Hopf Algebra

definition example

A bialgebra in a Symmetric Monoidal Category is an object carrying both a monoid $(\mu ,\eta )$ and a comonoid $(\delta ,ϵ)$ structure that interact by the bialgebra laws: $\delta \circ \mu =(\mu \otimes \mu )\circ (\mathrm{id}\otimes \sigma \otimes \mathrm{id})\circ (\delta \otimes \delta )$, $ϵ\circ \mu =ϵ\otimes ϵ$, $\delta \circ \eta =\eta \otimes \eta$, $ϵ\circ \eta ={\mathrm{id}}_I$ (the comultiplication is a monoid homomorphism). A Hopf algebra is a bialgebra with an antipode $s:H\to H$ satisfying $\mu \circ (s\otimes \mathrm{id})\circ \delta =\eta \circ ϵ=\mu \circ (\mathrm{id}\otimes s)\circ \delta$ — the analogue of inverses in a group.

Sources: 7 Sketches §5.3.3 (Theorem 5.60: the equations of Graphical Linear Algebra include the bialgebra laws for the black/white pairs; the antipode is the scalar $-1$), §5.4.2, §6.3.1 (contrast: Frobenius laws); DaoFP §5.3, §16.2 (comonoids).

• Contrast with a Frobenius Monoid: in a Frobenius structure the monoid and comonoid satisfy $\delta \circ \mu =(\mu \otimes \mathrm{id})\circ (\mathrm{id}\otimes \delta )$ (they “commute past each other”), whereas in a bialgebra they distribute. In the prop $\mathbf{Mat}(R)$, each colour of copy/add nodes is Frobenius on its own, and the two colours together form a bialgebra (7 Sketches Theorem 5.60); with the $-1$ scalar as antipode, $\mathrm{Mat}(\mathbb{Z})$ is a Hopf algebra — “graphical linear algebra”.
• Classical examples: the group algebra $k[G]$ with $\delta (g)=g\otimes g$, $s(g)=g^{-1}$; the universal enveloping algebra of a Lie algebra; the algebra of functions on a finite group.

JuliaLeanHaskell

Docs: Theories (Catlab)

using Catlab
# the bialgebra law inside the free hypergraph/symmetric monoidal theory of Mat(ℤ) is the
# "copy then add" = "add then copy" equation; here we check it on actual matrices:
copy_(v) = vcat(v, v); add_(v) = v[1:end÷2] .+ v[end÷2+1:end]
v = [1, 2]; w = [3, 4]
copy_(add_(vcat(v, w))) == let (v1, v2, w1, w2) = (v, v, w, w); vcat(add_(vcat(v1, w1)), add_(vcat(v2, w2))) end   # true

import Mathlib
#check @Bialgebra           -- class Bialgebra R A
#check @HopfAlgebra         -- with antipode
#check @MonoidAlgebra       -- the group algebra k[G], a Hopf algebra

-- bialgebra structure on Int-vectors: copy and add distribute
copy :: [Int] -> ([Int], [Int])
copy v = (v, v)
add :: ([Int], [Int]) -> [Int]
add (v, w) = zipWith (+) v w
-- copy (add (v, w)) == (add (v, w), add (v, w)) == let (v1,v2)=copy v; (w1,w2)=copy w in (add (v1,w1), add (v2,w2))