Math 2.0

Associative rings and algebras Hopf algebras and quantum groups Theorem

Nichols–Zoeller theorem

Builds on Hopf algebraFree moduleIndex of a subgroupLagrange's theorem

Statement

Let KK be a field and let HH be a finite-dimensional Hopf algebra over KK. Let B⊆HB \subseteq H be a Hopf subalgebra, that is a subalgebra with Δ(B)⊆B⊗B\Delta(B) \subseteq B \otimes B and S(B)⊆BS(B) \subseteq B. Then HH is a free left BB-module and a free right BB-module. In particular, dim⁡KB\dim_K B divides dim⁡KH\dim_K H.

Remarks

The proof views HH as a relative Hopf module: a right BB-module under multiplication with the compatible right HH-coaction Δ\Delta. One shows that a direct sum of finitely many copies of such a module is free over BB. A Krull–Schmidt argument for finite-dimensional modules then shows that HH itself is free. Example: for a finite group GG and a subgroup LL, the group algebra KGKG is a free right KLKL-module with basis a set of representatives of the left cosets gLgL. Then dim⁡KL=∣L∣\dim KL = \abs{L} divides dim⁡KG=∣G∣\dim KG = \abs{G}, which is Lagrange's theorem. The divisibility statement follows from freeness: if H≅BnH \cong B^n as BB-modules, then dim⁡H=ndim⁡B\dim H = n \dim B.

Going deeper

For a student

The theorem is a basic tool in the classification of finite-dimensional Hopf algebras of small dimension. For example, the group-like elements of HH span a Hopf subalgebra KG(H)KG(H), so the order of the group of group-likes divides dim⁡H\dim H. The index analogue is the integer dim⁡H/dim⁡B\dim H / \dim B, the rank of HH as a free BB-module.

Reading path

To understand this page from the ground up, read these 39 pages in order. The foundations come first. The path takes the proof Proof of Lagrange's theorem.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionFree module, concept, no evidence levelFree moduleconceptGroup, concept, no evidence levelGroupconceptHopf algebra, concept, no evidence levelHopf algebraconceptIndex of a subgroup, concept, no evidence levelIndex of a subgroupconceptLeft coset, concept, no evidence levelLeft cosetconceptProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionUnion, definition, no evidence levelUniondefinitionSet-builder notation, concept, no evidence levelSet-builder notationconceptExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionDisjoint sets, definition, no evidence levelDisjoint setsdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionCosets have the same cardinality as the subgroup, statement, no evidence levelCosets have the same …statementUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionBinary relation, definition, no evidence levelBinary relationdefinitionPartition of a set, definition, no evidence levelPartition of a setdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionSymmetric relation, definition, no evidence levelSymmetric relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionEquivalence relation, definition, no evidence levelEquivalence relationdefinitionEquivalence class, definition, no evidence levelEquivalence classdefinitionEquivalence classes form a partition, statement, no evidence levelEquivalence classes f…statementProof of Lagrange's theorem, proof, no evidence levelProof of Lagrange's t…proofLagrange's theorem, statement, no evidence levelLagrange's theoremstatementNichols–Zoeller theorem, statement, no evidence levelNichols–Zoeller theor…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Free module concept
  3. Group concept
  4. Hopf algebra concept
  5. Index of a subgroup concept
  6. Left coset concept
  7. Proposition concept
  8. Set definition
  9. Subgroup definition
  10. Conjunction definition
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Empty set object
  15. Family of sets definition
  16. Ordered pair definition
  17. Intersection definition
  18. Left coset definition
  19. Right coset definition
  20. Union definition
  21. Set-builder notation concept
  22. Existential quantifier definition
  23. Universal quantifier definition
  24. Disjoint sets definition
  25. Cartesian product definition
  26. Subset definition
  27. Cosets have the same cardinality as the subgroup statement
  28. Union of a family of sets definition
  29. Binary relation definition
  30. Partition of a set definition
  31. Reflexive relation definition
  32. Symmetric relation definition
  33. Transitive relation definition
  34. Equivalence relation definition
  35. Equivalence class definition
  36. Equivalence classes form a partition statement
  37. Proof of Lagrange's theorem proof
  38. Lagrange's theorem statement
  39. Nichols–Zoeller theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Hopf_algebra
About this page
Tag856B6C7EE4
URIm2://856B6C7EE4
Kindstatement, theorem
WritingW0 stub
MSC16T05
StatusThis copy of the site has no build data. Run m2 status 856B6C7EE4.
AboutHopf algebra
UsesHopf algebra, Free module, Index of a subgroup, Lagrange's theorem

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