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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) ).  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 1 (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