nLab pre-Lie algebra

Definition

A pre-Lie algebra is a vector space $A$ equipped with a bilinear operation $\cdot: A \times A \to A$ such that

(1)$$[L_a, L_b] = L_{[a,b]}$$

for all $a, b\in A$. Here $L_a$ is the operation of left multiplication by $a$:

$$L_a b = a \cdot b$$

and $[L_a, L_b]$ is the usual commutator of operators using composition:

$$[L_a, L_b] = L_a L_b - L_b L_a$$

while $[a,b]$ is the commutator defined using the $\cdot$ operation:

(2)$$[a,b] = a \cdot b - b \cdot a$$

Unravelling this, we see a pre-Lie algebra is vector space $A$ equipped with a bilinear operation $\cdot: A \times A \to A$ such that

(3)$$a \cdot (b \cdot c) - b \cdot (a \cdot c) = (a \cdot b) \cdot c - (b \cdot a) \cdot c$$

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 $A$ is an associative algebra, the operation

$$[a,b] = a \cdot b - b \cdot a$$

makes $A$ 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).

• Chengming Bai, An introduction to pre-Lie algebras. In: Algebra and Applications 1: Nonssociative Algebras and Categories, Wiley Online Library (2021), 245-273.

Examples

First, given a manifold with a flat torsion-free connection $\nabla$ on its tangent bundle, we can make the space of tangent vector fields into a pre-Lie algebra by defining

$$v \cdot w = \nabla_v w$$

The definition of ‘flat’ is precisely (1), whereas that of ‘torsion-free’ is precisely (2). The Lie algebra arising from this pre-Lie algebra is just the usual Lie algebra of vector fields.

Second, suppose $O$ is a linear operad, and let $A$ be the free $O$-algebra on one generator. As a vector space we have

$$A = \bigoplus_{n} O_n/S_n \, .$$

Here $S_n$ is the symmetric group, which acts on the space $O_n$ of $n$-ary operations of $O$. Moreover, $A$ becomes a pre-Lie algebra in a manner described here:

• Dominique Manchon, A short survey on pre-Lie algebras

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:

• Justin Thomas, Graduate student seminar: Hochschild cohomology

In fact it was Gerstenhaber who coined the term ‘pre-Lie algebra’, for this reason.

Relation to the work of Connes and Kreimer

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:

• John Baez, This Week’s Finds in Mathematical Physics, Week 299. (web) (blog)

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:

• Alain Connes and Dirk Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199 (1998), 203–242 (arXiv)

Pre-Lie algebras are algebras of a linear operad called $PL$. The space $PL_n$ has a basis given by labelled rooted trees with $n$ vertices, and the $i$th partial composite $s \circ_i t$ is given by summing all the possible ways of inserting the tree $t$ inside the tree $s$ at the vertex labelled $i$. For details see:

• Frédéric Chapoton, Muriel Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not. 2001 (2001), 395-408.

The free pre-Lie algebra on one generator is thus

$$\bigoplus_{n} PL_n /S_n \,$$

so the description of $PL_n$ in terms of rooted trees gives a kind of ‘explanation’ of the relation between the Connes–Kreimer Hopf algebra and rooted trees.

Self-referentiality

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 $O$, the free $O$-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:

• Dominique Manchon, A short survey on pre-Lie algebras. (pdf)

References

The best overall introduction to pre-Lie algebras seems to be that by Dominique Manchon, cited above. For two more introductions, try the following:

• John Baez, This Week’s Finds in Mathematical Physics, Week 299. (web) (blog)

• Frédéric Chapoton, Operadic point of view on the Hopf algebra of rooted trees. (pdf)