Library

In this page, I provide links to articles, monographs, lecture notes and maple programs produced by me which might be useful for other people.

Maxima Programs

  • SimpleLieAlgebra.mac (v2.11)

    • Maxima program to construct the root system, the weight systems of irreps, the irreducible decomposition of the product of two irreps, and the irrep branching rules w.r.t. subalgebras for any simple Lie algebra.

    This maxima program was transplanted from Maple.

  • Riemann.mac (v1.22)

    • Maxima program to calculate the connection coefficients, Riemann curvature tensor, Weyl tensor, Ricci tensor, Ricci scalar, Einstein tensor, connection form, curvature form, Kretschmann invariant, Newman-Penrose coefficients (4D only), and the contranction, inner product and covariant derivatives of tensors for a given metric in a curved space(time) of arbitrary dimensions.

    This maxima program was transplanted from Maple.

  • Clifford.mac (v1.00)

    • This small maxima program consists of functions defining the non-commutative product && of the gamma matrices and reducing polynomials of the gamma matrices to linear combination of the canonial basis $\Gamma^{a_1\,\cdots\, a_k}$. All calculations are symbolic and do not utilize any explicit matrix prepresentation of the gamma matrices. The space(-time) dimension can be arbitrary.

Maple Programs

  • SimpleLieAlgebra.mpl (v2.22)

    • Maple program to construct the root system, the weight systems of irreps, the irreducible decomposition of the product of two irreps, and the irrep branching rules w.r.t. subalgebras for any simple Lie algebra.

  • Riemann.mpl (v1.10)

    • Maple program to calculate the connection coefficients, Riemann curvature tensor, Weyl tensor, Ricci tensor, Ricci scalar, Einstein tensor, connection form, curvature form, Kretschmann invariant, Newman-Penrose coefficients (4D only), and the contranction, inner product and covariant derivatives of tensors for a given metric in a curved space(time) of arbitrary dimensions.

Monographic Notes (in Japanese)

The following are my private notes in Japanese on various topics which have been compiled through my reserch activities. They are far from complete and may contain erroneous statements.

  • LieGroup.pdf (v20230503)

    • An extensive note on Lie groups. It covers the structure and representation of general Lie algebras/groups, the maximal quasi-semi-simple subalgebras of simple Lie algebras with rank less than 6 (except for Sp5) and E6, their embedding/projection matrices and the corresponding branching rules of the fundamental irreps.

    Version note

    In this version, a kind of uniqueness theorem on the adjoint invariant metric in a simple Lie algebra is added.

  • Geometry.pdf (v20230503)

    • An extensive note on gemetry of manifolds and varieties. It covers basics and some topics of differential geometry, complex manifolds and algebraic geometry.

  • algebra.pdf (v20230203) New!

    • A private note on algebras. It includes basics of module, finite group, commutative rings and algebras. Topics covered are quite limited.

  • topology.pdf (v20230209) New!

    • A private note on algebraic and differential topology. It includes basics of homology and cohomology, manifolds, differential topology, fiber bundles, characteristic classes and knots and links. Topics covered are quite limited.

  • UT.pdf (v20231017)

    • A complehensive note on Ultimate Theory of Nature and related topics. It covers basics of spacetime symmetries and supersymmetries, quantum field theories, quantum anomalies, grand unified theories, supergravity theories in four and higher dimensions, maximal supergravities in four dimensions, superstring theories, M-theory, higher-dimensional unified theories, and inflation cosmology based on them.

    Version note

    In this version, §5.52 on the magnetic monopole is revised and expanded.

  • formula.pdf (v20230329) New!

    • Private collection of formulas mainly in the fields of classical general relativity and Clifford algebra.

Lecture Notes (in Japanese)

The following are the distributed notes of recent lectures I have given in varioius universities. They are far from complete and may contain out-of-date contents or incorrect statements. If you use their contents, you should check the validity of them in advance by yourself.

Articles on gravity, cosmology and unifield theories (in Japanese)

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