Alphabeta MathBialgebras, Convolution, and Antipode Identities
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 6 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 6 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
Categorification and Geometric Representation Theory
Linear Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Lemma
The tensor product of two coalgebras is a coalgebra
Definition
Bialgebras and bialgebra maps
Lemma
Convolution of linear maps is associative and unital
Lemma
The two standard bialgebra compatibility formulations are equivalent
Definition
Hopf algebras and antipodes
Theorem
Uniqueness and anti-multiplicativity and anti-comultiplicativity of the antipode
Lemma
Group algebras, finite-group function algebras, and polynomial Hopf algebras
Lemma