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A catalogue of 5 objects with explicit fields, dimensions, and category views.
5 separately identified objects and parameterized families. Real and complex scalar choices have distinct entries. An isomorphism is a recorded relationship, not a reason to collapse the entries.
Explore pairs and categories · Catalogue overview
This catalogue contains 659 separately identified objects and parameterized families, 42 category conventions, 689 recorded relationships, and 175 Hom/End/Aut records. Real and complex versions, split and compact forms, and the requested small sizes are explicit entries. A family entry states its parameter restrictions; it does not silently treat every parameter value as the same object.
Open the category explorer · List of newly created knowls · Explore the finite-group table · Explore the Lie-group table
Hom, End, and Aut depend on the chosen category. Select two objects in the explorer, choose a shared category, then inspect the recorded maps. End and Aut use a single object with a specified structure. A missing entry means not catalogued, not that its Hom-set is empty. Complete and partial descriptions are labeled separately.
A matrix Lie group such as is not itself a vector space. Use smooth or holomorphic group maps for that object, and real-linear or complex-linear maps for its Lie algebra or a specified ambient vector space. Changing the algebra product is a construction, not merely a change of scalar field.
The compact real Freudenthal magic square is backed by all 16 data cells, with both input algebras and each output Lie algebra identified. Changing a composition algebra to a split form requires a new real-form calculation; the compact table is not presented as a table of all real forms.
The low-dimensional entries retain their own identities even when isomorphic. This records, for example, the distinctions between Spin groups and their orthogonal quotients, between one-dimensional Hermitian constructions, and between the three Spin(8) representations related by triality.
Each recorded relationship names its category or identifies itself as a construction, states its conditions, and carries evidence. The catalogue does not infer arbitrary compositions or inverses. The present evidence is mathematical exposition and checked literature, not Lean proofs. Formal-proof references are reserved for future verified declarations.
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The orders and fractional ideals reading list covers the separate transcript request, including new definitions and existing prerequisites reused.
The real Lie algebra is the direct sum of three copies of . For triples, its bracket is
Its real dimension is , and its three simple factors commute.
The real Lie algebra is the direct sum of two copies of , with
It has real dimension . The two eight-dimensional factors are commuting simple ideals, so the sum is semisimple and not simple.
The negative half-spin representation of compact is its action on one of the two irreducible real modules of the even Clifford algebra
Here Clifford generators satisfy . For an oriented orthonormal basis, put . The negative module is the simple module on which acts by . The module has real dimension , and restriction to is irreducible.
The positive half-spin representation of compact is its action on one of the two irreducible real modules of the even Clifford algebra
Here Clifford generators satisfy . For an oriented orthonormal basis, put . The positive module is the simple module on which acts by . The module has real dimension , and restriction to is irreducible.