on two characterisations of lie algebras
play

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

Jan 28, 2024 •685 likes •2.2k views

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

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March

Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March

Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March 2018 1 We consider finiteness conditions and ques- tions of growth of noncommutative algebras, known as A con s. They appear in M.Wemyss work on

214 views • 17 slides
CHARACTERISATIONS OF HYBRID COMPOSITES A. Kalam 1* ,M. N Berhan 1, ,H. Ismail 2 1 Fakulti

CHARACTERISATIONS OF HYBRID COMPOSITES A. Kalam 1* ,M. N Berhan 1, ,H. Ismail 2 1 Fakulti

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS CHARACTERISATIONS OF HYBRID COMPOSITES A. Kalam 1* ,M. N Berhan 1, ,H. Ismail 2 1 Fakulti Kejuruteraan Mekanikal, Universiti Teknologi MARA, Shah Alam Selangor, Malaysia, 2 School of Materials

423 views • 4 slides
Tense MV-algebras and related functions Jan Paseka Michal Botur Department of Mathematics and

Tense MV-algebras and related functions Jan Paseka Michal Botur Department of Mathematics and

Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Tense MV-algebras

1.59k views • 121 slides
Logical Characterisations of Probabilistic Bisimilarity Yuxin Deng East China Normal University

Logical Characterisations of Probabilistic Bisimilarity Yuxin Deng East China Normal University

Logical Characterisations of Probabilistic Bisimilarity Yuxin Deng East China Normal University (Based on joint work with Hengyang Wu and Yuan Feng) IFIP Working Group 2.2 meeting, Bordeaux, September 18, 2017 1 Preliminaries 2 Labelled

549 views • 28 slides
Varieties of positive interior algebras: operation . The two presentations produce

Varieties of positive interior algebras: operation . The two presentations produce

Modal algebras form the algebraic semantics of normal modal logics. They are Boolean algebras with a unary operation s.t. 1 1 and ( x y ) x y . Modal algebras can be presented also as BAs with a unary

186 views • 5 slides
Block Decomposition of a Class of Integrable Representations of Toroidal Lie Algebras Tanusree

Block Decomposition of a Class of Integrable Representations of Toroidal Lie Algebras Tanusree

Block Decomposition of a Class of Integrable Representations of Toroidal Lie Algebras Tanusree Khandai Indian Institute of Science Education and Research, Mohali Interactions of quantum affine algebras with cluster algebras, current algebras

327 views • 28 slides
(Pre-)Algebras for Linguistics 2. Introducing Preordered Algebras Carl Pollard Linguistics 680:

(Pre-)Algebras for Linguistics 2. Introducing Preordered Algebras Carl Pollard Linguistics 680:

(Pre-)Algebras for Linguistics 2. Introducing Preordered Algebras Carl Pollard Linguistics 680: Formal Foundations Autumn 2010 Carl Pollard (Pre-)Algebras for Linguistics Algebras A (one-sorted) algebra is a set with one or more operations

497 views • 38 slides
Unified characterisations of resolution hardness measures Olaf Beyersdorff 1 Oliver Kullmann 2 1

Unified characterisations of resolution hardness measures Olaf Beyersdorff 1 Oliver Kullmann 2 1

Unified characterisations of resolution hardness measures Olaf Beyersdorff 1 Oliver Kullmann 2 1 School of Computing, University of Leeds, UK 2 Computer Science Department, Swansea University, UK 1 Hardness measures for resolution Historically

659 views • 23 slides
The Calabi-Yau property of Hopf algebras and braided Hopf algebras Xiaolan YU joint work with

The Calabi-Yau property of Hopf algebras and braided Hopf algebras Xiaolan YU joint work with

The Calabi-Yau property of Hopf algebras and braided Hopf algebras Xiaolan YU joint work with Yinhuo Zhang Hangzhou Normal University September 16th, 2011 1 / 35 Outline Motivation Braided Hopf algebras Calabi-Yau algebras The Calabi-Yau

639 views • 63 slides
Lattices from Octonion Algebras Octonion Algebras Lattices via Octonion Algebras Nelson G.

Lattices from Octonion Algebras Octonion Algebras Lattices via Octonion Algebras Nelson G.

Nelson, Cintya, Sueli Lattices Lattices from Octonion Algebras Octonion Algebras Lattices via Octonion Algebras Nelson G. Brasil Junior 1 Cintya W. O. Benedito 2 Examples Sueli I. R. Costa 1 1 Institute of Mathematics, Statistics and

650 views • 6 slides
Algebras birational to generic Sklyanin algebras Sue Sierra University of Edinburgh*

Algebras birational to generic Sklyanin algebras Sue Sierra University of Edinburgh*

Algebras birational to generic Sklyanin algebras Sue Sierra University of Edinburgh* Noncommutative and non-associative structures, braces and applications Malta 2018 Throughout, k is an algebraically closed field. Definition Let a , b , c

574 views • 20 slides
On SK 1 of Iwasawa algebras joint work with Peter Schneider Otmar Venjakob Mathematisches

On SK 1 of Iwasawa algebras joint work with Peter Schneider Otmar Venjakob Mathematisches

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras and SK 1 Application On SK 1 of Iwasawa algebras joint work with Peter Schneider Otmar Venjakob Mathematisches Institut

1.07k views • 16 slides
Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c

Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c

Notre Dame 2015 Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c University of Zagreb, Croatia Supported by CSF, grant. no. 2634 Conference on Lie algebras, vertex operator algebras and related

586 views • 39 slides
Relation algebras Robin Hirsch and Ian Hodkinson Thanks to the organisers for inviting us! And

Relation algebras Robin Hirsch and Ian Hodkinson Thanks to the organisers for inviting us! And

Relation algebras Robin Hirsch and Ian Hodkinson Thanks to the organisers for inviting us! And Happy New Year! Workshop outline 1. Introduction to relation algebras 2. Games 3. Monk algebras: completions, canonicity 4. Rainbow construction:

599 views • 37 slides
Homogeneous orthocomplete effect algebras are covered by MV-algebras Josef Niederle and Jan

Homogeneous orthocomplete effect algebras are covered by MV-algebras Josef Niederle and Jan

Homogeneous orthocomplete effect algebras are covered by MV-algebras Josef Niederle and Jan Paseka Department of Mathematics and Statistics Masaryk University Brno, Czech Republic niederle,paseka@math.muni.cz Supported by TACL 2011 July

951 views • 80 slides
Algebras of incidence structures: representing regular double p-algebras Christopher Taylor La

Algebras of incidence structures: representing regular double p-algebras Christopher Taylor La

Algebras of incidence structures: representing regular double p-algebras Christopher Taylor La Trobe University Victorian Algebra Conference 2015 Chris Taylor Algebras of incidence structures VAC 2015 1 / 24 Chris Taylor Algebras of

967 views • 69 slides
Hoare Logic and Model Checking Model Checking Lecture 11: Model checking for Computation Tree

Hoare Logic and Model Checking Model Checking Lecture 11: Model checking for Computation Tree

Hoare Logic and Model Checking Model Checking Lecture 11: Model checking for Computation Tree Logic Dominic Mulligan Based on previous slides by Alan Mycroft and Mike Gordon Programming, Logic, and Semantics Group University of Cambridge

402 views • 25 slides
Computational Social Choice: Autumn 2012 Ulle Endriss Approach 1: Axiomatic Method Institute for

Computational Social Choice: Autumn 2012 Ulle Endriss Approach 1: Axiomatic Method Institute for

Characterisation Results COMSOC 2012 Characterisation Results COMSOC 2012 Computational Social Choice: Autumn 2012 Ulle Endriss Approach 1: Axiomatic Method Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss

190 views • 7 slides
Towards an alternative characterisation of Boucherie product form Nigel Thomas University of

Towards an alternative characterisation of Boucherie product form Nigel Thomas University of

Towards an alternative characterisation of Boucherie product form Nigel Thomas University of Durham Contents Existing methods: Boucherie Motivation Mechanisms Characterisations Conditions for simple product form

306 views • 16 slides
Deadlocks: Characterisation & Prevention Summer 2013 Cornell University 1 Today What

Deadlocks: Characterisation & Prevention Summer 2013 Cornell University 1 Today What

CS 4410 Operating Systems Deadlocks: Characterisation & Prevention Summer 2013 Cornell University 1 Today What are the deadlocks and how are they created? System Model Deadlock examples Four conditions for deadlock

349 views • 15 slides
Approximation in the Zygmund Class Od Soler i Gibert Joint work with Artur Nicolau

Approximation in the Zygmund Class Od Soler i Gibert Joint work with Artur Nicolau

Approximation in the Zygmund Class Od Soler i Gibert Joint work with Artur Nicolau Universitat Aut` onoma de Barcelona New Developments in Complex Analysis and Function Theory, 02 July 2018 Approximation in the Zygmund Class CAFT 2018

664 views • 21 slides
Aspects of univalence Nicola Gambino School of Mathematics, University of Leeds Homotopy Type

Aspects of univalence Nicola Gambino School of Mathematics, University of Leeds Homotopy Type

Aspects of univalence Nicola Gambino School of Mathematics, University of Leeds Homotopy Type Theory and Univalent Foundations DMV 2015 September 24th, 2015 1 Overview of the talk Disclaimer: this is an expository talk, based on work by

437 views • 19 slides
Point sets in AG ( n , q ) (not) determining certain directions Jan De Beule Department of

Point sets in AG ( n , q ) (not) determining certain directions Jan De Beule Department of

Introduction point sets of AG ( 2 , q ) point sets in AG ( n , q ) Point sets in AG ( n , q ) (not) determining certain directions Jan De Beule Department of Mathematics Ghent University June 12, 2010 Baer Colloquium Summer 2010, Gent

254 views • 22 slides
The chain relation in sofic subshifts Alexandr Kazda Charles University, Prague

The chain relation in sofic subshifts Alexandr Kazda Charles University, Prague

Introduction Characterisation of the chain relation Summary The chain relation in sofic subshifts Alexandr Kazda Charles University, Prague alexak@atrey.karlin.mff.cuni.cz WSDC 2007 Alexandr Kazda The chain relation in sofic subshifts

611 views • 59 slides

More recommend