Download A brief introduction to pre

A brief introduction to pre-Lie algebras
Chengming Bai
Chern Institute of Mathematics, Nankai University
Vasteras, November 16, 2016
Chengming Bai
A brief introduction to pre-Lie algebras
Outline
What is a pre-Lie algebra?
Examples of pre-Lie algebras
Pre-Lie algebras and classical Yang-Baxter equation
Pre-Lie algebras and vertex (operator) algebras
Chengming Bai
A brief introduction to pre-Lie algebras
What is a pre-Lie algebra?
Definition
A pre-Lie algebra A is a vector space with a binary operation
(x, y) → xy satisfying
(xy)z − x(yz) = (yx)z − y(xz), ∀x, y, z ∈ A.
◦ Other names:
1
left-symmetric algebra
2
Koszul-Vinberg algebra
3
quasi-associative algebra
4
right-symmetric algebra
5
···
Chengming Bai
A brief introduction to pre-Lie algebras
(1)
What is a pre-Lie algebra?
Proposition
Let A be a pre-Lie algebra.
1
The commutator
[x, y] = xy − yx, ∀x, y ∈ A
(2)
defines a Lie algebra g(A), which is called the sub-adjacent
Lie algebra of A.
2
For any x, y ∈ A, let L(x) denote the left multiplication
operator. Then Eq. (1) is just
[L(x), L(y)] = L([x, y]), ∀x, y ∈ A,
(3)
which means that L : g(A) → gl(g(A)) with x → L(x) gives
a representation of the Lie algebra g(A).
Chengming Bai
A brief introduction to pre-Lie algebras
What is a pre-Lie algebra?
◦ The “left-symmetry” of associators.
That is, for any x, y, z ∈ A, the associator
(x, y, z) = (xy)z − x(yz)
(4)
(x, y, z) = (y, x, z)
(5)
satisfies
which is equivalent to Eq. (1).
Chengming Bai
A brief introduction to pre-Lie algebras
What is a pre-Lie algebra?
◦ A geometric interpretation of “left-symmetry”:
Let G be a Lie group with a left-invariant affine structure:
there is a flat torsion-free left-invariant affine connection ∇ on G,
namely, for all left-invariant vector fields X, Y, Z ∈ g = T (G),
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z = 0,
(6)
T (X, Y ) = ∇X Y − ∇Y X − [X, Y ] = 0.
(7)
This means both the curvature R(X, Y ) and torsion T (X, Y ) are
zero for the connection ∇. If we define
∇X Y = XY,
(8)
then the identity (1) for a pre-Lie algebra exactly amounts to Eq.
(6).
Chengming Bai
A brief introduction to pre-Lie algebras
What is a pre-Lie algebra?
◦ The origins and roles of pre-Lie algebras:
Convex homogeneous cones, affine manifolds and affine
structures on Lie groups: Koszul(1961), Vinberg (1963),
Auslander (1964), Medina (1981), Kim(1986), · · ·
Cohomology and deformations of associative algebras:
Gerstenhaber(1964), · · ·
Symplectic and Kähler structures on Lie groups: Chu(1973),
Shima(1980), Dardie and Medina(1995-6), · · ·
Complex structures on Lie groups: Andrada and Salamon
(2003), · · ·
Rooted trees: Cayley (1896), · · ·
Combinatorics: K. Ebrahimi-Fard (2002), · · ·
Operads: Chapoton and Livernet(2001), · · ·
Vertex algebras: Bakalov and Kac (2002), · · ·
Chengming Bai
A brief introduction to pre-Lie algebras
What is a pre-Lie algebra?
Phase spaces: Kuperschmidt(1994), · · ·
Integrable systems: Bordemann(1990), Winterhalder(1997),
···
Classical and quantum Yang-Baxter equations: Svinolupov
and Sokolov(1994), Etingof and Soloviev(1999), Golubschik
and Sokolov(2000), · · ·
Poisson brackets and infinite-dimensional Lie algebras:
Gel’fand and Dorfman(1979), Dubrovin and Novikov(1984),
Balinskii and Novikov(1985), · · ·
Quantum field theory and noncommutative geometry: Connes
and Kreimer(1998), · · ·
Dendriform algebras: Ronco (2000), Chapoton (2002), Aguiar
(2000), · · ·
Chengming Bai
A brief introduction to pre-Lie algebras
What is a pre-Lie algebra?
◦ Interesting structures:
Let A be a pre-Lie algebra and g(A) be the sub-adjacent Lie
algebra.
On A ⊕ A as a direct sum of vector spaces:
1
2
g(A) ⋉ad g(A) =⇒
g(A) ⋉L g(A) =⇒ Complex structures
On A ⊕ A∗ as a direct sum of vector spaces (A∗ is the dual
space of A):
1
2
g(A) ⋉ad∗ g(A)∗ =⇒ Manin Triple (Lie bialgebra)
g(A) ⋉L∗ g(A)∗ =⇒ Symplectic structures
Chengming Bai
A brief introduction to pre-Lie algebras
What is a pre-Lie algebra?
◦ Main Problems: Non-associativity!
1
There is not a suitable (and computable) representation
theory.
2
There is not a complete (and good) structure theory.
Example: There exists a transitive simple pre-Lie algebra
which combines “simplicity” and certain “nilpotence”:
e1 e2 = e2 , e1 e3 = −e3 , e2 e3 = e3 e2 = e1 .
◦ Ideas: Try to find more examples!
1
The relations with other topics (including application).
2
Realize by some known structures.
Chengming Bai
A brief introduction to pre-Lie algebras
Examples of pre-Lie algebras
◦ Construction from commutative associative algebras
Proposition
(S. Gel’fand) Let (A, ·) be a commutative associative algebra, and
D be its derivation. Then the new product
a ∗ b = a · Db, ∀a, b ∈ A
(9)
makes (A, ∗) become a pre-Lie algebra.
In fact, (A, ∗) is a Novikov algebra which is a pre-Lie algebra
satisfying an additional condition R(a)R(b) = R(b)R(a), where
R(a) is the right multiplication operator for any a ∈ A.
R(a)R(b) = R(b)R(a) defines a NAP algebra
Chengming Bai
A brief introduction to pre-Lie algebras
Examples of pre-Lie algebras
Background: Poisson brackets of hydrodynamic type and
Hamiltonian operators in the formal variational calculus.
{u(x1 ), v(x2 )} =
∂
x1
((uv)(x1 ))x−1
)
1 δ(
∂x1
x2
+(uv + vu)(x1 )
∂ −1 x1
x δ( ).
∂x1 1
x2
(10)
Generalization: (Bai-Meng’s conjecture) Every Novikov
algebra can be realized as the algebras (9) and their
(compatible) linear deformations.
Conclusion:
1
2
(Bai-Meng) The conjecture holds in dimension ≤ 3.
(Dzhumadil’daev-Lofwall) Any Novikov algebra is a quotient of
a subalgebra of an algebra given by Eq. (9).
Chengming Bai
A brief introduction to pre-Lie algebras
Examples of pre-Lie algebras
◦ Construction from linear functions
Proposition
Let V be a vector space over the complex field C with the ordinary
scalar product (, ) and a be a fixed vector in V , then
u ∗ v = (u, v)a + (u, a)v, ∀u, v ∈ V,
(11)
defines a pre-Lie algebra on V .
Background: The integrable (generalized) Burgers equation
Ut = Uxx + 2U ∗ Ux + (U ∗ (U ∗ U )) − ((U ∗ U ) ∗ U ). (12)
Generalization: Pre-Lie algebras from linear functions.
Conclusion: The pre-Lie algebra given by Eq. (11) is simple.
Chengming Bai
A brief introduction to pre-Lie algebras
Examples of pre-Lie algebras
◦ Construction from associative algebras
Proposition
Let (A, ·) be an associative algebra and R : A → A be a linear
map satisfying
R(x) · R(y) + R(x · y) = R(R(x) · y + x · R(y)), ∀x, y ∈ A. (13)
Then
x ∗ y = R(x) · y − y · R(x) − x · y, ∀x, y ∈ A
defines a pre-Lie algebra.
The above R is called Rota-Baxter map of weight 1.
Generalization: Approach from associative algebras.
Related to the so-called “modified classical Yang-Baxter
equation”.
Chengming Bai
A brief introduction to pre-Lie algebras
(14)
Examples of pre-Lie algebras
◦ Construction from Lie algebras (I)
1
Question 1: Whether there is a compatible pre-Lie algebra on
every Lie algebra?
Answer: No!
Necessary condition: The sub-adjacent Lie algebra of a
finite-dimensional pre-Lie algebra A over an algebraically
closed field with characteristic 0 satisfies
[g(A), g(A)] 6= g(A).
(15)
=⇒ g(A) can not be semisimple.
2
Question 2: How to construct a compatible pre-Lie algebra on
a Lie algebra?
Answer: etale affine representation ⇐⇒ bijective 1-cocycle.
Chengming Bai
A brief introduction to pre-Lie algebras
Examples of pre-Lie algebras
Let g be a Lie algebra and ρ : g → gl(V ) be a representation
of g. A 1-cocycle q associated to ρ (denoted by (ρ, q)) is defined
as a linear map from g to V satisfying
q[x, y] = ρ(x)q(y) − ρ(y)q(x), ∀x, y ∈ g.
(16)
Let (ρ, q) be a bijective 1-cocycle of g, then
x ∗ y = q −1 ρ(x)q(y), ∀x, y ∈ A,
(17)
defines a pre-Lie algebra structure on g. Conversely, for a pre-Lie
algebra A, (L, id) is a bijective 1-cocycle of g(A).
Proposition
There is a compatible pre-Lie algebra on a Lie algebra g if and only
if there exists a bijective 1-cocycle of g.
Chengming Bai
A brief introduction to pre-Lie algebras
Examples of pre-Lie algebras
Application: Providing a linearization procedure of
classification.
Conclusion: Classification of complex pre-Lie algebras up to
dimension 3.
Generalization: classical r-matrices! =⇒ a realization by Lie
algebras
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and classical Yang-Baxter equation
Definition
P
Let g be a Lie algebra and r =
ai ⊗ bi ∈ g ⊗ g. r is called a
i
solution of classical Yang-Baxter equation (CYBE) in g if
[r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] = 0 in U (g),
(18)
where U (g) is the universal enveloping algebra of g and
X
X
X
r12 =
ai ⊗bi ⊗1; r13 =
ai ⊗1⊗bi ; r23 =
1⊗ai ⊗bi . (19)
i
i
i
r is said to be skew-symmetric if
X
r=
(ai ⊗ bi − bi ⊗ ai ).
i
We also denote r 21 =
P
bi ⊗ a i .
i
Chengming Bai
A brief introduction to pre-Lie algebras
(20)
Pre-Lie algebras and classical Yang-Baxter equation
Definition
Let g be a Lie algebra and ρ : g → gl(V ) be a representation of
g. A linear map T : V → g is called an O-operator if T satisfies
[T (u), T (v)] = T (ρ(T (u))v − ρ(T (v))u), ∀u, v ∈ V.
Chengming Bai
A brief introduction to pre-Lie algebras
(21)
Pre-Lie algebras and classical Yang-Baxter equation
◦ Relationship between O-operators and CYBE
From CYBE to O-operators
Proposition
Let g be a Lie algebra and r ∈ g ⊗ g. Suppose that r is
skew-symmetric. Then r is a solution of CYBE in g if and only if r
regarded as a linear map from g∗ → g is an O-operator associated
to ad∗ .
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and classical Yang-Baxter equation
From O-operators to CYBE
Notation: let ρ : g → gl(V ) be a representation of the Lie
algebra g. On the vector space g ⊕ V , there is a natural Lie
algebra structure (denoted by g ⋉ρ V ) given as follows:
[x1 + v1 , x2 + v2 ] = [x1 , x2 ] + ρ(x1 )v2 − ρ(x2 )v1 ,
(22)
for any x1 , x2 ∈ g, v1 , v2 ∈ V .
Proposition
Let g be a Lie algebra. Let ρ : g → gl(V ) be a representation of g
and ρ∗ : g → gl(V ∗ ) be the dual representation. Let T : V → g be
a linear map which is identified as an element in
g ⊗ V ∗ ⊂ (g ⋉ρ∗ V ∗ ) ⊗ (g ⋉ρ∗ V ∗ ). Then r = T − T 21 is a
skew-symmetric solution of CYBE in g ⋉ρ∗ V ∗ if and only if T is
an O-operator.
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and classical Yang-Baxter equation
◦ Relationship between O-operators and pre-Lie algebras
Proposition
Let g be a Lie algebra and ρ : g → gl(V ) be a representation. Let
T : V → g be an O-operator associated to ρ, then
u ∗ v = ρ(T (u))v, ∀u, v ∈ V
defines a pre-Lie algebra on V .
Chengming Bai
A brief introduction to pre-Lie algebras
(23)
Pre-Lie algebras and classical Yang-Baxter equation
Lemma
Let g be a Lie algebra and (ρ, V ) is a representation. Suppose
f : g → V is invertible. Then f is a 1-cocycle of g associated to ρ
if and only if f −1 is an O-operator.
Corollary
Let g be a Lie algebra. There is a compatible pre-Lie algebra
structure on g if and only if there exists an invertible O-operator of
g.
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and classical Yang-Baxter equation
◦ Relationship between pre-Lie algebras and CYBE
From pre-Lie algebras to CYBE
Proposition
Let A be a pre-Lie algebra. Then
r=
n
X
(ei ⊗ e∗i − e∗i ⊗ ei )
i=1
is a solution of the classical Yang-Baxter equation in the Lie
algebra g(A) ⋉L∗ g(A)∗ , where {e1 , ..., en } is a basis of A and
{e∗1 , ..., e∗n } is the dual basis.
Chengming Bai
A brief introduction to pre-Lie algebras
(24)
Pre-Lie algebras and classical Yang-Baxter equation
From CYBE to pre-Lie algebras (Construction from Lie
algebras (II))
Lemma
Let g be a Lie algebra and f be a linear transformation on g. Then
on g the new product
x ∗ y = [f (x), y], ∀x, y ∈ g
(25)
defines a pre-Lie algebra if and only if
[f (x), f (y)] − f ([f (x), y] + [x, f (y)]) ∈ C(g), ∀x, y ∈ g,
where C(g) = {x ∈ g|[x, y] = 0, ∀y ∈ g} is the center of g.
Chengming Bai
A brief introduction to pre-Lie algebras
(26)
Pre-Lie algebras and classical Yang-Baxter equation
Corollary
Let g be a Lie algebra. Let r : g → g be an O-operator associated
to ad, that is,
[r(x), r(y)] − r([r(x), y] + [x, r(y)]) = 0, ∀x, y ∈ g.
(27)
Then Eq. (25) defines a pre-Lie algebra.
Remark
1 Eq. (27) is called operator form of CYBE;
2
Eq. (27) is the Rota-Baxter operator of weight zero in the
context of Lie algebras.
3
If there is a nondegenerate invariant bilinear form on g (that
is, as representations, ad is isomorphic to ad∗ ) and r is
skew-symmetric, then r satisfies CYBE if and only if r
regarded as a linear transformation on g satisfies Eq. (27).
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and classical Yang-Baxter equation
◦ An algebraic interpretation of “left-symmetry”:
Let {ei } be a basis of a Lie algebra g.PLet r : g → g be an
O-operator associated to ad. Set r(ei ) = j∈I rij ej . Then the
basis-interpretation of Eq. (25) is given as
X
ril [el , ej ].
(28)
ei ∗ ej =
l∈I
Such a construction of left-symmetric algebras can be regarded as
a Lie algebra “left-twisted” by a classical r-matrix.
On the other hand, let us consider the right-symmetry. We set
X
ei · ej = [ei , r(ej )] =
rjl [ei , el ].
(29)
l∈I
Then the above product defines a right-symmetric algebra on g,
which can be regarded as a Lie algebra “right-twisted” by a
classical r-matrix.
Application: Realization of pre-Lie algebras by Lie algebras.
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and vertex (operator) algebras
Definition
A vertex algebra is a vector space V equipped with a linear map
X
Y : V → Hom(V, V ((x))), v → Y (v, x) =
vn x−n−1
(30)
n∈Z
and equipped with a distinguished vector 1 ∈ V such that
Y (1, x) = 1;
Y (v, x)1 ∈ V [[x]] and Y (v, x)1|x=0 (= v−1 1) = v, ∀v ∈ V,
and for u, v ∈ V , there is Jacobi identity:
x−1
0 δ(
x1 − x2
x2 − x1
)Y (u, x1 )Y (v, x2 ) − x−1
)Y (v, x2 )Y (u, x1 )
0 δ(
x0
−x0
= x−1
2 δ(
x1 − x0
)Y (Y (u, x0 )v, x2 ).
x2
Chengming Bai
A brief introduction to pre-Lie algebras
(31)
Pre-Lie algebras and vertex (operator) algebras
Proposition
Let (V, Y, 1) be a vertex algebra. Then
a ∗ b = a−1 b
(32)
defines a pre-Lie algebra.
That is,
(a−1 b)−1 c − a−1 (b−1 c) = (b−1 a)−1 c − b−1 (a−1 c).
(33)
In fact, by Borcherd’s identities:
X
i
(am (b))n (c) =
(−1)i Cm
((am−i (bn+i (c)−(−1)m bm+n−i (ai (c))),
i≥0
(34)
let m = n = −1, we have
(a−1 b)−1 c − a−1 (b−1 c) =
X
(a−2−i (bi c) + b−2−i (ai c)).
i≥0
Chengming Bai
A brief introduction to pre-Lie algebras
(35)
Pre-Lie algebras and vertex (operator) algebras
Proposition
A vertex algebra is equivalent to a pre-Lie algebra and an algebra
names Lie conformal algebra with some compatible conditions.
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and vertex (operator) algebras
Definition
A Lie conformal algebra is a C[D]-module V endowed a C-linear
map V ⊗ V → C[λ] ⊗ V denoted by a ⊗ b → [aλ b], satisfying
[(Da)λ b] = −λ[aλ b];
[aλ (Db)] = (λ + D)[aλ b];
[bλ a] = [a−λ−T b];
[aλ b]λ+µ c = [aλ [bµ c] − [bµ [aλ c]].
Remark
Let (V, Y, 1) be a vertex algebra. Then
aλ b = Resx eλx Y (a, x)b =
X
λn an b/n!.
n≥0,finite
Chengming Bai
A brief introduction to pre-Lie algebras
(36)
Pre-Lie algebras and vertex (operator) algebras
Proposition
A vertex algebra is a pair (V, 1), where V is a C[D]-module and
1 ∈ V , endowed two operations:
1
V ⊗ V → C[λ] ⊗ V, a ⊗ b → [aλ b], Lie conformal algebra;
2
V ⊗ V → V , a ⊗ b → ab, a differential algebra with unit 1
and derivation D.
They satisfy the following conditions:
RD
RD
1 (ab)c − a(bc) = (
0 dλa)[bλ c] + ( 0 dλb)[aλ c];
R0
2 ab − ba =
−D dλa[aλ b];
Rλ
3 [a (bc)] = [a b]c + b[a c] +
λ
λ
λ
0 dµ[[aλ b] − µc].
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and vertex (operator) algebras
Roughly speaking,
Y (a, x)b = Y (a, x)+ b + Y (a, x)− b,
where
Y (a, x)+ b = (exD a)−1 b, Y (a, x)− b = (a−∂x b)(x−1 ).
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and vertex (operator) algebras
◦ Novikov algebras and infinite-dimensional Lie algebras and
vertex (operator) algebras
Proposition
Let A be a finite-dimensional algebra with a bilinear product
(a, b) → ab. Set
A = A ⊗ C[t, t−1 ].
(37)
Then the bracket
[a ⊗ tm , b ⊗ tn ] = (−mab + nba) ⊗ tm+n−1 , ∀a, b ∈ A, m, n ∈ Z
(38)
defines a Lie algebra structure on A if and only if A is a Novikov
algebra with the product ab.
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and vertex (operator) algebras
Let A be a Novikov algebra. There is a Z−grading on the Lie
algebra A defined by Eqs. (37) and (38):
M
A=
A(n) , A(n) = A ⊗ t−n+1 .
n∈Z
Moreover,
A(n≤1) =
M
A(n) =
n≤1
M
A(n) ⊕ A(1)
n≤0
is a Lie subalgebra of A. For any a ∈ A, we define the generating
function:
X
X
a(x) =
an x−n−1 =
(a ⊗ tn )x−n−1 ∈ A[[x, x−1 ]].
n∈Z
n∈Z
Then we have
[a(x1 ), b(x2 )] = (ab+ba)(x1 )
∂ −1 x1
∂
x1
x2 δ( )+[
ab(x1 )]x−1
).
2 δ(
∂x1
x2
∂x1
x2
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and vertex (operator) algebras
Let C be the trivial module of A(n≤1) and we can get the
following (Verma) module of A:
 = U (A) ⊗U (A(n≤1) ) C,
where U (A) (U (A(n≤1) )) is the universal enveloping algebra of A
(A(n≤1) ). Â is a natural Z-graded A-module:
 =
M
Â(n) ,
n≥0
where
(1)
(r)
Â(n) = {a−m1 · · · a−mr 1|m1 +· · ·+mr = n−r, m1 ≥ · · · ≥ mr ≥ 1, r ≥ 0
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and vertex (operator) algebras
Theorem
Let A be a Novikov algebra. Then there exists a unique vertex
algebra structure (Â, Y, 1) on  such that 1 = 1 ∈ C and
Y (a, x) = a(x), ∀a ∈ A, and
(1)
(r)
Y (a(n1 ) · · · a(nr ) 1, x) = a(1) (x)n1 · · · a(r) (x)nr 1Â
(39)
for any r ≥ 0, a(i) ∈ A, ni ∈ Z, where
a(x)n b(x) = Resx1 ((x1 − x)n a(x1 )b(x) − (−x + x1 )n b(x)a(x1 )).
(40)
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and vertex (operator) algebras
Theorem
Let (V, Y, 1) be a vertex algebra with the following properties:
1
V = ⊕n≥0 V(n) , V(0) = C1, V(1) = 0;
2
V is generated by V(2) with the following property
(1)
(r)
V = span{a−m1 · · · a−mr 1|mi ≥ 1, r ≥ 0, a(i) ∈ V(2) }
3
KerD = C, where D is a linear transformation of V given by
D(v) = v−2 1, ∀v ∈ V .
4
V is a graded (by the integers) vertex algebra, that is,
1 ∈ V(0) and V(i)n V(j) ⊂ V(i+j−n−1) .
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and vertex (operator) algebras
Theorem
(Continued) Then V is generated by V(2) with the following
property
(1)
(r)
V = span{a−m1 · · · a−mr 1|m1 ≥ · · · ≥ mr ≥ 1, r ≥ 0, a(i) ∈ V(2) }
and V(2) is a Novikov algebra with a product (a, b) → a ∗ b given by
a ∗ b = −D −1 (b0 a).
Chengming Bai
A brief introduction to pre-Lie algebras
Pre-Lie algebras and vertex (operator) algebras
Furthermore, if the Novikov algebra A has an identity e, then
−e corresponds to the Virasoro element which gives a vertex
operator algebra structure with zero central charge. With a
suitable central extension, the non-zero central charge will lead to
a commutative associative algebra with a nondegenerate
symmetric invariant bilinear form (the so-called Frobenius algebra)
Corollary
For the above vertex algebra (V, Y, 1), the algebra given by
a ∗ b = a−1 b
is a graded pre-Lie algebra, that is Vm ∗ Vn ⊂ Vm+n .
Chengming Bai
A brief introduction to pre-Lie algebras
The End
Thank You!
Chengming Bai
A brief introduction to pre-Lie algebras