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 Kcoalgebra 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 Kbialgebra 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 uprimitive 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 Kalgebra 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.


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