A brief introduction to pre-lie algebras

Transcription

1 Chern Institute of Mathematics, Nankai University Vasteras, November 16, 2016

2 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

3 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. (1) Other names: 1 left-symmetric algebra 2 Koszul-Vinberg algebra 3 quasi-associative algebra 4 right-symmetric algebra 5

4 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).

5 What is a pre-lie algebra? The left-symmetry of associators. That is, for any x,y,z A, the associator satisfies which is equivalent to Eq. (1). (x,y,z) = (xy)z x(yz) (4) (x,y,z) = (y,x,z) (5)

6 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).

7 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),

8 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),

9 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 g(a) ad g(a) = 2 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 g(a) ad g(a) = Manin Triple (Lie bialgebra) 2 g(a) L g(a) = Symplectic structures

10 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 : e 1 e 2 = e 2,e 1 e 3 = e 3,e 2 e 3 = e 3 e 2 = e 1. Ideas: Try to find more examples! 1 The relations with other topics (including application). 2 Realize by some known structures.

11 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 makes (A, ) become a pre-lie algebra. a b = a Db, a,b A (9) 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

12 Examples of pre-lie algebras Background: Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. {u(x 1 ),v(x 2 )} = x 1 ((uv)(x 1 ))x 1 1 δ(x 1 x 2 ) +(uv +vu)(x 1 ) x 1 x 1 1 δ(x 1 x 2 ). (10) Generalization: (Bai-Meng s conjecture) Every Novikov algebra can be realized as the algebras (9) and their (compatible) linear deformations. Conclusion: 1 (Bai-Meng) The conjecture holds in dimension 3. 2 (Dzhumadil daev-lofwall) Any Novikov algebra is a quotient of a subalgebra of an algebra given by Eq. (9).

13 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 defines a pre-lie algebra on V. u v = (u,v)a+(u,a)v, u,v V, (11) Background: The integrable (generalized) Burgers equation U t = U xx +2U U x +(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.

14 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 (14) 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.

15 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) can not be semisimple. [g(a),g(a)] g(a). (15) 2 Question 2: How to construct a compatible pre-lie algebra on a Lie algebra? Answer: etale affine representation bijective 1-cocycle.

16 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.

17 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

18 Pre-Lie algebras and classical Yang-Baxter equation Definition Let g be a Lie algebra and r = a i b i g g. r is called a i solution of classical Yang-Baxter equation (CYBE) in g if [r 12,r 13 ]+[r 12,r 23 ]+[r 13,r 23 ] = 0 in U(g), (18) where U(g) is the universal enveloping algebra of g and r 12 = i a i b i 1;r 13 = i a i 1 b i ;r 23 = i 1 a i b i. (19) r is said to be skew-symmetric if r = i (a i b i b i a i ). (20) We also denote r 21 = i b i a i.

19 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. (21)

20 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.

21 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: [x 1 +v 1,x 2 +v 2 ] = [x 1,x 2 ]+ρ(x 1 )v 2 ρ(x 2 )v 1, (22) for any x 1,x 2 g,v 1,v 2 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.

22 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 defines a pre-lie algebra on V. u v = ρ(t(u))v, u,v V (23)

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.

24 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 (e i e i e i e i) (24) i=1 is a solution of the classical Yang-Baxter equation in the Lie algebra g(a) L g(a), where {e 1,...,e n } is a basis of A and {e 1,...,e n } is the dual basis.

25 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 defines a pre-lie algebra if and only if x y = [f(x),y], x,y g (25) [f(x),f(y)] f([f(x),y]+[x,f(y)]) C(g), x,y g, (26) where C(g) = {x g [x,y] = 0, y g} is the center of g.

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).

27 Pre-Lie algebras and classical Yang-Baxter equation An algebraic interpretation of left-symmetry : Let {e i } be a basis of a Lie algebra g. Let r : g g be an O-operator associated to ad. Set r(e i ) = j I r ije j. Then the basis-interpretation of Eq. (25) is given as e i e j = l I r il [e l,e j ]. (28) 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 e i e j = [e i,r(e j )] = l I r jl [e i,e l ]. (29) 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.

28 Pre-Lie algebras and vertex (operator) algebras Definition A vertex algebra is a vector space V equipped with a linear map Y : V Hom(V,V((x))),v Y(v,x) = n Zv n x n 1 (30) 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(x 1 x 2 x 0 )Y(u,x 1 )Y(v,x 2 ) x 1 0 δ(x 2 x 1 x 0 )Y(v,x 2 )Y(u,x 1 ) = x 1 2 δ(x 1 x 0 x 2 )Y(Y(u,x 0 )v,x 2 ). (31)

29 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: (a m (b)) n (c) = i 0( 1) i Cm((a i m i (b n+i (c) ( 1) m b m+n i (a i (c))), (34) let m = n = 1, we have (a 1 b) 1 c a 1 (b 1 c) = 2 i (b i c)+b 2 i (a i c)). (35) i 0(a

30 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.

31 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 [a λ b], satisfying [(Da) λ b] = λ[a λ b]; [a λ (Db)] = (λ+d)[a λ b]; [b λ a] = [a λ T b]; [a λ b] λ+µ c = [a λ [b µ c] [b µ [a λ c]]. Remark Let (V,Y,1) be a vertex algebra. Then a λ b = Res x e λx Y(a,x)b = λ n a n b/n!. (36) n 0,finite

32 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 [a λ 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: 1 (ab)c a(bc) = ( D 0 dλa)[b λc]+( D 0 dλb)[a λc]; 2 ab ba = 0 D dλa[a λb]; 3 [a λ (bc)] = [a λ b]c+b[a λ c]+ λ 0 dµ[[a λb] µc].

33 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 = (e xd a) 1 b,y(a,x) b = (a x b)(x 1 ).

34 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 t m,b t n ] = ( mab+nba) t m+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.

35 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): Moreover, A = n ZA (n), A (n) = A t n+1. A (n 1) = n 1 A (n) = n 0A (n) A (1) is a Lie subalgebra of A. For any a A, we define the generating function: a(x) = n Z a n x n 1 = n Z(a t n )x n 1 A[[x,x 1 ]]. Then we have [a(x 1 ),b(x 2 )] = (ab+ba)(x 1 ) x 1 2 x δ(x 1 )+[ ab(x 1 )]x x 2 x δ(x 1 ). 1 x 2

36 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:  = n 0 (n), where  (n) = {a (1) m 1 a (r) m r 1 m 1 + +m r = n r, m 1 m r 1,r 0

37 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 Y(a (1) (n 1 ) a(r) (n r) 1,x) = a(1) (x) n1 a (r) (x) nr 1 (39) for any r 0,a (i) A,n i Z, where a(x) n b(x) = Res x1 ((x 1 x) n a(x 1 )b(x) ( x+x 1 ) n b(x)a(x 1 )). (40)

38 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 V = span{a (1) m 1 a (r) m r 1 m i 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).

39 Pre-Lie algebras and vertex (operator) algebras Theorem (Continued) Then V is generated by V (2) with the following property V = span{a (1) m 1 a (r) m r 1 m 1 m r 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 (b 0 a).

40 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 V m V n V m+n.

41 The End Thank You!