nLab
pre-Lie algebra

Pre-Lie algebras

Definition

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

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

for all a,bAa, b\in A. Here L aL_a is the operation of left multiplication by aa:

L ab=ab L_a b = a \cdot b

and [L a,L b][L_a, L_b] is the usual commutator of operators using composition:

[L a,L b]=L aL bL bL a [L_a, L_b] = L_a L_b - L_b L_a

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

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

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

(3)a(bc)b(ac)=(ab)c(ba)c 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 AA is an associative algebra, the operation

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

makes AA 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).

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

vw= vw 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 OO is a linear operad, and let AA be the free OO-algebra on one generator. As a vector space we have

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

Here S nS_n is the symmetric group, which acts on the space O nO_n of nn-ary operations of OO. Moreover, AA 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.

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 PLPL. The space PL nPL_n has a basis given by labelled rooted trees with nn vertices, and the iith partial composite s its \circ_i t is given by summing all the possible ways of inserting the tree tt inside the tree ss at the vertex labelled ii. 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

nPL n/S n\bigoplus_{n} PL_n /S_n \,

so the description of PL nPL_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 OO, the free OO-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:

  • 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)

Revised on September 25, 2014 16:14:05 by Blake Stacey (209.6.90.149)