on two characterisations of lie algebras
play

On two characterisations of Lie algebras Xabier Garca-Martnez Joint - PowerPoint PPT Presentation

On two characterisations of Lie algebras Xabier Garca-Martnez Joint work with Tim Van der Linden and Corentin Vienne Ottawa, August 1st-2nd, 2019 Ministerio de Economa, industria y Competitividad MTM2016-79661-P Agencia Estatal de


  1. � � Points and Actions Definition Let B and X be two Lie algebras. The split extension � E X � B induces an action of B on X . Moreover, any action of B on X , defines the split extension � X ¸ B X � B X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 10 / 45

  2. � � Points and Actions Definition Let B and X be two Lie algebras. The split extension � E X � B induces an action of B on X . Moreover, any action of B on X , defines the split extension � X ¸ B X � B In fact, there is an equivalence of categories Pt B p Lie K q – B - Act X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 10 / 45

  3. Representability of actions Definition Let B and X be two Lie algebras. An action of B on X is a Lie algebra homomorphism B Ñ Der K p X q X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 11 / 45

  4. Representability of actions Definition Let B and X be two Lie algebras. An action of B on X is a Lie algebra homomorphism B Ñ Der K p X q This means that the functor Act p B , ´q is representable. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 11 / 45

  5. Representability of actions Definition Let B and X be two Lie algebras. An action of B on X is a Lie algebra homomorphism B Ñ Der K p X q This means that the functor Act p B , ´q is representable. Definition (Broceux-Janelidze-Kelly, 2005) A category is action representable if for any objects B and X , there exists an object r X s such that Act p B , X q – Hom p B , r X sq X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 11 / 45

  6. Representability of actions Definition Let B and X be two Lie algebras. An action of B on X is a Lie algebra homomorphism B Ñ Der K p X q This means that the functor Act p B , ´q is representable. Definition (Broceux-Janelidze-Kelly, 2005) A category is action representable if for any objects B and X , there exists an object r X s such that Act p B , X q – Hom p B , r X sq Examples Groups X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 11 / 45

  7. Representability of actions Definition Let B and X be two Lie algebras. An action of B on X is a Lie algebra homomorphism B Ñ Der K p X q This means that the functor Act p B , ´q is representable. Definition (Broceux-Janelidze-Kelly, 2005) A category is action representable if for any objects B and X , there exists an object r X s such that Act p B , X q – Hom p B , r X sq Examples Groups Lie algebras X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 11 / 45

  8. Cartesian Closed Categories Definition A category with finite products is cartesian closed if for all objects B , the functor B ˆ p´q has a right adjoint. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 12 / 45

  9. Cartesian Closed Categories Definition A category with finite products is cartesian closed if for all objects B , the functor B ˆ p´q has a right adjoint. If C is cartesian closed and pointed, then Hom p X , Y q – Hom p X ˆ 0 , Y q – Hom p 0 , Y X q – 0 . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 12 / 45

  10. Locally Cartesian Closed Categories Let C be a finitely complete category, X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

  11. Locally Cartesian Closed Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , the change of base functor a ˚ : p C Ó B q Ñ p C Ó B 1 q sends X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

  12. � Locally Cartesian Closed Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , the change of base functor a ˚ : p C Ó B q Ñ p C Ó B 1 q sends B 1 ˆ B X π 2 � X f a � B B 1 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

  13. � Locally Cartesian Closed Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , the change of base functor a ˚ : p C Ó B q Ñ p C Ó B 1 q sends B 1 ˆ B X π 2 � X f a � B B 1 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

  14. � � Locally Cartesian Closed Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , the change of base functor a ˚ : p C Ó B q Ñ p C Ó B 1 q sends B 1 ˆ B X π 2 � X f π 1 a � B B 1 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

  15. � � Locally Cartesian Closed Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , the change of base functor a ˚ : p C Ó B q Ñ p C Ó B 1 q sends B 1 ˆ B X π 2 � X f π 1 a � B B 1 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

  16. � � Locally Cartesian Closed Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , the change of base functor a ˚ : p C Ó B q Ñ p C Ó B 1 q sends B 1 ˆ B X π 2 � X f π 1 a � B B 1 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

  17. � � Locally Cartesian Closed Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , the change of base functor a ˚ : p C Ó B q Ñ p C Ó B 1 q sends B 1 ˆ B X π 2 � X f π 1 a � B B 1 Definition C is locally cartesian closed if and only if all the change of base functors a ˚ have a right adjoint. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

  18. Locally Algebraically Cartesian Closed (LACC) Categories Let C be a finitely complete category, X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

  19. Locally Algebraically Cartesian Closed (LACC) Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , we can define a functor a ˚ : Pt B p C q Ñ Pt B 1 p C q that sends X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

  20. � � � Locally Algebraically Cartesian Closed (LACC) Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , we can define a functor a ˚ : Pt B p C q Ñ Pt B 1 p C q that sends B 1 ˆ B X π 2 � X s x 1 B 1 , s ˝ a y f a � B B 1 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

  21. � � � Locally Algebraically Cartesian Closed (LACC) Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , we can define a functor a ˚ : Pt B p C q Ñ Pt B 1 p C q that sends B 1 ˆ B X π 2 � X s x 1 B 1 , s ˝ a y f a � B B 1 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

  22. � � � � Locally Algebraically Cartesian Closed (LACC) Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , we can define a functor a ˚ : Pt B p C q Ñ Pt B 1 p C q that sends B 1 ˆ B X π 2 � X s x 1 B 1 , s ˝ a y f π 1 a � B B 1 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

  23. � � � � Locally Algebraically Cartesian Closed (LACC) Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , we can define a functor a ˚ : Pt B p C q Ñ Pt B 1 p C q that sends B 1 ˆ B X π 2 � X s x 1 B 1 , s ˝ a y f π 1 a � B B 1 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

  24. � � � � Locally Algebraically Cartesian Closed (LACC) Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , we can define a functor a ˚ : Pt B p C q Ñ Pt B 1 p C q that sends B 1 ˆ B X π 2 � X s x 1 B 1 , s ˝ a y f π 1 a � B B 1 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

  25. � � � � Locally Algebraically Cartesian Closed (LACC) Categories Let C be a finitely complete category, given a morphism a : B 1 Ñ B , we can define a functor a ˚ : Pt B p C q Ñ Pt B 1 p C q that sends B 1 ˆ B X π 2 � X s x 1 B 1 , s ˝ a y f π 1 a � B B 1 Definition (Gray, 2012) C is locally algebraically cartesian closed (LACC for short) if and only if all the induced functors a ˚ have a right adjoint. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

  26. Kernel functor Proposition (Gray, 2012) If C has zero object, then it is LACC if and only if ¡ ˚ � Pt 0 p C q – C B : Pt B p C q Ker f X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 15 / 45

  27. � � Kernel functor Proposition (Gray, 2012) If C has zero object, then it is LACC if and only if ¡ ˚ � Pt 0 p C q – C B : Pt B p C q E Ker f s f B X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 15 / 45

  28. � � � ✤ Kernel functor Proposition (Gray, 2012) If C has zero object, then it is LACC if and only if ¡ ˚ � Pt 0 p C q – C B : Pt B p C q E Ker f s f B X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 15 / 45

  29. � � � ✤ Kernel functor Proposition (Gray, 2012) If C has zero object, then it is LACC if and only if ¡ ˚ � Pt 0 p C q – C B : Pt B p C q E Ker f s f B has a right adjoint for all B . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 15 / 45

  30. (LACC) Examples Groups (over a cartesian closed category) (Gray, 2012) X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 16 / 45

  31. (LACC) Examples Groups (over a cartesian closed category) (Gray, 2012) Lie algebras (over some monoidal categories) (Gray, 2012, G.M.-Gray, in progress) X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 16 / 45

  32. Forgetful functor Proposition Let C be a category with finite limits and zero object. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 17 / 45

  33. Forgetful functor Proposition Let C be a category with finite limits and zero object. C is (LACC) if and only if the kernel functor ¡ ˚ B : B - Act » Pt B p V q Ñ C has a right adjoint for all B . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 17 / 45

  34. Forgetful functor Proposition Let C be a category with finite limits and zero object. C is (LACC) if and only if the kernel functor ¡ ˚ B : B - Act » Pt B p V q Ñ C has a right adjoint for all B . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 17 / 45

  35. � � ✤ Forgetful functor Proposition Let C be a category with finite limits and zero object. C is (LACC) if and only if the kernel functor ¡ ˚ B : B - Act » Pt B p V q Ñ C B 5 X X X has a right adjoint for all B . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 17 / 45

  36. Non-associative algebras Definition Let K be a field. A non-associative algebra is a K -vector space with a linear map A b A Ñ A . We denote the category by Alg K . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 18 / 45

  37. Non-associative algebras Definition Let K be a field. A non-associative algebra is a K -vector space with a linear map A b A Ñ A . We denote the category by Alg K . A subvariety of Alg K is any equationally defined class of algebras, considered as a full subcategory V of Alg K . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 18 / 45

  38. Non-associative algebras Example Lie algebras Lie K . They satisfy the equations xx “ 0 x p yz q ` y p zx q ` z p xy q “ 0 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 19 / 45

  39. Non-associative algebras Example Lie algebras Lie K . They satisfy the equations xx “ 0 x p yz q ` y p zx q ` z p xy q “ 0 Example Associative algebras AsAlg K . They satisfy the equation x p yz q ´ p xy q z “ 0 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 19 / 45

  40. Non-associative algebras Example Lie algebras Lie K . They satisfy the equations xx “ 0 x p yz q ` y p zx q ` z p xy q “ 0 Example Associative algebras AsAlg K . They satisfy the equation x p yz q ´ p xy q z “ 0 Example Abelian algebras Ab K . They satisfy the equation xy “ 0 X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 19 / 45

  41. Non-associative algebras Theorem If V is a variety of algebras over an infinite field K , all of its identities are of the form φ p x 1 , . . . , x n q , where φ is a non-associative polynomial. Moreover, each of its homogeneous components ψ p x i 1 , . . . , x i n q is also an identity. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 20 / 45

  42. Non-associative algebras Theorem If V is a variety of algebras over an infinite field K , all of its identities are of the form φ p x 1 , . . . , x n q , where φ is a non-associative polynomial. Moreover, each of its homogeneous components ψ p x i 1 , . . . , x i n q is also an identity. This means that if p xy q z ` x 2 is an identity of V , then X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 20 / 45

  43. Non-associative algebras Theorem If V is a variety of algebras over an infinite field K , all of its identities are of the form φ p x 1 , . . . , x n q , where φ is a non-associative polynomial. Moreover, each of its homogeneous components ψ p x i 1 , . . . , x i n q is also an identity. This means that if p xy q z ` x 2 is an identity of V , then x p yz q x 2 are also identities of V . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 20 / 45

  44. Non-associative algebras Theorem If V is a variety of algebras over an infinite field K , all of its identities are of the form φ p x 1 , . . . , x n q , where φ is a non-associative polynomial. Moreover, each of its homogeneous components ψ p x i 1 , . . . , x i n q is also an identity. This means that if x p yz q ` y p zx q ` z p xy q ` xy ` yx is an identity of V , then X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 20 / 45

  45. Non-associative algebras Theorem If V is a variety of algebras over an infinite field K , all of its identities are of the form φ p x 1 , . . . , x n q , where φ is a non-associative polynomial. Moreover, each of its homogeneous components ψ p x i 1 , . . . , x i n q is also an identity. This means that if x p yz q ` y p zx q ` z p xy q ` xy ` yx is an identity of V , then x p yz q ` y p zx q ` z p xy q xy ` yx are also identities of V . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 20 / 45

  46. Preservation of coproducts of B 5p´q Proposition (Gray, 2012) Let V be a variety of non-associative algebras. It is (LACC) if and only if the canonical comparison p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is an isomorphism. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 21 / 45

  47. Algebraic coherence Theorem The following are equivalent: V is algebraically coherent, i.e. the map p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is surjective. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 22 / 45

  48. Algebraic coherence Theorem The following are equivalent: V is algebraically coherent, i.e. the map p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is surjective. There exist λ 1 , . . . , λ 8 , µ 1 , . . . , µ 8 P K such that z p xy q “ λ 1 p zx q y ` λ 2 p zy q x ` ¨ ¨ ¨ ` λ 8 y p xz q p xy q z “ µ 1 p zx q y ` µ 2 p zy q x ` ¨ ¨ ¨ ` µ 8 y p xz q are identities of V . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 22 / 45

  49. Algebraic coherence Theorem The following are equivalent: V is algebraically coherent, i.e. the map p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is surjective. There exist λ 1 , . . . , λ 8 , µ 1 , . . . , µ 8 P K such that z p xy q “ λ 1 p zx q y ` λ 2 p zy q x ` ¨ ¨ ¨ ` λ 8 y p xz q p xy q z “ µ 1 p zx q y ` µ 2 p zy q x ` ¨ ¨ ¨ ` µ 8 y p xz q are identities of V . For any ideal I , I 2 is also an ideal. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 22 / 45

  50. Algebraic coherence Theorem The following are equivalent: V is algebraically coherent, i.e. the map p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is surjective. There exist λ 1 , . . . , λ 8 , µ 1 , . . . , µ 8 P K such that z p xy q “ λ 1 p zx q y ` λ 2 p zy q x ` ¨ ¨ ¨ ` λ 8 y p xz q p xy q z “ µ 1 p zx q y ` µ 2 p zy q x ` ¨ ¨ ¨ ` µ 8 y p xz q are identities of V . For any ideal I , I 2 is also an ideal. V is an Orzech category of interest. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 22 / 45

  51. Nilpotent algebras Proposition If V is (LACC) and x p yz q “ 0 is an identity in V , then V is abelian. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 23 / 45

  52. Nilpotent algebras Proposition If V is (LACC) and x p yz q “ 0 is an identity in V , then V is abelian. Proof: Let B , X , Y be free algebras on one generator. Since V is (LACC) , the morphism p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is an isomorphism. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 23 / 45

  53. Nilpotent algebras Proposition If V is (LACC) and x p yz q “ 0 is an identity in V , then V is abelian. Proof: Let B , X , Y be free algebras on one generator. Since V is (LACC) , the morphism p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is an isomorphism. The element x p yb q P B 5p X ` Y q comes from zero. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 23 / 45

  54. Nilpotent algebras Proposition If V is (LACC) and x p yz q “ 0 is an identity in V , then V is abelian. Proof: Let B , X , Y be free algebras on one generator. Since V is (LACC) , the morphism p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is an isomorphism. The element x p yb q P B 5p X ` Y q comes from zero. The expression yb plays the role of just “one element” in B 5p X ` Y q . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 23 / 45

  55. Nilpotent algebras Proposition If V is (LACC) and x p yz q “ 0 is an identity in V , then V is abelian. Proof: Let B , X , Y be free algebras on one generator. Since V is (LACC) , the morphism p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is an isomorphism. The element x p yb q P B 5p X ` Y q comes from zero. The expression yb plays the role of just “one element” in B 5p X ` Y q . Then if x p yb q is zero, either x p yb q “ 0 or yb “ 0 have to be rules of V . In both cases, it implies that the algebra is abelian. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 23 / 45

  56. Associative algebras Proposition The variety of associative algebras is not (LACC) . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 24 / 45

  57. Associative algebras Proposition The variety of associative algebras is not (LACC) . Proof: Consider again B , X , Y as free algebras on one generator. Assume that we have an isomorphism: p B 5 X ` B 5 Y q Ñ B 5p X ` Y q X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 24 / 45

  58. Associative algebras Proposition The variety of associative algebras is not (LACC) . Proof: Consider again B , X , Y as free algebras on one generator. Assume that we have an isomorphism: p B 5 X ` B 5 Y q Ñ B 5p X ` Y q Then p xb q y and x p by q go to the same element in B 5p X ` Y q but they are different in p B 5 X ` B 5 Y q . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 24 / 45

  59. Leibniz algebras Proposition The variety of Leibniz algebras is not (LACC) . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 25 / 45

  60. Leibniz algebras Proposition The variety of Leibniz algebras is not (LACC) . Proof: In the Leibniz case, we have they identities b p xy q “ p bx q y ` x p by q X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 25 / 45

  61. Leibniz algebras Proposition The variety of Leibniz algebras is not (LACC) . Proof: In the Leibniz case, we have they identities b p xy q “ p bx q y ` x p by q b p xy q “ ´p xb q y ` x p by q X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 25 / 45

  62. Leibniz algebras Proposition The variety of Leibniz algebras is not (LACC) . Proof: In the Leibniz case, we have they identities b p xy q “ p bx q y ` x p by q b p xy q “ ´p xb q y ` x p by q Then, we have that p bx q y ` p xb q y “ 0. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 25 / 45

  63. Leibniz algebras Proposition The variety of Leibniz algebras is not (LACC) . Proof: In the Leibniz case, we have they identities b p xy q “ p bx q y ` x p by q b p xy q “ ´p xb q y ` x p by q Then, we have that p bx q y ` p xb q y “ 0. Again, p bx q y ` p xb q y is zero in B 5p X ` Y q but it does not need to be in p B 5 X ` B 5 Y q . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 25 / 45

  64. Operations of degree 2 Theorem If V is a (LACC) anticommutative variety of algebras, i.e. xy “ ´ yx is an identity, then V is subvariety of Lie K . X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 26 / 45

  65. Operations of degree 2 Theorem If V is a (LACC) anticommutative variety of algebras, i.e. xy “ ´ yx is an identity, then V is subvariety of Lie K . Theorem Let K be an infinite field of char ‰ 2 . If V is a (LACC) commutative variety of algebras, i.e. xy “ yx is an identity, then V is abelian. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 26 / 45

  66. Operations of degree 2 Theorem If V is a (LACC) anticommutative variety of algebras, i.e. xy “ ´ yx is an identity, then V is subvariety of Lie K . Theorem Let K be an infinite field of char ‰ 2 . If V is a (LACC) commutative variety of algebras, i.e. xy “ yx is an identity, then V is abelian. Theorem If V is a proper (LACC) subvariety of Lie K , then it is abelian. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 26 / 45

  67. Non-commutative and non-anticommutative Let us assume that there are no operations of degree 2. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

  68. Non-commutative and non-anticommutative Let us assume that there are no operations of degree 2. We need to see if there is any variety such that the map p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is an isomorphism. X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

  69. Non-commutative and non-anticommutative Let us assume that there are no operations of degree 2. We need to see if there is any variety such that the map p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is an isomorphism. x p by q “ λ 1 p xb q y ` λ 2 p bx q y ` λ 3 y p xb q ` λ 4 y p bx q ` λ 5 p xy q b ` λ 6 p yx q b ` λ 7 b p xy q ` λ 8 b p yx q “ λ 1 p xb q y ` λ 2 p bx q y ` λ 3 y p xb q ` λ 4 y p bx q ` ˘ ` λ 5 µ 1 p bx q y ` µ 2 p xb q y ` µ 3 y p bx q ` ¨ ¨ ¨ ` µ 7 x p by q ` µ 8 x p yb q ` λ 6 ` µ 1 p by q x ` µ 2 p yb q x ` µ 3 x p by q ` ¨ ¨ ¨ ` µ 7 y p bx q ` µ 8 y p xb q ˘ ` ˘ ` λ 7 λ 1 p bx q y ` λ 2 p xb q y ` λ 3 y p bx q ` ¨ ¨ ¨ ` λ 7 x p by q ` λ 8 x p yb q ` ˘ ` λ 8 λ 1 p by q x ` λ 2 p yb q x ` λ 3 x p by q ` ¨ ¨ ¨ ` λ 7 y p bx q ` µ 8 y p xb q X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

  70. Non-commutative and non-anticommutative Let us assume that there are no operations of degree 2. We need to see if there is any variety such that the map p B 5 X ` B 5 Y q Ñ B 5p X ` Y q is an isomorphism. x p by q “ λ 1 p xb q y ` λ 2 p bx q y ` λ 3 y p xb q ` λ 4 y p bx q ` λ 5 p xy q b ` λ 6 p yx q b ` λ 7 b p xy q ` λ 8 b p yx q “ λ 1 p xb q y ` λ 2 p bx q y ` λ 3 y p xb q ` λ 4 y p bx q ` ˘ ` λ 5 µ 1 p bx q y ` µ 2 p xb q y ` µ 3 y p bx q ` ¨ ¨ ¨ ` µ 7 x p by q ` µ 8 x p yb q ` λ 6 ` µ 1 p by q x ` µ 2 p yb q x ` µ 3 x p by q ` ¨ ¨ ¨ ` µ 7 y p bx q ` µ 8 y p xb q ˘ ` ˘ ` λ 7 λ 1 p bx q y ` λ 2 p xb q y ` λ 3 y p bx q ` ¨ ¨ ¨ ` λ 7 x p by q ` λ 8 x p yb q ` ˘ ` λ 8 λ 1 p by q x ` λ 2 p yb q x ` λ 3 x p by q ` ¨ ¨ ¨ ` λ 7 y p bx q ` µ 8 y p xb q X. García-Martínez (UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

Recommend


More recommend