- Source: Pre-Lie algebra
In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space.
The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber in his work on deformations of algebras.
Pre-Lie algebras have been considered under some other names, among which one can cite left-symmetric algebras, right-symmetric algebras or Vinberg algebras.
Definition
A pre-Lie algebra
(
V
,
◃
)
{\displaystyle (V,\triangleleft )}
is a vector space
V
{\displaystyle V}
with a linear map
◃
:
V
⊗
V
→
V
{\displaystyle \triangleleft :V\otimes V\to V}
, satisfying the relation
(
x
◃
y
)
◃
z
−
x
◃
(
y
◃
z
)
=
(
x
◃
z
)
◃
y
−
x
◃
(
z
◃
y
)
.
{\displaystyle (x\triangleleft y)\triangleleft z-x\triangleleft (y\triangleleft z)=(x\triangleleft z)\triangleleft y-x\triangleleft (z\triangleleft y).}
This identity can be seen as the invariance of the associator
(
x
,
y
,
z
)
=
(
x
◃
y
)
◃
z
−
x
◃
(
y
◃
z
)
{\displaystyle (x,y,z)=(x\triangleleft y)\triangleleft z-x\triangleleft (y\triangleleft z)}
under the exchange of the two variables
y
{\displaystyle y}
and
z
{\displaystyle z}
.
Every associative algebra is hence also a pre-Lie algebra, as the associator vanishes identically. Although weaker than associativity, the defining relation of a pre-Lie algebra still implies that the commutator
x
◃
y
−
y
◃
x
{\displaystyle x\triangleleft y-y\triangleleft x}
is a Lie bracket. In particular, the Jacobi identity for the commutator follows from cycling the
x
,
y
,
z
{\displaystyle x,y,z}
terms in the defining relation for pre-Lie algebras, above.
Examples
= Vector fields on an affine space
=Let
U
⊂
R
n
{\displaystyle U\subset \mathbb {R} ^{n}}
be an open neighborhood of
R
n
{\displaystyle \mathbb {R} ^{n}}
, parameterised by variables
x
1
,
⋯
,
x
n
{\displaystyle x_{1},\cdots ,x_{n}}
. Given vector fields
u
=
u
i
∂
x
i
{\displaystyle u=u_{i}\partial _{x_{i}}}
,
v
=
v
j
∂
x
j
{\displaystyle v=v_{j}\partial _{x_{j}}}
we define
u
◃
v
=
v
j
∂
u
i
∂
x
j
∂
x
i
{\displaystyle u\triangleleft v=v_{j}{\frac {\partial u_{i}}{\partial x_{j}}}\partial _{x_{i}}}
.
The difference between
(
u
◃
v
)
◃
w
{\displaystyle (u\triangleleft v)\triangleleft w}
and
u
◃
(
v
◃
w
)
{\displaystyle u\triangleleft (v\triangleleft w)}
, is
(
u
◃
v
)
◃
w
−
u
◃
(
v
◃
w
)
=
v
j
w
k
∂
2
u
i
∂
x
j
∂
x
k
∂
x
i
{\displaystyle (u\triangleleft v)\triangleleft w-u\triangleleft (v\triangleleft w)=v_{j}w_{k}{\frac {\partial ^{2}u_{i}}{\partial x_{j}\partial x_{k}}}\partial _{x_{i}}}
which is symmetric in
v
{\displaystyle v}
and
w
{\displaystyle w}
. Thus
◃
{\displaystyle \triangleleft }
defines a pre-Lie algebra structure.
Given a manifold
M
{\displaystyle M}
and homeomorphisms
ϕ
,
ϕ
′
{\displaystyle \phi ,\phi '}
from
U
,
U
′
⊂
R
n
{\displaystyle U,U'\subset \mathbb {R} ^{n}}
to overlapping open neighborhoods of
M
{\displaystyle M}
, they each define a pre-Lie algebra structure
◃
,
◃
′
{\displaystyle \triangleleft ,\triangleleft '}
on vector fields defined on the overlap. Whilst
◃
{\displaystyle \triangleleft }
need not agree with
◃
′
{\displaystyle \triangleleft '}
, their commutators do agree:
u
◃
v
−
v
◃
u
=
u
◃
′
v
−
v
◃
′
u
=
[
v
,
u
]
{\displaystyle u\triangleleft v-v\triangleleft u=u\triangleleft 'v-v\triangleleft 'u=[v,u]}
, the Lie bracket of
v
{\displaystyle v}
and
u
{\displaystyle u}
.
= Rooted trees
=Let
T
{\displaystyle \mathbb {T} }
be the free vector space spanned by all rooted trees.
One can introduce a bilinear product
↶
{\displaystyle \curvearrowleft }
on
T
{\displaystyle \mathbb {T} }
as follows. Let
τ
1
{\displaystyle \tau _{1}}
and
τ
2
{\displaystyle \tau _{2}}
be two rooted trees.
τ
1
↶
τ
2
=
∑
s
∈
V
e
r
t
i
c
e
s
(
τ
1
)
τ
1
∘
s
τ
2
{\displaystyle \tau _{1}\curvearrowleft \tau _{2}=\sum _{s\in \mathrm {Vertices} (\tau _{1})}\tau _{1}\circ _{s}\tau _{2}}
where
τ
1
∘
s
τ
2
{\displaystyle \tau _{1}\circ _{s}\tau _{2}}
is the rooted tree obtained by adding to the disjoint union of
τ
1
{\displaystyle \tau _{1}}
and
τ
2
{\displaystyle \tau _{2}}
an edge going from the vertex
s
{\displaystyle s}
of
τ
1
{\displaystyle \tau _{1}}
to the root vertex of
τ
2
{\displaystyle \tau _{2}}
.
Then
(
T
,
↶
)
{\displaystyle (\mathbb {T} ,\curvearrowleft )}
is a free pre-Lie algebra on one generator. More generally, the free pre-Lie algebra on any set of generators is constructed the same way from trees with each vertex labelled by one of the generators.
References
Chapoton, F.; Livernet, M. (2001), "Pre-Lie algebras and the rooted trees operad", International Mathematics Research Notices, 2001 (8): 395–408, doi:10.1155/S1073792801000198, MR 1827084.
Szczesny, M. (2010), Pre-Lie algebras and incidence categories of colored rooted trees, vol. 1007, p. 4784, arXiv:1007.4784, Bibcode:2010arXiv1007.4784S.