Study of Lie algebras by using combinatorial structures

Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Study of Lie Algebras by Using Combinatorial Structures Manuel Ceballos, Juan Núñez and Ángel F. Tenorio University of Seville and Pablo de Olavide University 24 - 28 May 2010 Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Contents 1 Preliminaries on Lie algebras 2 Associating combinatorial structures with Lie algebras 3 Cycle Digraphs and Lie Algebras 4 Implementing the algorithm with Maple 5 2-step solvable Lie algebras and combinatorial structures 6 References Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Preliminaries Lie algebra ALie algebra g is a vector space with a second bilinear composition law ( [,] ) which satises: [ X , X ] = 0 , ∀ X ∈g and [[ X , Y ], Z ] + [[ Y , Z ], X ] + [[ Z , X ], Y ] = 0 , ∀ X , Y , Z ∈g . Structure constants A basis { e h } n h = 1 of g is characterized by its structure constants: [ e i , e j ] = P c h i , j e h , for 1 ≤ i , j ≤ n . Semisimple and simple Lie algebras A Lie algebra g is semisimple if it does not contain any proper abelian ideal. A simple Lie algebra is a non-abelian Lie algebra with no non-trivial ideals. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Preliminaries Lie algebra ALie algebra g is a vector space with a second bilinear composition law ( [,] ) which satises: [ X , X ] = 0 , ∀ X ∈g and [[ X , Y ], Z ] + [[ Y , Z ], X ] + [[ Z , X ], Y ] = 0 , ∀ X , Y , Z ∈g . Structure constants A basis { e h } n h = 1 of g is characterized by its structure constants: [ e i , e j ] = P c h i , j e h , for 1 ≤ i , j ≤ n . Semisimple and simple Lie algebras A Lie algebra g is semisimple if it does not contain any proper abelian ideal. A simple Lie algebra is a non-abelian Lie algebra with no non-trivial ideals. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Preliminaries Lie algebra ALie algebra g is a vector space with a second bilinear composition law ( [,] ) which satises: [ X , X ] = 0 , ∀ X ∈g and [[ X , Y ], Z ] + [[ Y , Z ], X ] + [[ Z , X ], Y ] = 0 , ∀ X , Y , Z ∈g . Structure constants A basis { e h } n h = 1 of g is characterized by its structure constants: [ e i , e j ] = P c h i , j e h , for 1 ≤ i , j ≤ n . Semisimple and simple Lie algebras A Lie algebra g is semisimple if it does not contain any proper abelian ideal. A simple Lie algebra is a non-abelian Lie algebra with no non-trivial ideals. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Preliminaries Upper central series The upper central series of a Lie algebra g is dened as C 1 (g) = g,C 2 (g)=[g,g],C 3 (g)=[C 2 (g),C 2 (g)], . . . , C k (g) = [C k − 1 (g),C k − 1 (g)], . . . Solvable Lie algebra If there exists m ∈N such that C m (g)≡ { 0 } , the Lie algebra g is solvable. A solvable Lie algebra is k -step if C k (g)6={ 0 } and C k + 1 (g)≡ { 0 } . Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Preliminaries Upper central series The upper central series of a Lie algebra g is dened as C 1 (g) = g,C 2 (g)=[g,g],C 3 (g)=[C 2 (g),C 2 (g)], . . . , C k (g) = [C k − 1 (g),C k − 1 (g)], . . . Solvable Lie algebra If there exists m ∈N such that C m (g)≡ { 0 } , the Lie algebra g is solvable. A solvable Lie algebra is k -step if C k (g)6={ 0 } and C k + 1 (g)≡ { 0 } . Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Preliminaries Lower central series The lower central series of a Lie algebra g is dened as: C 1 (g) = g,C 2 (g)=[g,g],C 3 (g)=[C 2 (g),g], . . . , C k (g)=[C k − 1 (g),g], . . . Nilpotent Lie algebra If there exists m ∈N such that C m (g)≡ { 0 } , the Lie algebra g is nilpotent. A nilpotent Lie algebra is k -step if C k (g)6={ 0 } and C k + 1 (g)≡ { 0 } . Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Preliminaries Lower central series The lower central series of a Lie algebra g is dened as: C 1 (g) = g,C 2 (g)=[g,g],C 3 (g)=[C 2 (g),g], . . . , C k (g)=[C k − 1 (g),g], . . . Nilpotent Lie algebra If there exists m ∈N such that C m (g)≡ { 0 } , the Lie algebra g is nilpotent. A nilpotent Lie algebra is k -step if C k (g)6={ 0 } and C k + 1 (g)≡ { 0 } . Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Combinatorial structures and Lie algebras Given two vertices i < j , if c i i , j 6= 0 or c j i , j 6= 0 , then a directed edge is drawn. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Combinatorial structures and Lie algebras Given two vertices i < j , if c i i , j 6= 0 or c j i , j 6= 0 , then a directed edge is drawn. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Combinatorial structures and Lie algebras Given two vertices i < j , if c i i , j 6= 0 or c j i , j 6= 0 , then a directed edge is drawn. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Combinatorial structures and Lie algebras Going-in and going-out vertex A vertex v is said to be a going-in (respectively going-out) vertex if all the directed incident edges with v are oriented towards v (respectively, from v ). Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Combinatorial structures and Lie algebras Going-in and going-out vertex A vertex v is said to be a going-in (respectively going-out) vertex if all the directed incident edges with v are oriented towards v (respectively, from v ). Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Combinatorial structures and Lie algebras Corollary Every Lie algebra with a selected basis is associated with a combinatorial structure. This association depends on the selected basis. Isolated vertex An isolated vertex corresponds to a vector from the center of g . Complete graph Acycle digraph, G , is dened as a cycle graph with directed edges. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Combinatorial structures and Lie algebras Corollary Every Lie algebra with a selected basis is associated with a combinatorial structure. This association depends on the selected basis. Isolated vertex An isolated vertex corresponds to a vector from the center of g . Complete graph Acycle digraph, G , is dened as a cycle graph with directed edges. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Combinatorial structures and Lie algebras Corollary Every Lie algebra with a selected basis is associated with a combinatorial structure. This association depends on the selected basis. Isolated vertex An isolated vertex corresponds to a vector from the center of g . Complete graph Acycle digraph, G , is dened as a cycle graph with directed edges. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Cycle Digraphs and Lie Algebras Cycle Digraphs ACycle Digraph is a cycle graph with directed edges. We consider a well-oriented weighted cycle digraph with double edges between their vertices. Given a combinatorial structure, T , of n vertices: Label all the vertices by 1 , 2 ,..., n , following the positive counterclockwise. The weight of the edge ij will be denoted by c i , j . Dene a vector space V with basis { e 1 ,..., e n } where e i corresponds to the vertex i of T and brackets [ e i , e j ] = c i i , j e i + c j i , j e j . Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Cycle Digraphs and Lie Algebras Cycle Digraphs ACycle Digraph is a cycle graph with directed edges. We consider a well-oriented weighted cycle digraph with double edges between their vertices. Given a combinatorial structure, T , of n vertices: Label all the vertices by 1 , 2 ,..., n , following the positive counterclockwise. The weight of the edge ij will be denoted by c i , j . Dene a vector space V with basis { e 1 ,..., e n } where e i corresponds to the vertex i of T and brackets [ e i , e j ] = c i i , j e i + c j i , j e j . Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Case n ≥ 4 We must solve the system of equations given by all the Jacobi identities. When imposing J ( e i , e j , e k ) = 0 , the following equation is obtained c j j , k c i i , j + c k j , k c i i , k = 0 , c k i , k c j j , k − c i i , k c j i , j = 0 , c i i , j c k i , k + c j i , j c k j , k = 0 . When imposing all the Jacobi identities and the restrictions: c p + 1 p , p + 1 = 1 , for all p ∈ { 1 ,..., n } and c 1 1 , n = 1 , we obtain the law of a particular Lie algebra, that can be considered over Z / 3 Z . [ e p , e q ] = − e p + e q [ e p , e n ] = e p + e n , where  1 ≤ p ≤ n − 2 ; p + 1 ≤ q ≤ n − 1 . Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Case n ≥ 4 We must solve the system of equations given by all the Jacobi identities. When imposing J ( e i , e j , e k ) = 0 , the following equation is obtained c j j , k c i i , j + c k j , k c i i , k = 0 , c k i , k c j j , k − c i i , k c j i , j = 0 , c i i , j c k i , k + c j i , j c k j , k = 0 . When imposing all the Jacobi identities and the restrictions: c p + 1 p , p + 1 = 1 , for all p ∈ { 1 ,..., n } and c 1 1 , n = 1 , we obtain the law of a particular Lie algebra, that can be considered over Z / 3 Z . [ e p , e q ] = − e p + e q [ e p , e n ] = e p + e n , where  1 ≤ p ≤ n − 2 ; p + 1 ≤ q ≤ n − 1 . Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References In this way, we can establish the following Proposition Let us consider a well-oriented, weighted cycle digraph G with double edges of 3 vertices. Then, G is associated with a 3 -dimensional Lie algebra if and only if the weights of its edges satisfy one of the following constraints (i) c 1 1 , 3 = 1 , c 2 1 , 2 = 1 , c 3 2 , 3 = 1 , c 2 2 , 3 = 1 , c 3 1 , 3 = 1 and c 1 1 , 2 =− 1 . In this case, the Lie algebra, denoted by g , is perfect. (ii) c 1 1 , 3 = 1 , c 2 1 , 2 = 1 , c 3 2 , 3 = 1 , c 2 2 , 3 =− 1 , c 3 1 , 3 =− 1 and c 1 1 , 2 = 1 . In this case, the Lie algebra, denoted by h , is 2 -step solvable and non-nilpotent. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References In this way, we can establish the following Proposition Let us consider a well-oriented, weighted cycle digraph G with double edges of 3 vertices. Then, G is associated with a 3 -dimensional Lie algebra if and only if the weights of its edges satisfy one of the following constraints (i) c 1 1 , 3 = 1 , c 2 1 , 2 = 1 , c 3 2 , 3 = 1 , c 2 2 , 3 = 1 , c 3 1 , 3 = 1 and c 1 1 , 2 =− 1 . In this case, the Lie algebra, denoted by g , is perfect. (ii) c 1 1 , 3 = 1 , c 2 1 , 2 = 1 , c 3 2 , 3 = 1 , c 2 2 , 3 =− 1 , c 3 1 , 3 =− 1 and c 1 1 , 2 = 1 . In this case, the Lie algebra, denoted by h , is 2 -step solvable and non-nilpotent. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Proposition Let us consider a well-oriented, weighted cycle digraph G with double edges of n ≥ 4 vertices. Then, G is associated with an n -dimensional Lie algebra if and only if the weights of the edges satisfy c p p , q =− 1 , c q p , q = 1 , c p p , n = c n p , n = 1 , where  1 ≤ p ≤ n − 2 ; p + 1 ≤ q ≤ n − 1 . The Lie algebra associated with the digraph is unique and is denoted by g n . Moreover, the Lie algebra g n is 2 -step solvable and non-nilpotent. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Next, for the sake of example, we show the digraph associated with a 4 -dimensional Lie algebra. Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Implementing the algorithm with Maple We implement the algorithm by using the symbolic computation package MAPLE with a subroutine and a main routine. Subroutine subr > subr:=proc(n,i,j,k) > S:=c[j,k,j]*c[i,j,i]+c[j,k,k]*c[i,k,i]=0, > c[i,k,k]*c[j,k,j]-c[i,k,i]*c[i,j,j]=0, > c[i,j,i]*c[i,k,k]+c[i,j,j]*c[j,k,k]=0; > for q from 1 to n-1 do > S:=op(S),c[q,q+1,q+1]=1;end do; > S:=op(S),c[1,n,1]=1; > return S; end proc: Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Implementing the algorithm with Maple We implement the algorithm by using the symbolic computation package MAPLE with a subroutine and a main routine. Subroutine subr > subr:=proc(n,i,j,k) > S:=c[j,k,j]*c[i,j,i]+c[j,k,k]*c[i,k,i]=0, > c[i,k,k]*c[j,k,j]-c[i,k,i]*c[i,j,j]=0, > c[i,j,i]*c[i,k,k]+c[i,j,j]*c[j,k,k]=0; > for q from 1 to n-1 do > S:=op(S),c[q,q+1,q+1]=1;end do; > S:=op(S),c[1,n,1]=1; > return S; end proc: Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Implementing the algorithm with Maple Routine mainr > mainr:=proc(n) > local L,T; > L:=choose(n,3); T:=; > for p from 1 to nops(L) do > T:=op(T),op(subr(n,L[p][1],L[p][2],L[p][3])); > end do; > return solve(T); > end proc; Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References We have proved that several digraphs are associated with 2 -step solvable non-nilpotent Lie algebras under some restrictions. Every 2 -step solvable non-nilpotent Lie algebra is associated with a digraph? g Lie brackets Parameters r 2 [ e 1 , e 2 ] = e 2 r 3 [ e 1 , e 2 ] = e 2 ,[ e 1 , e 3 ] = e 2 + e 3 r 3 , p [ e 1 , e 2 ] = e 2 ,[ e 1 , e 3 ] = pe 3 p ∈C∗,| p ≤ 1 r 4 , q [ e 1 , e 2 ] = e 2 ,[ e 1 , e 3 ] = e 3 ,[ e 1 , e 4 ] = qe 4 q ∈C∗ r 4 ,α,β [ e 1 , e 2 ] = e 3 ,[ e 1 , e 3 ] = e 4 ,[ e 1 , e 4 ] = α e 2 −β e 3 + e 4 α∈C∗ , β∈C or α, β = 0 gα 4 , 11 [ e 1 , e 2 ] = e 3 ,[ e 1 , e 3 ] = e 4 ,[ e 1 , e 4 ] = α( e 2 + e 3 )α∈C∗ g 4 , 12 [ e 1 , e 2 ] = e 3 ,[ e 1 , e 3 ] = e 4 ,[ e 1 , e 4 ] = e 2 Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures Preliminaries on Lie algebras Associating combinatorial structures with Lie algebras Cycle Digraphs and Lie Algebras Implementing the algorithm with Maple 2-step solvable Lie algebras and combinatorial structures References Study of Lie Algebras by Using Combinatorial Structures Manuel Ceballos, Juan Núñez and Ángel F. Tenorio University of Seville and Pablo de Olavide University 24 - 28 May 2010 Manuel Ceballos, Juan Núñez and Ángel F. Tenorio Study of Lie Algebras by Using Combinatorial Structures