# Lie Groups/Algebras, Notes 3: Bialgebras and Free Lie Algebras

Recall the construction of an associative algebra $A$ over a field $K.$  We have a module homomorphism $\nabla:A\otimes A\to A$ such that

$\nabla\circ (1\otimes\nabla)=\nabla\circ(\nabla\otimes 1).$

If the algebra is unital, then we also a canonical map $\eta:K\to A$ with $\eta(1)$ defined as the identity of $A$ and corresponding assumption that

$\nabla\circ(1\otimes\eta)=\nabla\circ(\eta\otimes 1)=1$

on $A$ (with canonical identifications $A\otimes K=K\otimes A=A$).

We can dually define a $K$coalgebra as a $K$-module $C$ together with a comultiplication map $\Delta:C\to C\otimes C$ such that

$(1\otimes\Delta)\circ\Delta=\Delta\circ(\Delta\otimes 1).$

We also call it unital if there is a map $\varepsilon:C\to K$ such that

$(1\otimes\varepsilon)\circ\Delta=(\varepsilon\otimes 1)\circ\Delta=1$

on $C$ (with the canonical identification $C\otimes K=K\otimes C=C$).

Definition 1.  A $K$bialgebra is a unital $K$-algebra and unital $K$-coalgebra such that

1. $\Delta\circ\nabla=(\nabla\otimes\nabla)\circ(1\otimes\tau\otimes 1)\circ(\Delta\otimes\Delta)$ ($\tau(x\otimes y)=y\otimes x$),
2. $\varepsilon\circ\nabla=\varepsilon\otimes\varepsilon,$
3. $\Delta\circ\eta=\eta\otimes\eta,$ and
4. $\varepsilon\circ\eta=1.$

Definition 2.  An element $x$ of a coalgebra is $u$primitive if $\Delta(x)=x\otimes u+u\otimes x.$  Elements are primitive if they are 1-primitive.

If $\sigma:\mathfrak{g}\to U(\mathfrak{g})$ is the embedding into the enveloping algebra, then the map $\Delta(\sigma(x))=\sigma(x)\otimes 1+1\otimes\sigma(x)$ gives $U(\mathfrak{g})$ a $K$-bialgebra structure.

Definition 3.  Let $X$ be a set and $A_K(X)$ be the free algebra over $X.$  We define the free Lie $K$algebra over $X$ as the quotient $A_K(X)/I$ where $I$ is the ideal generated by elements of the form $xy+yx$ and $x(yz)+z(xy)+y(zx).$  We will denote this Lie algebra by $L(X)$ and its product by $[x,y].$

The enveloping algebra $U(L(X))$ of the free Lie algebra $L(X)$ ends up being the free associative algebra over $X.$

[1]  Bourbaki, Nicholas.  Lie Groups and Lie Algebras, Chapters 1-3.  Springer-Verlag.  1971.