Mathematical object catalogue

Core idea

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

Long lists of objects

• Lie groups catalogue — 180 entries.
• Lie algebras catalogue — 152 entries.
• Scalar, associative, and Jordan algebras catalogue — 109 entries.
• Fields, local objects, and orders catalogue — 69 entries.
• Magic-square outputs and triality representations catalogue — 5 entries.
• Sporadic finite simple groups catalogue — 26 entries.
• Finite groups of Lie type catalogue — 68 entries.
• Elementary finite groups and families catalogue — 50 entries.

How to compare objects

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 $SL(2,\mathbb C)$ 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.

Worked comparisons

• Jord, UJord, and Jord1: units change Hom and End.
• Jordan maps from the scalar algebra are idempotents.
• The complex numbers as a real or complex vector space and algebra.
• Real and complex linear endomorphisms of sl(2,C).
• Real versus holomorphic maps on SL(2,C).
• K-linear, F-linear, K-algebra, and F-algebra endomorphisms.
• Q(i): linear maps versus algebra maps.
• Empty field Hom-sets versus the zero ring map.
• Finite-field linear maps versus Frobenius automorphisms.

Low dimensions and the magic square

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.

Category conventions

• Category of abelian groups — Abelian groups.
• Category of associative algebras with arbitrary homomorphisms — R-associative algebras; arbitrary maps; C-associative algebras; arbitrary maps; Q-associative algebras; arbitrary maps; F-associative algebras; arbitrary maps; K-associative algebras; arbitrary maps.
• Category of complex Lie groups — C Lie groups.
• Category of fields — Fields.
• Category of finite groups — Finite groups.
• Category of finite simple groups — Finite simple groups.
• Category of groups — Groups.
• Category Jord of Jordan algebras — Jord over R; Jord over C.
• Category of Lie algebras over a field — R-Lie algebras; C-Lie algebras.
• Category of nonassociative algebras — R-algebras; no associativity requirement; C-algebras; no associativity requirement.
• Category of real Lie groups — R Lie groups.
• Category of rings with arbitrary ring homomorphisms — Rings; units not required.
• Category of sets — Sets.
• Category of real Spin(8) representations — Real Spin(8) representations.
• Category of topological fields — Hausdorff topological fields.
• Category of topological groups — Hausdorff topological groups.
• Category of unital topological rings — Unital Hausdorff topological rings.
• Category of unital associative algebras — R-unital associative algebras; C-unital associative algebras; Q-unital associative algebras; F-unital associative algebras; K-unital associative algebras.
• Category UJord of unital Jordan algebras with arbitrary maps — UJord over R; UJord over C.
• Category Jord1 of unital Jordan algebras and unital maps — Jord1 over R; Jord1 over C.
• Category of unital rings — Unital rings.
• Category of vector spaces over a field — R-vector spaces; C-vector spaces; Q-vector spaces; F-vector spaces; K-vector spaces; F_2-vector spaces; F_3-vector spaces; F_p-vector spaces.

Data and proof status

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.

Download the indexed JSON · Download the SQLite catalogue

Separate algebra-class additions

The orders and fractional ideals reading list covers the separate transcript request, including new definitions and existing prerequisites reused.