A pre-Lie algebra is a vector space equipped with a bilinear operation such that
for all . Here is the operation of left multiplication by :
and is the usual commutator of operators using composition:
while is the commutator defined using the operation:
Unravelling this, we see a pre-Lie algebra is vector space equipped with a bilinear operation such that
More precisely, this is a left pre-Lie algebra. We can also define right pre-Lie algebras.
Every associative algebra is a pre-Lie algebra, but not conversely. The reason pre-Lie algebras have the name they do is that this weakening of the concept of associative algebra is still enough to give a Lie algebra! In other words: it is well-known that if is an associative algebra, the operation
makes into a Lie algebra. But this is also true for pre-Lie algebras! It is a fun exercise to derive the Jacobi identity from equation (3).
Here is the symmetric group, which acts on the space of -ary operations of . Moreover, becomes a pre-Lie algebra in a manner described here:
Third, the Hochschild chain complex of any associative algebra, with grading shifted down by one, can be given the structure of a ‘graded pre-Lie algebra’, as discovered by Gerstenhaber and described here:
In fact it was Gerstenhaber who coined the term ‘pre-Lie algebra’, for this reason.
Connes and Kreimer formalized the process of renormalization using a certain Hopf algebra built from Feynman diagrams. More abstractly we can understand the essence of their construction using a Hopf algebra built from rooted trees, as explained here:
The key is to form the free pre-Lie algebra on one generator, then turn this into a Lie algebra as described above, and then form the universal enveloping of that, which is a cocommutative Hopf algebra. Finally, the restricted dual of this cocommutative Hopf algebra is the commutative Hopf algebra considered by Connes and Kreimer here:
Pre-Lie algebras are algebras of a linear operad called . The space has a basis given by labelled rooted trees with vertices, and the th partial composite is given by summing all the possible ways of inserting the tree inside the tree at the vertex labelled . For details see:
The free pre-Lie algebra on one generator is thus
so the description of in terms of rooted trees gives a kind of ‘explanation’ of the relation between the Connes–Kreimer Hopf algebra and rooted trees.
Pre-Lie algebras have a strange self-referential feature. Every operad of a large class gives a pre-Lie algebra, but the operad for pre-Lie algebras is one of this class! This raises the following interesting puzzle.
As we have seen above, for any linear operad , the free -algebra with one generator becomes a pre-Lie algebra. But the operad for pre-Lie algebra is an operad of this type. So, the free pre-Lie algebra on one generator becomes a pre-Lie algebra in this way. But of course it already is a pre-Lie algebra! Do these pre-Lie structures agree?
The answer is no. For an explanation, see page 7 here:
The best overall introduction to pre-Lie algebras seems to be that by Dominique Manchon, cited above. For two more introductions, try the following: