions. The DirectSum.jl multivector parametric type polymorphism is based on tangent bundle vector spaces and conformal projective geometry to make the dispatch highly extensible for many applications. Additionally, the universal interoperability between different sub-algebras is enabled by AbstractTensors.jl, on which the type system is built.
This Grassmann package for the Julia language was created by github.com/chakravala for mathematics and computer algebra research with differential geometric algebras. These projects and repositories were started entirely independently and are available as free software to help spread the ideas to a wider audience. Please consider donating to show your thanks and appreciation to this project at liberapay, GitHub Sponsors, Patreon, Tidelift, Bandcamp or contribute (documentation, tests, examples) in the repositories.
TensorAlgebra design, Manifold code generationMathematical foundations and definitions specific to the Grassmann.jl implementation provide an extensible platform for computing with geometric algebra at high dimensions, along with the accompanying support packages. The design is based on the TensorAlgebra abstract type interoperability from AbstractTensors.jl with a TensorBundle parameter from DirectSum.jl. Abstract tangent vector space type operations happen at compile-time, resulting in a differential conformal geometric algebra of hyper-dual multivector forms.
Mathematics of Grassmann can be used to study unitary groups used in quantum computing by building efficient computational representations of their algebras. Applicability of the Grassmann computational package not only maps to quantum computing, but has the potential of impacting countless other engineering and scientific computing applications. It can be used to work with automatic differentiation and differential geometry, algebraic forms and invariant theory, electric circuits and wave scattering, spacetime geometry and relativity, computer graphics and photogrammetry, and much more.
using Grassmann, Makie; @basis S"∞+++"
streamplot(vectorfield(exp((π/4)*(v12+v∞3)),V(2,3,4),V(1,2,3)),-1.5..1.5,-1.5..1.5,-1.5..1.5,gridsize=(10,10))
Thus, computations involving fully general rotational algebras and Lie bivector groups are possible with a full trigonometric suite. Conformal geometric algebra is possible with the Minkowski plane, based on the null-basis. In general, multivalued quantum logic is enabled by the ∧,∨,⋆ Grassmann lattice. Mixed-symmetry algebra with Leibniz.jl and Grassmann.jl, having the geometric algebraic product chain rule, yields automatic differentiation and Hodge-DeRahm co/homology as unveiled by Grassmann. Most importantly, the Dirac-Clifford product yields generalized Hodge-Laplacian and the Betti numbers with Euler characteristic χ.
The Grassmann.jl package and its accompanying support packages provide an extensible platform for high performance computing with geometric algebra at high dimensions. This enables the usage of many different types of TensorAlgebra along with various Manifold parameters and interoperability for a wide range of scientific and research applications.
More information and tutorials are available at https://grassmann.crucialflow.com/dev
Grassmann.jl is a package for the Julia language, which can be obtained from their website or the recommended method for your operating system (GNU/Linux/Mac/Windows). Go to docs.julialang.org for documentation. Availability of this package and its subpackages can be automatically handled with the Julia package manager using Pkg and Pkg.add("Grassmann") or by entering:
pkg> add Grassmann
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