A short survey on pre-Lie algebras. (English) Zbl 1278.17001
Carey, Alan (ed.), Noncommutative geometry and physics: Renormalisation, motives, index theory. Based on the workshop “Number theory and physics”, Vienna, Austria, March 2009. Edited with the assistance of Harald Grosse and Steve Rosenberg. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-008-1/pbk). ESI Lectures in Mathematics and Physics, 89-102 (2011).
A (left) pre-Lie algebra over a field \(k\) is a pair \((A,\vartriangleright)\), consisting of a \(k\)-vector space \(A\) and a \(k\)-bilinear map \(A\otimes A\to A\), satisfying
\[
a\vartriangleright(b\vartriangleright c)- b\vartriangleright (a\vartriangleright c) - (a\vartriangleright b)\vartriangleright c + (b\vartriangleright) \vartriangleright c =0
\]
for all \(a,b,c\in V\). This is equivalent to requiring that
\[
\left[ L_a, L_b\right]= L_{[a,b]},
\]
where \(L_a=a\vartriangleright: A\to A\) denotes left multiplication by \(a\), and \([a,b]:=a\vartriangleright b-b\vartriangleright a\). Consequently, the commutator on the left is the commutator in \(\text{End}A\), i.e., \(L_a\circ L_b-L_b\circ L_a\). The commutator \([,]\) turns \(A\) into a Lie algebra, \(A_{Lie}\).
If \(V\) is a graded vector space, one can define a graded pre-Lie algebra by replacing commutators with graded commutators.
One can define similarly a right pre-Lie algebra, considering \((A,\vartriangleleft)\), where the right multiplication \(R_b(a)= a\vartriangleleft b\) is required to satisfy \([R_c,R_b]=R_{[b,c]}\), for all \(b,c\). Since passing from a left to right pre-Lie algebra is achieved by setting \(a\vartriangleright b= b\vartriangleleft a\), we will predominantly stick to left pre-Lie structures.
The name “pre-Lie algebra” was coined by M. Gerstenhaber, who observed in [Ann. Math. (2) 78, 267–288 (1963; Zbl 0131.27302)] that if \(A\) is an associative algebra, then its Hochschild cochain complex \(CH^\bullet(A)\), \(CH^m=\operatorname{Hom}(A^{\otimes m},A)\), carries the structure of a graded pre-Lie algebra. Pre-Lie algebras also appeared in the work of E. B. Vinberg [Trans. Mosc. Math. Soc. 12, 340–403 (1963); translation from Tr. Mosk. Mat. O.-va 12, 303–358 (1963; Zbl 0138.43301)], under the name “left-symmetric algebras”, hence this structure is sometimes called Vinberg algebra.
The paper under review is a short survey on the topic, including both modern and classical aspects of Vinberg algebras.
Section one is devoted to some classical properties. First, the author discusses the group of formal group laws, following A. Agrachev and R. Gamkrelidze [“Chronological algebras and nonstationary vector fields”, J. Sov. Math. 17, 1650–1675 (1981); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 11, 135–176 (1980; Zbl 0473.58021)].
If \(A\) admits a complete decreasing filtration, the Baker–Campbell–Hausdorff formula endows \(A\) with the structure of a pro-unipotent group. Namely, let \(W: A\to A\) be the bijection \(W(a)=e^{L_a}1-1\), where \(1\) is a “formal unit element”. One then uses \(W\) to transport the BCH product \(C(a,b)= a+b+ \frac{1}{2}[a,b]+\ldots\) to \(a\# b= W(C(\Omega(a),\Omega(b)))\), where \(\Omega=W^{-1}\). If the pre-Lie structure is associative, this reduces to \(a\#b = a+ b+ a\vartriangleright b\) . In Subsection 1.2 is given a version of the Poincar��–Birkhoff–Witt theorem (Theorem 1.1), which claims that there exists a canonical left \(\mathcal{U}(A_{Lie})\)-module structure on the symmetric algebra \(S(A)\), and a canonical left \(\mathcal{U}(A)\)-module isomorphism \(\eta: \mathcal{U}(A_{Lie})\to S(A)\), such that associated graded linear map is an isomorphism of commutative graded algebras. Subsection 1.3 discusses the Loday–Ronco theorem (Theorem 5.3) from [“Combinatorial Hopf Algebras”, (2010 Zbl 1217.16033)].
SarXiv:math/0209104], is discussed in 2.1, while 2.2 is devoted to the pre-Lie operad of F. Chapoton and M. Livernet, see [Int. Math. Res. Not. 2001, No. 8, 395–408 (2001; Zbl 1053.17001)]. Hopf algebras of rooted trees and Feynman graphs are discussed in 2.3 and 2.4, respectively.
One of the classical examples of a pre-Lie structure comes from geometry. Let \(M\) be a smooth manifold, and \(\nabla\) a flat, torsion-free connection. Then the space of global vector fields \(A=\Gamma(M,T_M)\) is a left pre-Lie algebra via \(X\vartriangleright Y:= \nabla_X Y\). Subsection 3.2 involves a discussion of an interesting relation between rooted trees and vector fields on \(\mathbb{R}^n\), discovered by A. Cayley [“On the theory of analytical forms called trees”, Phil. Mag. 13, 172–176 (1857)]. Subsection 3.4 contains a discussion of the application of rooted trees to Runge–Kutta methods for numerical solution of ODE.
The closing section 4 treats the links between pre-Lie structures and other algebraic structures introduced by J-L. Loday in [“Dialgebras”, Dialgebras and related operads. Berlin: Springer. Lect. Notes Math. 1763, 7-66 (2001; Zbl 0999.17002)], namely, dendriform algebras, Zinbiel algebras, and (multi) brace algebras.
For the entire collection see [Zbl 1217.00020].
If \(V\) is a graded vector space, one can define a graded pre-Lie algebra by replacing commutators with graded commutators.
One can define similarly a right pre-Lie algebra, considering \((A,\vartriangleleft)\), where the right multiplication \(R_b(a)= a\vartriangleleft b\) is required to satisfy \([R_c,R_b]=R_{[b,c]}\), for all \(b,c\). Since passing from a left to right pre-Lie algebra is achieved by setting \(a\vartriangleright b= b\vartriangleleft a\), we will predominantly stick to left pre-Lie structures.
The name “pre-Lie algebra” was coined by M. Gerstenhaber, who observed in [Ann. Math. (2) 78, 267–288 (1963; Zbl 0131.27302)] that if \(A\) is an associative algebra, then its Hochschild cochain complex \(CH^\bullet(A)\), \(CH^m=\operatorname{Hom}(A^{\otimes m},A)\), carries the structure of a graded pre-Lie algebra. Pre-Lie algebras also appeared in the work of E. B. Vinberg [Trans. Mosc. Math. Soc. 12, 340–403 (1963); translation from Tr. Mosk. Mat. O.-va 12, 303–358 (1963; Zbl 0138.43301)], under the name “left-symmetric algebras”, hence this structure is sometimes called Vinberg algebra.
The paper under review is a short survey on the topic, including both modern and classical aspects of Vinberg algebras.
Section one is devoted to some classical properties. First, the author discusses the group of formal group laws, following A. Agrachev and R. Gamkrelidze [“Chronological algebras and nonstationary vector fields”, J. Sov. Math. 17, 1650–1675 (1981); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 11, 135–176 (1980; Zbl 0473.58021)].
If \(A\) admits a complete decreasing filtration, the Baker–Campbell–Hausdorff formula endows \(A\) with the structure of a pro-unipotent group. Namely, let \(W: A\to A\) be the bijection \(W(a)=e^{L_a}1-1\), where \(1\) is a “formal unit element”. One then uses \(W\) to transport the BCH product \(C(a,b)= a+b+ \frac{1}{2}[a,b]+\ldots\) to \(a\# b= W(C(\Omega(a),\Omega(b)))\), where \(\Omega=W^{-1}\). If the pre-Lie structure is associative, this reduces to \(a\#b = a+ b+ a\vartriangleright b\) . In Subsection 1.2 is given a version of the Poincar��–Birkhoff–Witt theorem (Theorem 1.1), which claims that there exists a canonical left \(\mathcal{U}(A_{Lie})\)-module structure on the symmetric algebra \(S(A)\), and a canonical left \(\mathcal{U}(A)\)-module isomorphism \(\eta: \mathcal{U}(A_{Lie})\to S(A)\), such that associated graded linear map is an isomorphism of commutative graded algebras. Subsection 1.3 discusses the Loday–Ronco theorem (Theorem 5.3) from [“Combinatorial Hopf Algebras”, (2010 Zbl 1217.16033)].
SarXiv:math/0209104], is discussed in 2.1, while 2.2 is devoted to the pre-Lie operad of F. Chapoton and M. Livernet, see [Int. Math. Res. Not. 2001, No. 8, 395–408 (2001; Zbl 1053.17001)]. Hopf algebras of rooted trees and Feynman graphs are discussed in 2.3 and 2.4, respectively.
One of the classical examples of a pre-Lie structure comes from geometry. Let \(M\) be a smooth manifold, and \(\nabla\) a flat, torsion-free connection. Then the space of global vector fields \(A=\Gamma(M,T_M)\) is a left pre-Lie algebra via \(X\vartriangleright Y:= \nabla_X Y\). Subsection 3.2 involves a discussion of an interesting relation between rooted trees and vector fields on \(\mathbb{R}^n\), discovered by A. Cayley [“On the theory of analytical forms called trees”, Phil. Mag. 13, 172–176 (1857)]. Subsection 3.4 contains a discussion of the application of rooted trees to Runge–Kutta methods for numerical solution of ODE.
The closing section 4 treats the links between pre-Lie structures and other algebraic structures introduced by J-L. Loday in [“Dialgebras”, Dialgebras and related operads. Berlin: Springer. Lect. Notes Math. 1763, 7-66 (2001; Zbl 0999.17002)], namely, dendriform algebras, Zinbiel algebras, and (multi) brace algebras.
For the entire collection see [Zbl 1217.00020].
Reviewer: Peter Dalakov (Trieste)
