Finitely presented Lie rings The algorithm n -Engel Lie rings Gr¨ obner bases in A Z ( X ) Let G ⊂ A Z ( X ) be a finite set and let f ∈ A Z ( X ). Let g 1 , . . . , g s ∈ G be all elements of G such that LM ( g i ) is a factor of LM ( f ). Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Gr¨ obner bases in A Z ( X ) Let G ⊂ A Z ( X ) be a finite set and let f ∈ A Z ( X ). Let g 1 , . . . , g s ∈ G be all elements of G such that LM ( g i ) is a factor of LM ( f ). Suppose that LC ( f ) is divisible by d = gcd( c 1 , . . . , c s ), where c i = LC ( g i ). Let e i > 0 be such that e 1 c 1 + . . . + e s c s = d . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Gr¨ obner bases in A Z ( X ) Let G ⊂ A Z ( X ) be a finite set and let f ∈ A Z ( X ). Let g 1 , . . . , g s ∈ G be all elements of G such that LM ( g i ) is a factor of LM ( f ). Suppose that LC ( f ) is divisible by d = gcd( c 1 , . . . , c s ), where c i = LC ( g i ). Let e i > 0 be such that e 1 c 1 + . . . + e s c s = d . Let c be such that LC ( f ) = cd . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Gr¨ obner bases in A Z ( X ) Let G ⊂ A Z ( X ) be a finite set and let f ∈ A Z ( X ). Let g 1 , . . . , g s ∈ G be all elements of G such that LM ( g i ) is a factor of LM ( f ). Suppose that LC ( f ) is divisible by d = gcd( c 1 , . . . , c s ), where c i = LC ( g i ). Let e i > 0 be such that e 1 c 1 + . . . + e s c s = d . Let c be such that LC ( f ) = cd . Let α i be a product prescription such that P α i ( LM ( g i )) = LM ( f ). Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Gr¨ obner bases in A Z ( X ) Let G ⊂ A Z ( X ) be a finite set and let f ∈ A Z ( X ). Let g 1 , . . . , g s ∈ G be all elements of G such that LM ( g i ) is a factor of LM ( f ). Suppose that LC ( f ) is divisible by d = gcd( c 1 , . . . , c s ), where c i = LC ( g i ). Let e i > 0 be such that e 1 c 1 + . . . + e s c s = d . Let c be such that LC ( f ) = cd . Let α i be a product prescription such that P α i ( LM ( g i )) = LM ( f ). We say that f reduces modulo G to f ′ where f ′ = f − c ( e 1 P α 1 ( g 1 ) + . . . + e s P α s ( g s )) . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Gr¨ obner bases in A Z ( X ) Let G ⊂ A Z ( X ) be a finite set and let f ∈ A Z ( X ). Let g 1 , . . . , g s ∈ G be all elements of G such that LM ( g i ) is a factor of LM ( f ). Suppose that LC ( f ) is divisible by d = gcd( c 1 , . . . , c s ), where c i = LC ( g i ). Let e i > 0 be such that e 1 c 1 + . . . + e s c s = d . Let c be such that LC ( f ) = cd . Let α i be a product prescription such that P α i ( LM ( g i )) = LM ( f ). We say that f reduces modulo G to f ′ where f ′ = f − c ( e 1 P α 1 ( g 1 ) + . . . + e s P α s ( g s )) . Let J ⊂ A Z ( X ) be an ideal. We call a G ⊂ J a Gr¨ obner basis of J if every f ∈ J reduces to zero modulo G . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The algorithm FpLieRing Let L be a Lie ring given by a finite set of generators that satisfy a set R of relations . We assume that L is finite-dimensional . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The algorithm FpLieRing Let L be a Lie ring given by a finite set of generators that satisfy a set R of relations . We assume that L is finite-dimensional . Let X be a set of symbols, in bijection with the generators of L . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The algorithm FpLieRing Let L be a Lie ring given by a finite set of generators that satisfy a set R of relations . We assume that L is finite-dimensional . Let X be a set of symbols, in bijection with the generators of L . Let J be the ideal of A Z ( X ) generated by ( m , m ) for m ∈ M ( X ), ( m , n ) + ( n , m ) for m , n ∈ M ( X ), Jac ( m , n , p ) for m , n , p ∈ M ( X ), where Jac ( m , n , p ) = ( m , ( n , p )) + ( n , ( p , m )) + ( p , ( m , n )) , the elements of R . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The algorithm FpLieRing Let L be a Lie ring given by a finite set of generators that satisfy a set R of relations . We assume that L is finite-dimensional . Let X be a set of symbols, in bijection with the generators of L . Let J be the ideal of A Z ( X ) generated by ( m , m ) for m ∈ M ( X ), ( m , n ) + ( n , m ) for m , n ∈ M ( X ), Jac ( m , n , p ) for m , n , p ∈ M ( X ), where Jac ( m , n , p ) = ( m , ( n , p )) + ( n , ( p , m )) + ( p , ( m , n )) , the elements of R . Then L ∼ = A Z ( X ) / J . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The algorithm FpLieRing The idea of the algorithm FpLieRing is to treat the monic and non-monic elements of G differentely: Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The algorithm FpLieRing The idea of the algorithm FpLieRing is to treat the monic and non-monic elements of G differentely: the set of monic elements G mon will be self-reduced then it is a Gr¨ obner basis; Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The algorithm FpLieRing The idea of the algorithm FpLieRing is to treat the monic and non-monic elements of G differentely: the set of monic elements G mon will be self-reduced then it is a Gr¨ obner basis; the set of non-monic elements B is closed under multiplication (that means if b ∈ B and x is a generator then ( x , b ) lies in the span of B ). Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example Let X = { x , y } , with x < y , and R = { h 1 , h 2 , h 3 } , where h 1 = [ x , [ x , y ]] + [ x , y ] , h 2 = 3[ y , [ y , [ x , y ]]] + 6[ y , [ x , y ]] + 2 y , = [ y , [ y , [ y , [ x , y ]]]] + 2[ y , [ y , [ x , y ]]] . h 3 Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example Let X = { x , y } , with x < y , and R = { h 1 , h 2 , h 3 } , where h 1 = [ x , [ x , y ]] + [ x , y ] , h 2 = 3[ y , [ y , [ x , y ]]] + 6[ y , [ x , y ]] + 2 y , = [ y , [ y , [ y , [ x , y ]]]] + 2[ y , [ y , [ x , y ]]] . h 3 Using the algorithm FpLieRing , we can calculate a Gr¨ obner basis G of the ideal J of A Z ( X ), generated by ( m , m ), ( m , n ) + ( n , m ), Jac ( m , n , p ), for m , n , p ∈ M ( X ), and the elements of R . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example Let X = { x , y } , with x < y , and R = { h 1 , h 2 , h 3 } , where h 1 = [ x , [ x , y ]] + [ x , y ] , h 2 = 3[ y , [ y , [ x , y ]]] + 6[ y , [ x , y ]] + 2 y , = [ y , [ y , [ y , [ x , y ]]]] + 2[ y , [ y , [ x , y ]]] . h 3 Using the algorithm FpLieRing , we can calculate a Gr¨ obner basis G of the ideal J of A Z ( X ), generated by ( m , m ), ( m , n ) + ( n , m ), Jac ( m , n , p ), for m , n , p ∈ M ( X ), and the elements of R . Also we want to determine the set B of normal monomials modulo G . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example The only relation of degree ≤ 3 is, by h 1 , g 1 = ( x , ( x , y )) + ( x , y ) , then we have G 3 = { g 1 } , B 3 = ∅ M ≤ 3 = { x , y , ( x , y ) , ( y , ( x , y )) } . and Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example In degree 4, by h 2 , we take g 2 = 3( y , ( y , ( x , y ))) + 6( y , ( x , y )) + 2 y . Also we have Jac ( x , y , ( x , y )) = ( x , ( y , ( x , y ))) − ( y , ( x , ( x , y ))) , and because we can reduce that modulo g 1 , we obtain the new relation g 3 = ( x , ( y , ( x , y ))) + ( y , ( x , y )) . Then G 4 = { g 1 , g 2 , g 3 } , B 4 = { g 2 } M 4 = { ( y , ( y , ( x , y ))) } . and Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example In degree 5, by h 3 , we put g 4 = ( y , ( y , ( y , ( x , y )))) + 2( y , ( y , ( x , y ))) . Also, Jac ( x , y , ( y , ( x , y ))) reduce to g 5 = (( x , y ) , ( y , ( x , y ))) − ( x , ( y , ( y , ( x , y )))) − ( y , ( y , ( x , y ))) . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example In degree 5, by h 3 , we put g 4 = ( y , ( y , ( y , ( x , y )))) + 2( y , ( y , ( x , y ))) . Also, Jac ( x , y , ( y , ( x , y ))) reduce to g 5 = (( x , y ) , ( y , ( x , y ))) − ( x , ( y , ( y , ( x , y )))) − ( y , ( y , ( x , y ))) . Now we must to consider ( x , g 2 ) and ( y , g 2 ). Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example In degree 5, by h 3 , we put g 4 = ( y , ( y , ( y , ( x , y )))) + 2( y , ( y , ( x , y ))) . Also, Jac ( x , y , ( y , ( x , y ))) reduce to g 5 = (( x , y ) , ( y , ( x , y ))) − ( x , ( y , ( y , ( x , y )))) − ( y , ( y , ( x , y ))) . Now we must to consider ( x , g 2 ) and ( y , g 2 ). The first implies the new relation g 6 = 3( x , ( y , ( y , ( x , y )))) − 6( y , ( x , y )) + 2( x , y ) , while the second reduce to zero modulo g 4 . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example In degree 5, by h 3 , we put g 4 = ( y , ( y , ( y , ( x , y )))) + 2( y , ( y , ( x , y ))) . Also, Jac ( x , y , ( y , ( x , y ))) reduce to g 5 = (( x , y ) , ( y , ( x , y ))) − ( x , ( y , ( y , ( x , y )))) − ( y , ( y , ( x , y ))) . Now we must to consider ( x , g 2 ) and ( y , g 2 ). The first implies the new relation g 6 = 3( x , ( y , ( y , ( x , y )))) − 6( y , ( x , y )) + 2( x , y ) , while the second reduce to zero modulo g 4 . Then G 5 = { g 1 , . . . , g 6 } , B 5 = { g 2 , g 6 } and M 5 = { ( x , ( y , ( y , ( x , y )))) } . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example If we proceed in this way we obtain that G = { f 1 , f 2 , f 3 , f 4 , f 5 , f 6 , f 7 } and B = { f 1 , f 2 , f 4 } , where = 8 y , f 1 f 2 = 4( x , y ) + 4 y , = ( x , ( x , y )) + ( x , y ) , f 3 f 4 = 4( y , ( x , y )) , = ( x , ( y , ( x , y ))) + ( y , ( x , y )) , f 5 f 6 = ( y , ( y , ( x , y ))) + 2( y , ( x , y )) + 6 y , f 7 = (( x , y ) , ( y , ( x , y ))) + 6( x , y ) + 6 y . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example The set of normal monomials modulo G is then B = { e 1 , e 2 , e 3 , e 4 } with relations 4 e 1 = 0 , 4 e 2 + 4 e 3 = 0 , 8 e 3 = 0 , where we put e 1 = ( y , ( x , y )) , e 2 = ( x , y ) , e 3 = y , e 4 = x . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example The set of normal monomials modulo G is then B = { e 1 , e 2 , e 3 , e 4 } with relations 4 e 1 = 0 , 4 e 2 + 4 e 3 = 0 , 8 e 3 = 0 , where we put e 1 = ( y , ( x , y )) , e 2 = ( x , y ) , e 3 = y , e 4 = x . We want to calculate a multiplication table of the Lie ring L . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example The set B forms also a basis of the non-associative ring A = A Z ( X ) / � J , where � J is the ideal of A Z ( X ) generated by G mon . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example The set B forms also a basis of the non-associative ring A = A Z ( X ) / � J , where � J is the ideal of A Z ( X ) generated by G mon . We can prove that { e 1 , e 2 + e 3 , e 3 , e 4 } is a basis of A such that 4 e 1 = 0 , 4( e 2 + e 3 ) = 0 , 8 e 3 = 0 . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example The set B forms also a basis of the non-associative ring A = A Z ( X ) / � J , where � J is the ideal of A Z ( X ) generated by G mon . We can prove that { e 1 , e 2 + e 3 , e 3 , e 4 } is a basis of A such that 4 e 1 = 0 , 4( e 2 + e 3 ) = 0 , 8 e 3 = 0 . There is a homomorphism σ of A onto Z / 4 Z ⊕ Z / 4 Z ⊕ Z / 8 Z ⊕ Z taking α 1 e 1 + α 2 ( e 2 + e 3 ) + α 3 e 3 + α 4 e 4 , where α 1 , . . . , α 4 ∈ Z , to ( α 1 mod 4 , α 2 mod 4 , α 3 mod 8 , α 4 ) . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example Let now v 1 , . . . , v 4 such that = σ ( e 1 ) , v 1 v 2 = σ ( e 2 + e 3 ) , v 3 = σ ( e 3 ) , v 4 = σ ( e 4 ) . Then { v 1 , v 2 , v 3 , v 4 } is a basis of L with 4 v 1 = 0, 4 v 2 = 0 and 8 v 3 = 0. Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example Let now v 1 , . . . , v 4 such that = σ ( e 1 ) , v 1 v 2 = σ ( e 2 + e 3 ) , v 3 = σ ( e 3 ) , v 4 = σ ( e 4 ) . Then { v 1 , v 2 , v 3 , v 4 } is a basis of L with 4 v 1 = 0, 4 v 2 = 0 and 8 v 3 = 0. We can calculate all products [ v i , v j ], for all i , j = 1 , . . . , 4, i < j , by means of [ v i , v j ] = σ ( σ − 1 ( v i ) · σ − 1 ( v j )) . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example The multiplication table of L is [ v 1 , v 2 ] = 2 v 1 + 2 v 2 + 6 v 3 , [ v 1 , v 3 ] = 2 v 1 + 6 v 3 , [ v 1 , v 4 ] = v 1 , [ v 2 , v 3 ] = 3 v 1 , [ v 2 , v 4 ] = 0 , [ v 3 , v 4 ] = 3 v 2 + v 3 , 4 v 1 = 0 , 4 v 2 = 0 , 8 v 3 = 0 . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Remarks FpLieRing will terminate whenever the input defines a finite-dimensional Lie ring. Otherwise it will run forever . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Remarks FpLieRing will terminate whenever the input defines a finite-dimensional Lie ring. Otherwise it will run forever . FpLieRing is similar to known algorithms where Gr¨ obner bases are used to construct finitely presented Lie algebras . In our we extend these methods to deal with finitely presented Lie rings. The fact that we work over Z and not over a field causes a lot of additional problems . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Remarks FpLieRing will terminate whenever the input defines a finite-dimensional Lie ring. Otherwise it will run forever . FpLieRing is similar to known algorithms where Gr¨ obner bases are used to construct finitely presented Lie algebras . In our we extend these methods to deal with finitely presented Lie rings. The fact that we work over Z and not over a field causes a lot of additional problems . It is also possible that a finitely presented Lie ring is defined by an infinite set of relations. FpLieRing can deal with this provided that we can only have a finite number of relations of a given degree. Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The algorithm LieNQ In a paper of 1997, C. Schneider described an algorithm, called LieNQ , to compute so-called nilpotent quotients of finitely presented Lie rings. Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The algorithm LieNQ In a paper of 1997, C. Schneider described an algorithm, called LieNQ , to compute so-called nilpotent quotients of finitely presented Lie rings. When the input relations are homogeneous (i.e., each relation has monomials of the same degree), then it is possible to reformulate the algorithm FpLieRing in such a way that it becomes very similar to Schneider’s algorithm. So for this case the two approaches yield similar algorithms. Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The algorithm LieNQ In a paper of 1997, C. Schneider described an algorithm, called LieNQ , to compute so-called nilpotent quotients of finitely presented Lie rings. When the input relations are homogeneous (i.e., each relation has monomials of the same degree), then it is possible to reformulate the algorithm FpLieRing in such a way that it becomes very similar to Schneider’s algorithm. So for this case the two approaches yield similar algorithms. However, the approach via Gr¨ obner bases leads to a more general algorithm, that will work whenever the finitely presented Lie ring is finite-dimensional. Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 1 Finitely presented Lie rings 2 The algorithm 3 n -Engel Lie rings Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Preliminars We have applied the algorithm FpLieRing to construct the biggest Lie ring that is generated by t elements; satisfies the n -Engel condition for various t and n . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The n -Engel condition A Lie ring L satisfies the n -Engel condition if [ x , [ x , [ . . . , [ x , y ] . . . ]]] = 0 � �� � n for all x , y in L . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings The n -Engel condition A Lie ring L satisfies the n -Engel condition if [ x , [ x , [ . . . , [ x , y ] . . . ]]] = 0 � �� � n for all x , y in L . We will use the right normed convention for iterated commutators. For example, [ xxxxy ] will be the element [ x [ x [ x [ xy ]]]] of L . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Lower central series The lower central series of L is defined as: L 1 = L ; L r +1 = [ L , L r ], for r ≥ 1; where [ L , L r ] is the subring of L generated (as an abelian group) by all [ x , y ] for x ∈ L and y ∈ L r . L is nilpotent if L s +1 = 0 for some s , and the smallest such s is called the nilpotency class . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings E ( t , n ) In 1989 Zelmanov shows that: a finitely-generated Lie ring that satisfies an n-Engel condition is nilpotent . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings E ( t , n ) In 1989 Zelmanov shows that: a finitely-generated Lie ring that satisfies an n-Engel condition is nilpotent . Let E ( t , n ) be the biggest Lie ring with t generators which satisfies the n -Engel condition. Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings E ( t , n ) In 1989 Zelmanov shows that: a finitely-generated Lie ring that satisfies an n-Engel condition is nilpotent . Let E ( t , n ) be the biggest Lie ring with t generators which satisfies the n -Engel condition. What is the structure of E ( t , n ) ? Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings E ( t , n ) In 1989 Zelmanov shows that: a finitely-generated Lie ring that satisfies an n-Engel condition is nilpotent . Let E ( t , n ) be the biggest Lie ring with t generators which satisfies the n -Engel condition. What is the structure of E ( t , n ) ? Higgins and Traustason have studied the structure of E ( t , n ) over fields . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Results over fields Let L be an algebra over a field k . In 1953 Higgins shows that: 2-Engel condition implies L 4 = 0; 3-Engel condition implies L 7 = 0 if char k � = 2 , 5 . In 1993 Traustason shows that: 3-Engel condition implies L 5 = 0 if char k � = 2 , 5; 4-Engel condition implies L c = 0 , if char k � = 2 , 3 , 5 . c < 9 Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Main problem We have studied the structure of E ( t , n ) over Z rather than over a field. For this, we have applied our algorithm for various t and n . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Main problem We have studied the structure of E ( t , n ) over Z rather than over a field. For this, we have applied our algorithm for various t and n . One problem when dealing with the n -Engel condition is: The condition [ x . . . xy ] = 0 is not multilinear . In fact, it is only linear in y . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Main problem We have studied the structure of E ( t , n ) over Z rather than over a field. For this, we have applied our algorithm for various t and n . One problem when dealing with the n -Engel condition is: The condition [ x . . . xy ] = 0 is not multilinear . In fact, it is only linear in y . In order to establish whether a Lie ring L is n -Engel it is not sufficient to check this condition for the elements of a basis. Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Main problem We have studied the structure of E ( t , n ) over Z rather than over a field. For this, we have applied our algorithm for various t and n . One problem when dealing with the n -Engel condition is: The condition [ x . . . xy ] = 0 is not multilinear . In fact, it is only linear in y . In order to establish whether a Lie ring L is n -Engel it is not sufficient to check this condition for the elements of a basis. Let L be generated as an abelian group by B = { x 1 , . . . , x m } . We have determined several sets of conditions on the elements of B only, that are necessary and sufficient for L to satisfy the n -Engel condition. Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 2-Engel Lie rings Let L be generated as an abelian group by B = { x 1 , . . . , x m } . The 2 -Engel condition is [ xxy ] = 0 for all x , y ∈ L . Then [ x i x i y ] = 0 for all x i ∈ B and y ∈ L . Let x = x i + p j x j with p j = ± 1. We have 0 = [( x i + p j x j )( x i + p j x j ) y ] = [ x i x i y ]+ p j ([ x i x j y ]+[ x j x i y ])+[ x j x j y ] then [ x i x j y ] + [ x j x i y ] = 0 ∀ x i , x j ∈ B , i < j , y ∈ L . Let x = p j 1 x j 1 + . . . + p j s x j s , x j r ∈ B and p j r = ± 1. We have � � p 2 0 = [ xxy ] = j r [ x j r x j r y ] + p j r p j t ([ x j r x j t y ] + [ x j t x j r y ]) . r r � = t Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 2-Engel Lie rings A Lie ring L is 2 -Engel if and only if [ x i x i y ] = 0; [ x i x j y ] + [ x j x i y ] = 0; for all y ∈ L , x i , x j ∈ B and i < j . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 2-Engel Lie rings A Lie ring L is 2 -Engel if and only if [ x i x i y ] = 0; [ x i x j y ] + [ x j x i y ] = 0; for all y ∈ L , x i , x j ∈ B and i < j . We have showed that E ( t , 2) has dimension: � t � � t � dim( E ( t , 2)) = t + + 2 3 and nilpotency class 3, for t > 2. Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 3-Engel Lie rings The 3 -Engel condition is [ xxxy ] = 0 for all x , y ∈ L . Then [ x i x i x i y ] = 0 for all x i ∈ B and y ∈ L . If x = x i + p j x j with p j = ± 1 0 = [ x i x i x j y ] + [ x i x j x i y ] + [ x j x i x i y ] ± ([ x i x j x j y ] + [ x j x i x j y ] + [ x j x j x i y ]) . [( x (2) x j ) ∗ y ] = [ x i x i x j y ] + [ x i x j x i y ] + [ x j x i x i y ] i [( x i x (2) ) ∗ y ] = [ x i x j x j y ] + [ x j x i x j y ] + [ x j x j x i y ] j [( x (2) x j ) ∗ y ] ± [( x i x (2) ) ∗ y ] = 0 i j ∀ x i , x j ∈ B , i < j , y ∈ L . If x = x i + p j x j + p k x k with p j , p k = ± 1 [( x i x j x k ) ∗ y ] = 0 ∀ x i , x j , x k ∈ B , i ≤ j ≤ k , y ∈ L . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 3-Engel Lie rings A Lie ring L is 3 -Engel if and only if [( x (3) ) ∗ y ] = 0 i [( x (2) x j ) ∗ y ] + [( x i x (2) ) ∗ y ] = 0 i j [( x (2) x j ) ∗ y ] − [( x i x (2) ) ∗ y ] = 0 i j [( x i x j x k ) ∗ y ] = 0 for all y ∈ L , x i , x j , x k ∈ B and i ≤ j ≤ k . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 3-Engel Lie rings A Lie ring L is 3 -Engel if and only if [( x (3) ) ∗ y ] = 0 i [( x (2) x j ) ∗ y ] + [( x i x (2) ) ∗ y ] = 0 i j [( x i x j x k ) ∗ y ] = 0 2[( x i x (2) ) ∗ y ] = 0 j for all y ∈ L , x i , x j , x k ∈ B and i ≤ j ≤ k . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 3-Engel Lie rings A Lie ring L is 3 -Engel if and only if [( x (3) ) ∗ y ] = 0 i [( x (2) x j ) ∗ y ] + [( x i x (2) ) ∗ y ] = 0 i j [( x i x j x k ) ∗ y ] = 0 2[( x i x (2) ) ∗ y ] = 0 j for all y ∈ L , x i , x j , x k ∈ B and i ≤ j ≤ k . If in the third relation [ x i x j x k y ]+[ x i x k x j y ]+[ x j x i x k y ]+[ x j x k x i y ]+[ x k x i x j y ]+[ x k x j x i y ] = 0 Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 3-Engel Lie rings A Lie ring L is 3 -Engel if and only if [( x (3) ) ∗ y ] = 0 i [( x (2) x j ) ∗ y ] + [( x i x (2) ) ∗ y ] = 0 i j [( x i x j x k ) ∗ y ] = 0 2[( x i x (2) ) ∗ y ] = 0 j for all y ∈ L , x i , x j , x k ∈ B and i ≤ j ≤ k . If in the third relation [ x i x j x k y ]+[ x i x k x j y ]+[ x j x i x k y ]+[ x j x k x i y ]+[ x k x i x j y ]+[ x k x j x i y ] = 0 we put i ≤ j = k we obtain Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 3-Engel Lie rings A Lie ring L is 3 -Engel if and only if [( x (3) ) ∗ y ] = 0 i [( x (2) x j ) ∗ y ] + [( x i x (2) ) ∗ y ] = 0 i j [( x i x j x k ) ∗ y ] = 0 2[( x i x (2) ) ∗ y ] = 0 j for all y ∈ L , x i , x j , x k ∈ B and i ≤ j ≤ k . If in the third relation [ x i x j x k y ]+[ x i x k x j y ]+[ x j x i x k y ]+[ x j x k x i y ]+[ x k x i x j y ]+[ x k x j x i y ] = 0 we put i ≤ j = k we obtain [ x i x j x j y ]+ [ x i x j x j y ]+ [ x j x i x j y ]+ [ x j x j x i y ] + [ x j x i x j y ]+ [ x j x j x i y ] = 0 Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 3-Engel Lie rings A Lie ring L is 3 -Engel if and only if [( x (3) ) ∗ y ] = 0 i [( x (2) x j ) ∗ y ] + [( x i x (2) ) ∗ y ] = 0 i j [( x i x j x k ) ∗ y ] = 0 for all y ∈ L , x i , x j , x k ∈ B and i ≤ j ≤ k . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 4-Engel Lie rings A Lie ring L is 4 -Engel if and only if [( x (4) ) ∗ y ] = 0; i [( x (3) x j ) ∗ y ] + [( x (2) x (2) ) ∗ y ] + [( x i x (3) ) ∗ y ] = 0; i i j j [( x (3) x j ) ∗ y ] − [( x (2) x (2) ) ∗ y ] + [( x i x (3) ) ∗ y ] = 0; i i j j [( x (2) x j x k ) ∗ y ] + [( x i x (2) x k ) ∗ y ] + [( x i x j x (2) k ) ∗ y ] = 0; i j [( x (2) x j x k ) ∗ y ] − [( x i x (2) x k ) ∗ y ] + [( x i x j x (2) k ) ∗ y ] = 0; i j [( x (2) x j x k ) ∗ y ] + [( x i x (2) x k ) ∗ y ] − [( x i x j x (2) k ) ∗ y ] = 0; i j [( x (2) x j x k ) ∗ y ] − [( x i x (2) x k ) ∗ y ] − [( x i x j x (2) k ) ∗ y ] = 0; i j [( x i x j x k x r ) ∗ y ] = 0; for all y ∈ L , x i , x j , x k , x r ∈ B and i ≤ j ≤ k ≤ r . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 4-Engel Lie rings A Lie ring L is 4 -Engel if and only if (1) [( x (4) ) ∗ y ] = 0; i (2) [( x (3) x j ) ∗ y ] + [( x (2) x (2) ) ∗ y ] + [( x i x (3) ) ∗ y ] = 0; i i j j (3) [( x (2) x j x k ) ∗ y ] + [( x i x (2) x k ) ∗ y ] + [( x i x j x (2) k ) ∗ y ] = 0; i j (4) [( x i x j x k x r ) ∗ y ] = 0; (5) 2[( x (2) x (2) ) ∗ y ] = 0; i j (6) 2[( x i x (2) x k ) ∗ y ] = 0; j (7) 2[( x i x j x (2) k ) ∗ y ] = 0; for all y ∈ L , x i , x j , x k , x r ∈ B and i ≤ j ≤ k ≤ r . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 4-Engel Lie rings A Lie ring L is 4 -Engel if and only if (1) [( x (4) ) ∗ y ] = 0; i (2) [( x (3) x j ) ∗ y ] + [( x (2) x (2) ) ∗ y ] + [( x i x (3) ) ∗ y ] = 0; i i j j (3) [( x (2) x j x k ) ∗ y ] + [( x i x (2) x k ) ∗ y ] + [( x i x j x (2) k ) ∗ y ] = 0; i j (4) [( x i x j x k x r ) ∗ y ] = 0; (5) 2[( x (2) x (2) ) ∗ y ] = 0; i j (6) 2[( x i x (2) x k ) ∗ y ] = 0; j (7) 2[( x i x j x (2) k ) ∗ y ] = 0; for all y ∈ L , x i , x j , x k , x r ∈ B and i ≤ j ≤ k ≤ r . (4) with i ≤ j = k ≤ r implies (6). Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 4-Engel Lie rings A Lie ring L is 4 -Engel if and only if (1) [( x (4) ) ∗ y ] = 0; i (2) [( x (3) x j ) ∗ y ] + [( x (2) x (2) ) ∗ y ] + [( x i x (3) ) ∗ y ] = 0; i i j j (3) [( x (2) x j x k ) ∗ y ] + [( x i x (2) x k ) ∗ y ] + [( x i x j x (2) k ) ∗ y ] = 0; i j (4) [( x i x j x k x r ) ∗ y ] = 0; (5) 2[( x (2) x (2) ) ∗ y ] = 0; i j (6) 2[( x i x (2) x k ) ∗ y ] = 0; j (7) 2[( x i x j x (2) k ) ∗ y ] = 0; for all y ∈ L , x i , x j , x k , x r ∈ B and i ≤ j ≤ k ≤ r . (4) with 1 ≤ j ≤ k = r implies (7). Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 4-Engel Lie rings A Lie ring L is 4 -Engel if and only if (1) [( x (4) ) ∗ y ] = 0; i (2) [( x (3) x j ) ∗ y ] + [( x (2) x (2) ) ∗ y ] + [( x i x (3) ) ∗ y ] = 0; i i j j (3) [( x (2) x j x k ) ∗ y ] + [( x i x (2) x k ) ∗ y ] + [( x i x j x (2) k ) ∗ y ] = 0; i j (4) [( x i x j x k x r ) ∗ y ] = 0; (5) 2[( x (2) x (2) ) ∗ y ] = 0; i j (6) 2[( x i x (2) x k ) ∗ y ] = 0; j (7) 2[( x i x j x (2) k ) ∗ y ] = 0; for all y ∈ L , x i , x j , x k , x r ∈ B and i ≤ j ≤ k ≤ r . (3) with i ≤ j = k is 2[( x (2) x (2) ) ∗ y ] + 6[( x i x (3) ) ∗ y ] = 0. i j j Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 4-Engel Lie rings A Lie ring L is 4 -Engel if and only if (1) [( x (4) ) ∗ y ] = 0; i (2) [( x (3) x j ) ∗ y ] + [( x (2) x (2) ) ∗ y ] + [( x i x (3) ) ∗ y ] = 0; i i j j (3) [( x (2) x j x k ) ∗ y ] + [( x i x (2) x k ) ∗ y ] + [( x i x j x (2) k ) ∗ y ] = 0; i j (4) [( x i x j x k x r ) ∗ y ] = 0; (5) 2[( x (2) x (2) ) ∗ y ] = 0; i j (6) 2[( x i x (2) x k ) ∗ y ] = 0; j (7) 2[( x i x j x (2) k ) ∗ y ] = 0; for all y ∈ L , x i , x j , x k , x r ∈ B and i ≤ j ≤ k ≤ r . (3) with i ≤ j = k is 2[( x (2) x (2) ) ∗ y ] + 6[( x i x (3) ) ∗ y ] = 0. i j j (4) with 1 ≤ j = k = r is 6[( x i x (3) ) ∗ y ] = 0. j Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 4-Engel Lie rings A Lie ring L is 4 -Engel if and only if (1) [( x (4) ) ∗ y ] = 0; i (2) [( x (3) x j ) ∗ y ] + [( x (2) x (2) ) ∗ y ] + [( x i x (3) ) ∗ y ] = 0; i i j j (3) [( x (2) x j x k ) ∗ y ] + [( x i x (2) x k ) ∗ y ] + [( x i x j x (2) k ) ∗ y ] = 0; i j (4) [( x i x j x k x r ) ∗ y ] = 0; (5) 2[( x (2) x (2) ) ∗ y ] = 0; i j (6) 2[( x i x (2) x k ) ∗ y ] = 0; j (7) 2[( x i x j x (2) k ) ∗ y ] = 0; for all y ∈ L , x i , x j , x k , x r ∈ B and i ≤ j ≤ k ≤ r . (3) with i ≤ j = k is 2[( x (2) x (2) ) ∗ y ] + 6[( x i x (3) ) ∗ y ] = 0. i j j (4) with 1 ≤ j = k = r is 6[( x i x (3) ) ∗ y ] = 0. j Subtracting we obtain (5). Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings 4-Engel Lie rings A Lie ring L is 4 -Engel if and only if [( x (4) ) ∗ y ] = 0; i [( x (3) x j ) ∗ y ] + [( x (2) x (2) ) ∗ y ] + [( x i x (3) ) ∗ y ] = 0; i i j j [( x (2) x j x k ) ∗ y ] + [( x i x (2) x k ) ∗ y ] + [( x i x j x (2) k ) ∗ y ] = 0; i j [( x i x j x k x r ) ∗ y ] = 0; for all y ∈ L , x i , x j , x k , x r ∈ B and i ≤ j ≤ k ≤ r . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings n -Engel Lie rings A Lie ring L is n -Engel if and only if � [( x ( k 1 ) · · · x ( k s ) ) ∗ y ] = 0 j 1 j s k 1 ,..., k s ≥ 1 k 1 + ... + k s = n for all y ∈ L , 1 ≤ s ≤ n , 1 ≤ j 1 ≤ . . . ≤ j s ≤ m . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Example A Lie ring L is 4 -Engel if and only if [( x (4) ) ∗ y ] = 0; i [( x (3) x j ) ∗ y ] + [( x (2) x (2) ) ∗ y ] + [( x i x (3) ) ∗ y ] = 0; i i j j [( x (2) x j x k ) ∗ y ] + [( x i x (2) x k ) ∗ y ] + [( x i x j x (2) k ) ∗ y ] = 0; i j [( x i x j x k x r ) ∗ y ] = 0; 2[( x (2) x (2) ) ∗ y ] = 0; i j 6[( x i x (3) ) ∗ y ] = 0; j 2[( x i x (2) x k ) ∗ y ] = 0; j 2[( x i x j x (2) k ) ∗ y ] = 0; for all y ∈ L , x i , x j , x k , x r ∈ B and i < j < k < r . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Implementation We have implemented the algorithms in the computer algebra systems GAP4 and Magma . Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Implementation We have implemented the algorithms in the computer algebra systems GAP4 and Magma . Using this implementation we have obtained the lower central series of 3-Engel Lie rings with 2, 3 and 4 generators, 4-Engel Lie rings with 2 generators and 5-Engel Lie rings with 2 generators. Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Lower central series dimensions 3 -Engel 2 gens 3 gens 4 gens L 1 8 60 541 L 2 6 57 537 L 3 5 54 531 L 4 3 46 511 L 5 2 36 472 L 6 - 18 388 L 7 - 9 293 L 8 - 3 173 L 9 - - 62 L 10 - - 18 L 11 - - 4 Table: Lower central series dimensions Serena Cical` o University of Trento
Finitely presented Lie rings The algorithm n -Engel Lie rings Lower central series dimensions 4 -Engel 2 gens 5 -Engel 2 gens L 1 L 1 34 72 L 2 L 2 32 70 L 3 L 3 31 69 L 4 L 4 29 67 L 5 L 5 26 64 L 6 L 6 24 58 L 7 L 7 20 52 L 8 L 8 16 40 L 9 L 9 12 32 L 10 L 10 6 24 L 11 L 11 3 12 L 12 L 12 1 6 L 13 2 Table: Lower central series dimensions Serena Cical` o University of Trento
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