Topic Hopf Algebras
topic admitted

Hopf Algebras

A Hopf algebra is the algebra structure on the coordinate ring of an affine group scheme. The five-fold definition consists of an \(R\)-module \(A\) equipped with:

name type role
multiplication \(\mu\) \(A \otimes_R A \to A\) algebra product
unit \(\eta\) \(R \to A\) algebra unit
comultiplication \(\Delta\) \(A \to A \otimes_R A\) coalgebra coproduct
counit \(\varepsilon\) \(A \to R\) coalgebra counit
antipode \(S\) \(A \to A\) inverse map

subject to associativity, unit, coassociativity, counit, bialgebra compatibility (\(\Delta\) and \(\varepsilon\) are algebra maps), and the antipode axiom

\[ \mu \circ (S \otimes \operatorname{id}_A) \circ \Delta \;=\; \mu \circ (\operatorname{id}_A \otimes S) \circ \Delta \;=\; \eta \circ \varepsilon. \]

The geometric content of the antipode axiom is the group-axiom \(g \cdot g^{-1} = e\).

Hierarchy of structures

Level Structure Adds
Algebra \(\mu, \eta\) associativity, unit
Coalgebra \(\Delta, \varepsilon\) coassociativity, counit
Bialgebra both bialgebra compatibility
Hopf algebra bialgebra + \(S\) antipode axiom

A Hopf algebra is commutative if \(\mu \circ \tau = \mu\) (algebra multiplication is commutative), and cocommutative if \(\tau \circ \Delta = \Delta\) (comul is symmetric).

The dictionary with group schemes:

Hopf algebra over \(R\) Affine group scheme over \(\operatorname{Spec} R\)
commutative affine
cocommutative abelian (commutative group law)

Canonical examples

  1. Group algebra \(R[G]\) for a group \(G\): group-like Hopf structure \(\Delta(g) = g \otimes g\), \(\varepsilon(g) = 1\), \(S(g) = g^{-1}\). Corresponds to the constant group scheme \(\underline{G}\) (or to the diagonalisable group of which \(G\) is the character group, depending on convention).

  2. Polynomial Hopf algebra \(R[T]\) with primitive structure: \(\Delta(T) = T \otimes 1 + 1 \otimes T\), \(\varepsilon(T) = 0\), \(S(T) = -T\). Corresponds to \(\mathbb{G}_a\).

  3. Laurent polynomial Hopf algebra \(R[T, T^{-1}]\) with group-like structure: \(\Delta(T) = T \otimes T\), \(\varepsilon(T) = 1\), \(S(T) = T^{-1}\). Corresponds to \(\mathbb{G}_m\).

  4. Coordinate Hopf algebra \(R[X_{ij}, \det^{-1}]\) of \(\operatorname{GL}_n\) with matrix-multiplication coproduct \(\Delta(X_{ij}) = \sum_k X_{ik} \otimes X_{kj}\).

  5. Universal enveloping algebra \(U(\mathfrak{g})\) of a Lie algebra: cocommutative non-commutative Hopf algebra; primitive elements are the Lie algebra.

Key constructions

  • Tensor product: \(A \otimes_R B\) of two Hopf algebras is a Hopf algebra (with diagonal structure).
  • Restriction of scalars: for a ring map \(R \to R'\), an \(R'\)-Hopf algebra restricts to an \(R\)-Hopf algebra.
  • Base change: \(A \otimes_R R'\) is an \(R'\)-Hopf algebra.
  • Quotients by Hopf ideals: ideals closed under comul and antipode give quotient Hopf algebras (corresponding to closed subgroups).

Antipode anti-multiplicativity

For ANY Hopf algebra \(A\) (commutative or not), \(S(a \cdot b) = S(b) \cdot S(a)\): the antipode is an anti-homomorphism for the multiplication. For commutative \(A\), this reduces to \(S(ab) = S(a) S(b)\), making \(S\) an algebra homomorphism — see [[node:affine_group_schemes.antipode_multiplicativity_commutative_hopf|the commutative version]].

Mathlib formalisation

  • Mathlib.RingTheory.HopfAlgebra.Basic — the typeclass + antipode axioms.
  • Mathlib.RingTheory.Bialgebra.Basic — the Bialgebra layer.
  • Mathlib.RingTheory.Coalgebra.Basic — the Coalgebra layer.
  • Mathlib.RingTheory.HopfAlgebra.TensorProduct, MonoidAlgebra, etc. — specific instances.
  • Mathlib.RingTheory.Coalgebra.Convolution — the convolution semiring structure on linear maps, used to formulate the antipode axiom and study convolution inverses.

In this project, <code>hopfSpec</code> makes Hopf algebras into affine group schemes via Spec; the inverse direction (coordinate Hopf algebra of a given affine group scheme) is the content of [[node:affine_group_schemes.coordinate_hopf_algebra_of_group_scheme|the coordinate Hopf algebra]] node.

References

  • Sweedler, Hopf Algebras (1969).
  • Montgomery, Hopf Algebras and Their Actions on Rings (CBMS, 1993).
  • Underwood, Fundamentals of Hopf Algebras (Universitext, 2015).

Prerequisite Topics

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