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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

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On two characterisations of Lie algebras

Xabier García-Martínez

Joint work with Tim Van der Linden and Corentin Vienne

Ottawa, August 1st-2nd, 2019

Ministerio de Economía, industria y Competitividad MTM2016-79661-P Agencia Estatal de Investigación ED431C 2019/10 Xunta de Galicia

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 1 / 45

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Aim of the talk Give two difficult answers, to a very easy question.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 2 / 45

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Aim of the talk Give two difficult answers, to a very easy question.

Question:

Among all the varieties of non-associative algebras, which one is LieK?

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 2 / 45

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Semi-abelian categories

Definition (Janelidze-Márki-Tholen, 2002)

A category C is semi-abelian when it is: Pointed Has finite coproducts (denoted by `) Barr-exact Bourn-Protomodular

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 3 / 45

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SLIDE 5

Semi-abelian categories

Definition (Janelidze-Márki-Tholen, 2002)

A category C is semi-abelian when it is: Pointed Has finite coproducts (denoted by `) Barr-exact Bourn-Protomodular

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 3 / 45

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Split extensions

Definition

A split extension is a diagram K

k

E

f

B

s

  • where k “ Kerpf q, and f ˝ s “ 1B.
  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 4 / 45

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SLIDE 7

Split extensions

Definition

A split extension is a diagram K

k

E

f

B

s

  • where k “ Kerpf q, and f ˝ s “ 1B.

Abelian groups or Modules

K

k

K ‘ B

f

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 4 / 45

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SLIDE 8

Split extensions

Definition

A split extension is a diagram K

k

E

f

B

s

  • where k “ Kerpf q, and f ˝ s “ 1B.

Groups or Lie algebras

K

k

K ¸ B

f

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 4 / 45

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Split extensions

Definition

A split extension is a diagram K

k

E

f

B

s

  • where k “ Kerpf q, and f ˝ s “ 1B.

Monoids

N ˆ N

`

N

x1,0y

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 4 / 45

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Protomodularity

Let C be a category. For a fixed object B P C, we form the category of points over B, denoted by PtBpCq.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 5 / 45

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Protomodularity

Let C be a category. For a fixed object B P C, we form the category of points over B, denoted by PtBpCq. ‚ Objects are split extensions E 1

f 1

  • ϕ

E

f

  • B

s1

  • s
  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 5 / 45

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Protomodularity

Let C be a category. For a fixed object B P C, we form the category of points over B, denoted by PtBpCq. ‚ Objects are split extensions ‚ Morphisms are arrows making the (reasonable) diagrams commute. E 1

f 1

  • ϕ

E

f

  • B

s1

  • s
  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 5 / 45

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Protomodularity

If C has pullbacks, for any arrow a: B1 Ñ B there is an induced functor a˚ : PtBpCq Ñ PtB1pCq

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 6 / 45

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Protomodularity

If C has pullbacks, for any arrow a: B1 Ñ B there is an induced functor a˚ : PtBpCq Ñ PtB1pCq B1 ˆB E

π1

  • π2

E

f

  • B1

x1B1,s˝ay

  • B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 6 / 45

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SLIDE 15

Protomodularity

If C has pullbacks, for any arrow a: B1 Ñ B there is an induced functor a˚ : PtBpCq Ñ PtB1pCq B1 ˆB E

π1

  • π2

E

f

  • B1

x1B1,s˝ay

  • a

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 6 / 45

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SLIDE 16

Protomodularity

If C has pullbacks, for any arrow a: B1 Ñ B there is an induced functor a˚ : PtBpCq Ñ PtB1pCq B1 ˆB E

π1

  • π2

E

f

  • B1

x1B1,s˝ay

  • a

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 6 / 45

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SLIDE 17

Protomodularity

If C has pullbacks, for any arrow a: B1 Ñ B there is an induced functor a˚ : PtBpCq Ñ PtB1pCq B1 ˆB E

π1

  • π2

E

f

  • B1

x1B1,s˝ay

  • a

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 6 / 45

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SLIDE 18

Protomodularity

If C has pullbacks, for any arrow a: B1 Ñ B there is an induced functor a˚ : PtBpCq Ñ PtB1pCq B1 ˆB E

π1

  • π2

E

f

  • B1

x1B1,s˝ay

  • a

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 6 / 45

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Protomodularity

If C has pullbacks, for any arrow a: B1 Ñ B there is an induced functor a˚ : PtBpCq Ñ PtB1pCq B1 ˆB E

π1

  • π2

E

f

  • B1

x1B1,s˝ay

  • a

B

s

  • Definition (Bourn, 1991)

A category with finite limits C is protomodular if all a˚ : PtBpCq Ñ PtB1pCq reflect isomorphisms.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 6 / 45

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Protomodularity

Proposition

If C is pointed, it is protomodular if and only if for every B ¡˚

B : PtBpCq Ñ Pt0pCq – C

reflect isomorphisms.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 7 / 45

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Points and actions

Let C be a semi-abelian category, the kernel functor has a left adjoint

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 8 / 45

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Points and actions

Let C be a semi-abelian category, the kernel functor has a left adjoint PtBpCq

¡˚

B

  • J

C,

L

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 8 / 45

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Points and actions

Let C be a semi-abelian category, the kernel functor has a left adjoint PtBpCq

¡˚

B

  • J

C,

L

  • sending X to B ` X

p1B 0q B ιB

  • .
  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 8 / 45

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Points and actions

Let C be a semi-abelian category, the kernel functor has a left adjoint PtBpCq

¡˚

B

  • J

C,

L

  • sending X to B ` X

p1B 0q B ιB

  • .

This defines a monad B5p´q: C Ñ C that sends any X to B5X B ` X

p1B 0q

B

ι1

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 8 / 45

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Points and actions

Let C be a semi-abelian category, the kernel functor has a left adjoint PtBpCq

¡˚

B

  • J

C,

L

  • sending X to B ` X

p1B 0q B ιB

  • .

This defines a monad B5p´q: C Ñ C that sends any X to B5X B ` X

p1B 0q

B

ι1

  • Bourn-Janelidze, 1998

An action of B on X (or a B-action) is an algebra over the monad B5p´q.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 8 / 45

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Points and actions

Let C be a semi-abelian category, the kernel functor has a left adjoint PtBpCq

¡˚

B

  • J

C,

L

  • sending X to B ` X

p1B 0q B ιB

  • .

This defines a monad B5p´q: C Ñ C that sends any X to B5X B ` X

p1B 0q

B

ι1

  • Bourn-Janelidze, 1998

An action of B on X (or a B-action) is an algebra over the monad B5p´q. There is an equivalence of categories PtBpCq » B-ActpCq

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 8 / 45

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Definition

Let B and X be two Lie algebras. An action of B on X is a bilinear map B ˆ X Ñ X satisfying that “ x, rm, m1s ‰ “ “ rx, ms, m1‰ ´ “ rx, m1s, m ‰ “ rx, ys, m ‰ “ “ x, ry, ms ‰ ´ “ y, rx, ms ‰

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 9 / 45

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Definition

Let B and X be two Lie algebras. An action of B on X is a bilinear map B ˆ X Ñ X satisfying that “ x, rm, m1s ‰ “ “ rx, ms, m1‰ ´ “ rx, m1s, m ‰ “ rx, ys, m ‰ “ “ x, ry, ms ‰ ´ “ y, rx, ms ‰

Definition

Let B and X be two Lie algebras. An action of B on X is a Lie bracket defined on B ‘ X where B is a subalgebra and X an ideal.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 9 / 45

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Definition

Let B and X be two Lie algebras. An action of B on X is a bilinear map B ˆ X Ñ X satisfying that “ x, rm, m1s ‰ “ “ rx, ms, m1‰ ´ “ rx, m1s, m ‰ “ rx, ys, m ‰ “ “ x, ry, ms ‰ ´ “ y, rx, ms ‰

Definition

Let B and X be two Lie algebras. An action of B on X is a Lie bracket defined on B ‘ X where B is a subalgebra and X an ideal.

Definition

Let B and X be two Lie algebras. An action of B on X is a Lie algebras homomorphism B Ñ DerKpXq

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 9 / 45

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Points and Actions

Definition

Let B and X be two Lie algebras. The split extension X E B

  • induces an action of B on X.
  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 10 / 45

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Points and Actions

Definition

Let B and X be two Lie algebras. The split extension X E B

  • induces an action of B on X.

Moreover, any action of B on X, defines the split extension X X ¸ B B

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 10 / 45

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Points and Actions

Definition

Let B and X be two Lie algebras. The split extension X E B

  • induces an action of B on X.

Moreover, any action of B on X, defines the split extension X X ¸ B B

  • In fact, there is an equivalence of categories

PtBpLieKq – B-Act

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 10 / 45

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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 DerKpXq

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 11 / 45

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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 DerKpXq This means that the functor ActpB, ´q is representable.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 11 / 45

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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 DerKpXq This means that the functor ActpB, ´q is representable.

Definition (Broceux-Janelidze-Kelly, 2005)

A category is action representable if for any objects B and X, there exists an

  • bject rXs such that

ActpB, Xq – HompB, rXsq

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 11 / 45

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SLIDE 36

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 DerKpXq This means that the functor ActpB, ´q is representable.

Definition (Broceux-Janelidze-Kelly, 2005)

A category is action representable if for any objects B and X, there exists an

  • bject rXs such that

ActpB, Xq – HompB, rXsq

Examples

Groups

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 11 / 45

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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 DerKpXq This means that the functor ActpB, ´q is representable.

Definition (Broceux-Janelidze-Kelly, 2005)

A category is action representable if for any objects B and X, there exists an

  • bject rXs such that

ActpB, Xq – HompB, rXsq

Examples

Groups Lie algebras

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 11 / 45

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SLIDE 38

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

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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 HompX, Y q – HompX ˆ 0, Y q – Homp0, Y Xq – 0.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 12 / 45

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Locally Cartesian Closed Categories

Let C be a finitely complete category,

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

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SLIDE 41

Locally Cartesian Closed Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, the change of base functor a˚ : pC Ó Bq Ñ pC Ó B1q sends

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

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SLIDE 42

Locally Cartesian Closed Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, the change of base functor a˚ : pC Ó Bq Ñ pC Ó B1q sends B1 ˆB X

π2

X

f

  • B1

a

B

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

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SLIDE 43

Locally Cartesian Closed Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, the change of base functor a˚ : pC Ó Bq Ñ pC Ó B1q sends B1 ˆB X

π2

X

f

  • B1

a

B

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

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SLIDE 44

Locally Cartesian Closed Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, the change of base functor a˚ : pC Ó Bq Ñ pC Ó B1q sends B1 ˆB X

π1

  • π2

X

f

  • B1

a

B

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

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SLIDE 45

Locally Cartesian Closed Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, the change of base functor a˚ : pC Ó Bq Ñ pC Ó B1q sends B1 ˆB X

π1

  • π2

X

f

  • B1

a

B

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

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SLIDE 46

Locally Cartesian Closed Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, the change of base functor a˚ : pC Ó Bq Ñ pC Ó B1q sends B1 ˆB X

π1

  • π2

X

f

  • B1

a

B

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 13 / 45

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SLIDE 47

Locally Cartesian Closed Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, the change of base functor a˚ : pC Ó Bq Ñ pC Ó B1q sends B1 ˆB X

π1

  • π2

X

f

  • B1

a

B

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

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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

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SLIDE 49

Locally Algebraically Cartesian Closed (LACC) Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, we can define a functor a˚ : PtBpCq Ñ PtB1pCq that sends

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

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SLIDE 50

Locally Algebraically Cartesian Closed (LACC) Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, we can define a functor a˚ : PtBpCq Ñ PtB1pCq that sends B1 ˆB X

π2

X

f

  • B1

x1B1,s˝ay

  • a

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

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SLIDE 51

Locally Algebraically Cartesian Closed (LACC) Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, we can define a functor a˚ : PtBpCq Ñ PtB1pCq that sends B1 ˆB X

π2

X

f

  • B1

x1B1,s˝ay

  • a

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

slide-52
SLIDE 52

Locally Algebraically Cartesian Closed (LACC) Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, we can define a functor a˚ : PtBpCq Ñ PtB1pCq that sends B1 ˆB X

π1

  • π2

X

f

  • B1

x1B1,s˝ay

  • a

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

slide-53
SLIDE 53

Locally Algebraically Cartesian Closed (LACC) Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, we can define a functor a˚ : PtBpCq Ñ PtB1pCq that sends B1 ˆB X

π1

  • π2

X

f

  • B1

x1B1,s˝ay

  • a

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

slide-54
SLIDE 54

Locally Algebraically Cartesian Closed (LACC) Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, we can define a functor a˚ : PtBpCq Ñ PtB1pCq that sends B1 ˆB X

π1

  • π2

X

f

  • B1

x1B1,s˝ay

  • a

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 14 / 45

slide-55
SLIDE 55

Locally Algebraically Cartesian Closed (LACC) Categories

Let C be a finitely complete category, given a morphism a: B1 Ñ B, we can define a functor a˚ : PtBpCq Ñ PtB1pCq that sends B1 ˆB X

π1

  • π2

X

f

  • B1

x1B1,s˝ay

  • a

B

s

  • 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

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SLIDE 56

Kernel functor

Proposition (Gray, 2012)

If C has zero object, then it is LACC if and only if ¡˚

B : PtBpCq

Pt0pCq – C Ker f

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 15 / 45

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SLIDE 57

Kernel functor

Proposition (Gray, 2012)

If C has zero object, then it is LACC if and only if ¡˚

B : PtBpCq

Pt0pCq – C E

f

  • Ker f

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 15 / 45

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SLIDE 58

Kernel functor

Proposition (Gray, 2012)

If C has zero object, then it is LACC if and only if ¡˚

B : PtBpCq

Pt0pCq – C E

f

  • ✤
  • Ker f

B

s

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 15 / 45

slide-59
SLIDE 59

Kernel functor

Proposition (Gray, 2012)

If C has zero object, then it is LACC if and only if ¡˚

B : PtBpCq

Pt0pCq – C E

f

  • ✤
  • Ker f

B

s

  • has a right adjoint for all B.
  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 15 / 45

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SLIDE 60

(LACC)

Examples

Groups (over a cartesian closed category) (Gray, 2012)

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 16 / 45

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SLIDE 61

(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

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SLIDE 62

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

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SLIDE 63

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 » PtBpVq Ñ C

has a right adjoint for all B.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 17 / 45

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SLIDE 64

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 » PtBpVq Ñ C

has a right adjoint for all B.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 17 / 45

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SLIDE 65

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 » PtBpVq Ñ C

B5X

  • ✤
  • X

X has a right adjoint for all B.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 17 / 45

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SLIDE 66

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 AlgK.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 18 / 45

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SLIDE 67

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 AlgK. A subvariety of AlgK is any equationally defined class of algebras, considered as a full subcategory V of AlgK.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 18 / 45

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SLIDE 68

Non-associative algebras

Example

Lie algebras LieK. They satisfy the equations xx “ 0 xpyzq ` ypzxq ` zpxyq “ 0

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 19 / 45

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SLIDE 69

Non-associative algebras

Example

Lie algebras LieK. They satisfy the equations xx “ 0 xpyzq ` ypzxq ` zpxyq “ 0

Example

Associative algebras AsAlgK. They satisfy the equation xpyzq ´ pxyqz “ 0

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 19 / 45

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SLIDE 70

Non-associative algebras

Example

Lie algebras LieK. They satisfy the equations xx “ 0 xpyzq ` ypzxq ` zpxyq “ 0

Example

Associative algebras AsAlgK. They satisfy the equation xpyzq ´ pxyqz “ 0

Example

Abelian algebras AbK. They satisfy the equation xy “ 0

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 19 / 45

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SLIDE 71

Non-associative algebras

Theorem

If V is a variety of algebras over an infinite field K, all of its identities are of the form φpx1, . . . , xnq, where φ is a non-associative polynomial. Moreover, each of its homogeneous components ψpxi1, . . . , xinq is also an identity.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 20 / 45

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SLIDE 72

Non-associative algebras

Theorem

If V is a variety of algebras over an infinite field K, all of its identities are of the form φpx1, . . . , xnq, where φ is a non-associative polynomial. Moreover, each of its homogeneous components ψpxi1, . . . , xinq is also an identity. This means that if pxyqz ` x2 is an identity of V, then

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 20 / 45

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SLIDE 73

Non-associative algebras

Theorem

If V is a variety of algebras over an infinite field K, all of its identities are of the form φpx1, . . . , xnq, where φ is a non-associative polynomial. Moreover, each of its homogeneous components ψpxi1, . . . , xinq is also an identity. This means that if pxyqz ` x2 is an identity of V, then xpyzq x2 are also identities of V.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 20 / 45

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SLIDE 74

Non-associative algebras

Theorem

If V is a variety of algebras over an infinite field K, all of its identities are of the form φpx1, . . . , xnq, where φ is a non-associative polynomial. Moreover, each of its homogeneous components ψpxi1, . . . , xinq is also an identity. This means that if xpyzq ` ypzxq ` zpxyq ` xy ` yx is an identity of V, then

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 20 / 45

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SLIDE 75

Non-associative algebras

Theorem

If V is a variety of algebras over an infinite field K, all of its identities are of the form φpx1, . . . , xnq, where φ is a non-associative polynomial. Moreover, each of its homogeneous components ψpxi1, . . . , xinq is also an identity. This means that if xpyzq ` ypzxq ` zpxyq ` xy ` yx is an identity of V, then xpyzq ` ypzxq ` zpxyq xy ` yx are also identities of V.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 20 / 45

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SLIDE 76

Preservation of coproducts of B5p´q

Proposition (Gray, 2012)

Let V be a variety of non-associative algebras. It is (LACC) if and only if the canonical comparison pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 21 / 45

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SLIDE 77

Algebraic coherence

Theorem

The following are equivalent: V is algebraically coherent, i.e. the map pB5X ` B5Y q Ñ B5pX ` Y q is surjective.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 22 / 45

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SLIDE 78

Algebraic coherence

Theorem

The following are equivalent: V is algebraically coherent, i.e. the map pB5X ` B5Y q Ñ B5pX ` Y q is surjective. There exist λ1, . . . , λ8, µ1, . . . , µ8 P K such that zpxyq “ λ1pzxqy ` λ2pzyqx ` ¨ ¨ ¨ ` λ8ypxzq pxyqz “ µ1pzxqy ` µ2pzyqx ` ¨ ¨ ¨ ` µ8ypxzq are identities of V.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 22 / 45

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SLIDE 79

Algebraic coherence

Theorem

The following are equivalent: V is algebraically coherent, i.e. the map pB5X ` B5Y q Ñ B5pX ` Y q is surjective. There exist λ1, . . . , λ8, µ1, . . . , µ8 P K such that zpxyq “ λ1pzxqy ` λ2pzyqx ` ¨ ¨ ¨ ` λ8ypxzq pxyqz “ µ1pzxqy ` µ2pzyqx ` ¨ ¨ ¨ ` µ8ypxzq 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

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SLIDE 80

Algebraic coherence

Theorem

The following are equivalent: V is algebraically coherent, i.e. the map pB5X ` B5Y q Ñ B5pX ` Y q is surjective. There exist λ1, . . . , λ8, µ1, . . . , µ8 P K such that zpxyq “ λ1pzxqy ` λ2pzyqx ` ¨ ¨ ¨ ` λ8ypxzq pxyqz “ µ1pzxqy ` µ2pzyqx ` ¨ ¨ ¨ ` µ8ypxzq 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

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SLIDE 81

Nilpotent algebras

Proposition

If V is (LACC) and xpyzq “ 0 is an identity in V, then V is abelian.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 23 / 45

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SLIDE 82

Nilpotent algebras

Proposition

If V is (LACC) and xpyzq “ 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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 23 / 45

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SLIDE 83

Nilpotent algebras

Proposition

If V is (LACC) and xpyzq “ 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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. The element xpybq P B5pX ` Y q comes from zero.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 23 / 45

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SLIDE 84

Nilpotent algebras

Proposition

If V is (LACC) and xpyzq “ 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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. The element xpybq P B5pX ` Y q comes from zero. The expression yb plays the role of just “one element” in B5pX ` Y q.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 23 / 45

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SLIDE 85

Nilpotent algebras

Proposition

If V is (LACC) and xpyzq “ 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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. The element xpybq P B5pX ` Y q comes from zero. The expression yb plays the role of just “one element” in B5pX ` Y q. Then if xpybq is zero, either xpybq “ 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

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SLIDE 86

Associative algebras

Proposition

The variety of associative algebras is not (LACC).

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 24 / 45

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SLIDE 87

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: pB5X ` B5Y q Ñ B5pX ` Y q

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 24 / 45

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SLIDE 88

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: pB5X ` B5Y q Ñ B5pX ` Y q Then pxbqy and xpbyq go to the same element in B5pX ` Y q but they are different in pB5X ` B5Y q.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 24 / 45

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SLIDE 89

Leibniz algebras

Proposition

The variety of Leibniz algebras is not (LACC).

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 25 / 45

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SLIDE 90

Leibniz algebras

Proposition

The variety of Leibniz algebras is not (LACC). Proof: In the Leibniz case, we have they identities bpxyq “ pbxqy ` xpbyq

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 25 / 45

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SLIDE 91

Leibniz algebras

Proposition

The variety of Leibniz algebras is not (LACC). Proof: In the Leibniz case, we have they identities bpxyq “ pbxqy ` xpbyq bpxyq “ ´pxbqy ` xpbyq

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 25 / 45

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SLIDE 92

Leibniz algebras

Proposition

The variety of Leibniz algebras is not (LACC). Proof: In the Leibniz case, we have they identities bpxyq “ pbxqy ` xpbyq bpxyq “ ´pxbqy ` xpbyq Then, we have that pbxqy ` pxbqy “ 0.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 25 / 45

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SLIDE 93

Leibniz algebras

Proposition

The variety of Leibniz algebras is not (LACC). Proof: In the Leibniz case, we have they identities bpxyq “ pbxqy ` xpbyq bpxyq “ ´pxbqy ` xpbyq Then, we have that pbxqy ` pxbqy “ 0. Again, pbxqy ` pxbqy is zero in B5pX ` Y q but it does not need to be in pB5X ` B5Y q.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 25 / 45

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SLIDE 94

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 LieK.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 26 / 45

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SLIDE 95

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 LieK.

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

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SLIDE 96

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 LieK.

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 LieK, then it is abelian.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 26 / 45

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SLIDE 97

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

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SLIDE 98

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 99

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 100

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 101

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 102

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 103

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 104

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 105

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 106

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 107

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 108

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 109

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

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SLIDE 110

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

slide-111
SLIDE 111

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 pB5X ` B5Y q Ñ B5pX ` Y q is an isomorphism. xpbyq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5pxyqb ` λ6pyxqb ` λ7bpxyq ` λ8bpyxq “ λ1pxbqy ` λ2pbxqy ` λ3ypxbq ` λ4ypbxq ` λ5 ` µ1pbxqy ` µ2pxbqy ` µ3ypbxq ` ¨ ¨ ¨ ` µ7xpbyq ` µ8xpybq ˘ ` λ6 ` µ1pbyqx ` µ2pybqx ` µ3xpbyq ` ¨ ¨ ¨ ` µ7ypbxq ` µ8ypxbq ˘ ` λ7 ` λ1pbxqy ` λ2pxbqy ` λ3ypbxq ` ¨ ¨ ¨ ` λ7xpbyq ` λ8xpybq ˘ ` λ8 ` λ1pbyqx ` λ2pybqx ` λ3xpbyq ` ¨ ¨ ¨ ` λ7ypbxq ` µ8ypxbq ˘

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 27 / 45

slide-112
SLIDE 112

f33 “ µ1µ1µ5 ` µ2λ1µ5 ` µ3µ1λ5 ` µ4λ1λ5 ` µ1µ3µ1 ` µ2λ3µ1 ` µ3µ3λ1 ` µ4λ3λ1 ` µ5µ5µ7 ` µ6λ5µ7 ` µ7µ5λ7 ` µ8λ5λ7 ` µ5µ7µ3 ` µ6λ7µ3 ` µ7µ7λ3 ` µ8λ7λ3 f34 “ µ1µ1µ6 ` µ2λ1µ6 ` µ3µ1λ6 ` µ4λ1λ6 ` µ1µ3µ2 ` µ2λ3µ2 ` µ3µ3λ2 ` µ4λ3λ2 ` µ5µ6µ7 ` µ6λ6µ7 ` µ7µ6λ7 ` µ8λ6λ7 ` µ5µ8µ3 ` µ6λ8µ3 ` µ7µ8λ3 ` µ8λ8λ3 f35 “ µ1µ1µ7 ` µ2λ1µ7 ` µ3µ1λ7 ` µ4λ1λ7 ` µ1µ3µ3 ` µ2λ3µ3 ` µ3µ3λ3 ` µ4λ3λ3 ` µ5µ5µ5 ` µ6λ5µ5 ` µ7µ5λ5 ` µ8λ5λ5 ` µ5µ7µ1 ` µ6λ7µ1 ` µ7µ7λ1 ` µ8λ7λ1 f36 “ µ1µ1µ8 ` µ2λ1µ8 ` µ3µ1λ8 ` µ4λ1λ8 ` µ1µ3µ4 ` µ2λ3µ4 ` µ3µ3λ4 ` µ4λ3λ4 ` µ5µ6µ5 ` µ6λ6µ5 ` µ7µ6λ5 ` µ8λ6λ5 ` µ5µ8µ1 ` µ6λ8µ1 ` µ7µ8λ1 ` µ8λ8λ1 f37 “ µ1µ2µ5 ` µ2λ2µ5 ` µ3µ2λ5 ` µ4λ2λ5 ` µ1µ4µ1 ` µ2λ4µ1 ` µ3µ4λ1 ` µ4λ4λ1 ` µ5µ5µ8 ` µ6λ5µ8 ` µ7µ5λ8 ` µ8λ5λ8 ` µ5µ7µ4 ` µ6λ7µ4 ` µ7µ7λ4 ` µ8λ7λ4 f38 “ µ1µ2µ6 ` µ2λ2µ6 ` µ3µ2λ6 ` µ4λ2λ6 ` µ1µ4µ2 ` µ2λ4µ2 ` µ3µ4λ2 ` µ4λ4λ2 ` µ5µ6µ8 ` µ6λ6µ8 ` µ7µ6λ8 ` µ8λ6λ8 ` µ5µ8µ4 ` µ6λ8µ4 ` µ7µ8λ4 ` µ8λ8λ4 f39 “ µ1µ2µ7 ` µ2λ2µ7 ` µ3µ2λ7 ` µ4λ2λ7 ` µ1µ4µ3 ` µ2λ4µ3 ` µ3µ4λ3 ` µ4λ4λ3 ` µ5µ5µ6 ` µ6λ5µ6 ` µ7µ5λ6 ` µ8λ5λ6 ` µ5µ7µ2 ` µ6λ7µ2 ` µ7µ7λ2 ` µ8λ7λ2 f40 “ µ1µ2µ8 ` µ2λ2µ8 ` µ3µ2λ8 ` µ4λ2λ8 ` µ1µ4µ4 ` µ2λ4µ4 ` µ3µ4λ4 ` µ4λ4λ4 ` µ5µ6µ6 ` µ6λ6µ6 ` µ7µ6λ6 ` µ8λ6λ6 ` µ5µ8µ2 ` µ6λ8µ2 ` µ7µ8λ2 ` µ8λ8λ2

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 28 / 45

slide-113
SLIDE 113

f41 “ µ1µ5µ5 ` µ2λ5µ5 ` µ3µ5λ5 ` µ4λ5λ5 ` µ1µ7µ1 ` µ2λ7µ1 ` µ3µ7λ1 ` µ4λ7λ1 ` µ5µ1µ7 ` µ6λ1µ7 ` µ7µ1λ7 ` µ8λ1λ7 ` µ5µ3µ3 ` µ6λ3µ3 ` µ7µ3λ3 ` µ8λ3λ3 f42 “ µ1µ5µ6 ` µ2λ5µ6 ` µ3µ5λ6 ` µ4λ5λ6 ` µ1µ7µ2 ` µ2λ7µ2 ` µ3µ7λ2 ` µ4λ7λ2 ` µ5µ2µ7 ` µ6λ2µ7 ` µ7µ2λ7 ` µ8λ2λ7 ` µ5µ4µ3 ` µ6λ4µ3 ` µ7µ4λ3 ` µ8λ4λ3 f43 “ µ1µ5µ7 ` µ2λ5µ7 ` µ3µ5λ7 ` µ4λ5λ7 ` µ1µ7µ3 ` µ2λ7µ3 ` µ3µ7λ3 ` µ4λ7λ3 ` µ5µ1µ5 ` µ6λ1µ5 ` µ7µ1λ5 ` µ8λ1λ5 ` µ5µ3µ1 ` µ6λ3µ1 ` µ7µ3λ1 ` µ8λ3λ1 f44 “ µ1µ5µ8 ` µ2λ5µ8 ` µ3µ5λ8 ` µ4λ5λ8 ` µ1µ7µ4 ` µ2λ7µ4 ` µ3µ7λ4 ` µ4λ7λ4 ` µ5µ2µ5 ` µ6λ2µ5 ` µ7µ2λ5 ` µ8λ2λ5 ` µ5µ4µ1 ` µ6λ4µ1 ` µ7µ4λ1 ` µ8λ4λ1 f45 “ µ1µ6µ5 ` µ2λ6µ5 ` µ3µ6λ5 ` µ4λ6λ5 ` µ1µ8µ1 ` µ2λ8µ1 ` µ3µ8λ1 ` µ4λ8λ1 ` µ5µ1µ8 ` µ6λ1µ8 ` µ7µ1λ8 ` µ8λ1λ8 ` µ5µ3µ4 ` µ6λ3µ4 ` µ7µ3λ4 ` µ8λ3λ4 f46 “ µ1µ6µ6 ` µ2λ6µ6 ` µ3µ6λ6 ` µ4λ6λ6 ` µ1µ8µ2 ` µ2λ8µ2 ` µ3µ8λ2 ` µ4λ8λ2 ` µ5µ2µ8 ` µ6λ2µ8 ` µ7µ2λ8 ` µ8λ2λ8 ` µ5µ4µ4 ` µ6λ4µ4 ` µ7µ4λ4 ` µ8λ4λ4 f47 “ µ1µ6µ7 ` µ2λ6µ7 ` µ3µ6λ7 ` µ4λ6λ7 ` µ1µ8µ3 ` µ2λ8µ3 ` µ3µ8λ3 ` µ4λ8λ3 ` µ5µ1µ6 ` µ6λ1µ6 ` µ7µ1λ6 ` µ8λ1λ6 ` µ5µ3µ2 ` µ6λ3µ2 ` µ7µ3λ2 ` µ8λ3λ2 f48 “ µ1µ6µ8 ` µ2λ6µ8 ` µ3µ6λ8 ` µ4λ6λ8 ` µ1µ8µ4 ` µ2λ8µ4 ` µ3µ8λ4 ` µ4λ8λ4 ` µ5µ2µ6 ` µ6λ2µ6 ` µ7µ2λ6 ` µ8λ2λ6 ` µ5µ4µ2 ` µ6λ4µ2 ` µ7µ4λ2 ` µ8λ4λ2

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 29 / 45

slide-114
SLIDE 114

f49 “ ´λ1µ1 ´ λ2λ1 ` µ1µ2µ2 ` µ2λ2µ2 ` µ3µ2λ2 ` µ4λ2λ2 ` µ1µ4µ6 ` µ2λ4µ6 ` µ3µ4λ6 ` µ4λ4λ6 f50 “ ´λ1µ2 ´ λ2λ2 ` µ1µ1µ2 ` µ2λ1µ2 ` µ3µ1λ2 ` µ4λ1λ2 ` µ1µ3µ6 ` µ2λ3µ6 ` µ3µ3λ6 ` µ4λ3λ6 f51 “ ´λ1µ3 ´ λ2λ3 ` µ1µ2µ1 ` µ2λ2µ1 ` µ3µ2λ1 ` µ4λ2λ1 ` µ1µ4µ5 ` µ2λ4µ5 ` µ3µ4λ5 ` µ4λ4λ5 f52 “ ´λ1µ4 ´ λ2λ4 ` µ1µ1µ1 ` µ2λ1µ1 ` µ3µ1λ1 ` µ4λ1λ1 ` µ1µ3µ5 ` µ2λ3µ5 ` µ3µ3λ5 ` µ4λ3λ5 f53 “ ´λ1µ5 ´ λ2λ5 ` µ5µ2µ2 ` µ6λ2µ2 ` µ7µ2λ2 ` µ8λ2λ2 ` µ5µ4µ6 ` µ6λ4µ6 ` µ7µ4λ6 ` µ8λ4λ6 f54 “ ´λ1µ6 ´ λ2λ6 ` µ5µ1µ2 ` µ6λ1µ2 ` µ7µ1λ2 ` µ8λ1λ2 ` µ5µ3µ6 ` µ6λ3µ6 ` µ7µ3λ6 ` µ8λ3λ6 f55 “ ´λ1µ7 ´ λ2λ7 ` µ5µ2µ1 ` µ6λ2µ1 ` µ7µ2λ1 ` µ8λ2λ1 ` µ5µ4µ5 ` µ6λ4µ5 ` µ7µ4λ5 ` µ8λ4λ5 f56 “ ´λ1µ8 ´ λ2λ8 ` µ5µ1µ1 ` µ6λ1µ1 ` µ7µ1λ1 ` µ8λ1λ1 ` µ5µ3µ5 ` µ6λ3µ5 ` µ7µ3λ5 ` µ8λ3λ5 f57 “ ´λ3µ1 ´ λ4λ1 ` µ1µ2µ4 ` µ2λ2µ4 ` µ3µ2λ4 ` µ4λ2λ4 ` µ1µ4µ8 ` µ2λ4µ8 ` µ3µ4λ8 ` µ4λ4λ8 f58 “ ´λ3µ2 ´ λ4λ2 ` µ1µ1µ4 ` µ2λ1µ4 ` µ3µ1λ4 ` µ4λ1λ4 ` µ1µ3µ8 ` µ2λ3µ8 ` µ3µ3λ8 ` µ4λ3λ8 f59 “ ´λ3µ3 ´ λ4λ3 ` µ1µ2µ3 ` µ2λ2µ3 ` µ3µ2λ3 ` µ4λ2λ3 ` µ1µ4µ7 ` µ2λ4µ7 ` µ3µ4λ7 ` µ4λ4λ7 f60 “ ´λ3µ4 ´ λ4λ4 ` µ1µ1µ3 ` µ2λ1µ3 ` µ3µ1λ3 ` µ4λ1λ3 ` µ1µ3µ7 ` µ2λ3µ7 ` µ3µ3λ7 ` µ4λ3λ7 f61 “ ´λ3µ5 ´ λ4λ5 ` µ5µ2µ4 ` µ6λ2µ4 ` µ7µ2λ4 ` µ8λ2λ4 ` µ5µ4µ8 ` µ6λ4µ8 ` µ7µ4λ8 ` µ8λ4λ8 f62 “ ´λ3µ6 ´ λ4λ6 ` µ5µ1µ4 ` µ6λ1µ4 ` µ7µ1λ4 ` µ8λ1λ4 ` µ5µ3µ8 ` µ6λ3µ8 ` µ7µ3λ8 ` µ8λ3λ8 f63 “ ´λ3µ7 ´ λ4λ7 ` µ5µ2µ3 ` µ6λ2µ3 ` µ7µ2λ3 ` µ8λ2λ3 ` µ5µ4µ7 ` µ6λ4µ7 ` µ7µ4λ7 ` µ8λ4λ7 f64 “ ´λ3µ8 ´ λ4λ8 ` µ5µ1µ3 ` µ6λ1µ3 ` µ7µ1λ3 ` µ8λ1λ3 ` µ5µ3µ7 ` µ6λ3µ7 ` µ7µ3λ7 ` µ8λ3λ7

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 30 / 45

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SLIDE 115

f65 “ ´λ5µ1 ´ λ6λ1 ` µ1µ6µ2 ` µ2λ6µ2 ` µ3µ6λ2 ` µ4λ6λ2 ` µ1µ8µ6 ` µ2λ8µ6 ` µ3µ8λ6 ` µ4λ8λ6 f66 “ ´λ5µ2 ´ λ6λ2 ` µ1µ5µ2 ` µ2λ5µ2 ` µ3µ5λ2 ` µ4λ5λ2 ` µ1µ7µ6 ` µ2λ7µ6 ` µ3µ7λ6 ` µ4λ7λ6 f67 “ ´λ5µ3 ´ λ6λ3 ` µ1µ6µ1 ` µ2λ6µ1 ` µ3µ6λ1 ` µ4λ6λ1 ` µ1µ8µ5 ` µ2λ8µ5 ` µ3µ8λ5 ` µ4λ8λ5 f68 “ ´λ5µ4 ´ λ6λ4 ` µ1µ5µ1 ` µ2λ5µ1 ` µ3µ5λ1 ` µ4λ5λ1 ` µ1µ7µ5 ` µ2λ7µ5 ` µ3µ7λ5 ` µ4λ7λ5 f69 “ ´λ5µ5 ´ λ6λ5 ` µ5µ6µ2 ` µ6λ6µ2 ` µ7µ6λ2 ` µ8λ6λ2 ` µ5µ8µ6 ` µ6λ8µ6 ` µ7µ8λ6 ` µ8λ8λ6 f70 “ ´λ5µ6 ´ λ6λ6 ` µ5µ5µ2 ` µ6λ5µ2 ` µ7µ5λ2 ` µ8λ5λ2 ` µ5µ7µ6 ` µ6λ7µ6 ` µ7µ7λ6 ` µ8λ7λ6 f71 “ ´λ5µ7 ´ λ6λ7 ` µ5µ6µ1 ` µ6λ6µ1 ` µ7µ6λ1 ` µ8λ6λ1 ` µ5µ8µ5 ` µ6λ8µ5 ` µ7µ8λ5 ` µ8λ8λ5 f72 “ ´λ5µ8 ´ λ6λ8 ` µ5µ5µ1 ` µ6λ5µ1 ` µ7µ5λ1 ` µ8λ5λ1 ` µ5µ7µ5 ` µ6λ7µ5 ` µ7µ7λ5 ` µ8λ7λ5 f73 “ ´λ7µ1 ´ λ8λ1 ` µ1µ6µ4 ` µ2λ6µ4 ` µ3µ6λ4 ` µ4λ6λ4 ` µ1µ8µ8 ` µ2λ8µ8 ` µ3µ8λ8 ` µ4λ8λ8 f74 “ ´λ7µ2 ´ λ8λ2 ` µ1µ5µ4 ` µ2λ5µ4 ` µ3µ5λ4 ` µ4λ5λ4 ` µ1µ7µ8 ` µ2λ7µ8 ` µ3µ7λ8 ` µ4λ7λ8 f75 “ ´λ7µ3 ´ λ8λ3 ` µ1µ6µ3 ` µ2λ6µ3 ` µ3µ6λ3 ` µ4λ6λ3 ` µ1µ8µ7 ` µ2λ8µ7 ` µ3µ8λ7 ` µ4λ8λ7 f76 “ ´λ7µ4 ´ λ8λ4 ` µ1µ5µ3 ` µ2λ5µ3 ` µ3µ5λ3 ` µ4λ5λ3 ` µ1µ7µ7 ` µ2λ7µ7 ` µ3µ7λ7 ` µ4λ7λ7 f77 “ ´λ7µ5 ´ λ8λ5 ` µ5µ6µ4 ` µ6λ6µ4 ` µ7µ6λ4 ` µ8λ6λ4 ` µ5µ8µ8 ` µ6λ8µ8 ` µ7µ8λ8 ` µ8λ8λ8 f78 “ ´λ7µ6 ´ λ8λ6 ` µ5µ5µ4 ` µ6λ5µ4 ` µ7µ5λ4 ` µ8λ5λ4 ` µ5µ7µ8 ` µ6λ7µ8 ` µ7µ7λ8 ` µ8λ7λ8 f79 “ ´λ7µ7 ´ λ8λ7 ` µ5µ6µ3 ` µ6λ6µ3 ` µ7µ6λ3 ` µ8λ6λ3 ` µ5µ8µ7 ` µ6λ8µ7 ` µ7µ8λ7 ` µ8λ8λ7 f80 “ ´λ7µ8 ´ λ8λ8 ` µ5µ5µ3 ` µ6λ5µ3 ` µ7µ5λ3 ` µ8λ5λ3 ` µ5µ7µ7 ` µ6λ7µ7 ` µ7µ7λ7 ` µ8λ7λ7

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 31 / 45

slide-116
SLIDE 116

f81 “ λ1µ1µ5 ` λ2λ1µ5 ` λ3µ1λ5 ` λ4λ1λ5 ` λ1µ3µ1 ` λ2λ3µ1 ` λ3µ3λ1 ` λ4λ3λ1 ` λ5µ5µ7 ` λ6λ5µ7 ` λ7µ5λ7 ` λ8λ5λ7 ` λ5µ7µ3 ` λ6λ7µ3 ` λ7µ7λ3 ` λ8λ7λ3 f82 “ λ1µ1µ6 ` λ2λ1µ6 ` λ3µ1λ6 ` λ4λ1λ6 ` λ1µ3µ2 ` λ2λ3µ2 ` λ3µ3λ2 ` λ4λ3λ2 ` λ5µ6µ7 ` λ6λ6µ7 ` λ7µ6λ7 ` λ8λ6λ7 ` λ5µ8µ3 ` λ6λ8µ3 ` λ7µ8λ3 ` λ8λ8λ3 f83 “ λ1µ1µ7 ` λ2λ1µ7 ` λ3µ1λ7 ` λ4λ1λ7 ` λ1µ3µ3 ` λ2λ3µ3 ` λ3µ3λ3 ` λ4λ3λ3 ` λ5µ5µ5 ` λ6λ5µ5 ` λ7µ5λ5 ` λ8λ5λ5 ` λ5µ7µ1 ` λ6λ7µ1 ` λ7µ7λ1 ` λ8λ7λ1 f84 “ λ1µ1µ8 ` λ2λ1µ8 ` λ3µ1λ8 ` λ4λ1λ8 ` λ1µ3µ4 ` λ2λ3µ4 ` λ3µ3λ4 ` λ4λ3λ4 ` λ5µ6µ5 ` λ6λ6µ5 ` λ7µ6λ5 ` λ8λ6λ5 ` λ5µ8µ1 ` λ6λ8µ1 ` λ7µ8λ1 ` λ8λ8λ1 f85 “ λ1µ2µ5 ` λ2λ2µ5 ` λ3µ2λ5 ` λ4λ2λ5 ` λ1µ4µ1 ` λ2λ4µ1 ` λ3µ4λ1 ` λ4λ4λ1 ` λ5µ5µ8 ` λ6λ5µ8 ` λ7µ5λ8 ` λ8λ5λ8 ` λ5µ7µ4 ` λ6λ7µ4 ` λ7µ7λ4 ` λ8λ7λ4 f86 “ λ1µ2µ6 ` λ2λ2µ6 ` λ3µ2λ6 ` λ4λ2λ6 ` λ1µ4µ2 ` λ2λ4µ2 ` λ3µ4λ2 ` λ4λ4λ2 ` λ5µ6µ8 ` λ6λ6µ8 ` λ7µ6λ8 ` λ8λ6λ8 ` λ5µ8µ4 ` λ6λ8µ4 ` λ7µ8λ4 ` λ8λ8λ4 f87 “ λ1µ2µ7 ` λ2λ2µ7 ` λ3µ2λ7 ` λ4λ2λ7 ` λ1µ4µ3 ` λ2λ4µ3 ` λ3µ4λ3 ` λ4λ4λ3 ` λ5µ5µ6 ` λ6λ5µ6 ` λ7µ5λ6 ` λ8λ5λ6 ` λ5µ7µ2 ` λ6λ7µ2 ` λ7µ7λ2 ` λ8λ7λ2 f88 “ λ1µ2µ8 ` λ2λ2µ8 ` λ3µ2λ8 ` λ4λ2λ8 ` λ1µ4µ4 ` λ2λ4µ4 ` λ3µ4λ4 ` λ4λ4λ4 ` λ5µ6µ6 ` λ6λ6µ6 ` λ7µ6λ6 ` λ8λ6λ6 ` λ5µ8µ2 ` λ6λ8µ2 ` λ7µ8λ2 ` λ8λ8λ2

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 32 / 45

slide-117
SLIDE 117

f89 “ λ1µ5µ5 ` λ2λ5µ5 ` λ3µ5λ5 ` λ4λ5λ5 ` λ1µ7µ1 ` λ2λ7µ1 ` λ3µ7λ1 ` λ4λ7λ1 ` λ5µ1µ7 ` λ6λ1µ7 ` λ7µ1λ7 ` λ8λ1λ7 ` λ5µ3µ3 ` λ6λ3µ3 ` λ7µ3λ3 ` λ8λ3λ3 f90 “ λ1µ5µ6 ` λ2λ5µ6 ` λ3µ5λ6 ` λ4λ5λ6 ` λ1µ7µ2 ` λ2λ7µ2 ` λ3µ7λ2 ` λ4λ7λ2 ` λ5µ2µ7 ` λ6λ2µ7 ` λ7µ2λ7 ` λ8λ2λ7 ` λ5µ4µ3 ` λ6λ4µ3 ` λ7µ4λ3 ` λ8λ4λ3 f91 “ λ1µ5µ7 ` λ2λ5µ7 ` λ3µ5λ7 ` λ4λ5λ7 ` λ1µ7µ3 ` λ2λ7µ3 ` λ3µ7λ3 ` λ4λ7λ3 ` λ5µ1µ5 ` λ6λ1µ5 ` λ7µ1λ5 ` λ8λ1λ5 ` λ5µ3µ1 ` λ6λ3µ1 ` λ7µ3λ1 ` λ8λ3λ1 f92 “ λ1µ5µ8 ` λ2λ5µ8 ` λ3µ5λ8 ` λ4λ5λ8 ` λ1µ7µ4 ` λ2λ7µ4 ` λ3µ7λ4 ` λ4λ7λ4 ` λ5µ2µ5 ` λ6λ2µ5 ` λ7µ2λ5 ` λ8λ2λ5 ` λ5µ4µ1 ` λ6λ4µ1 ` λ7µ4λ1 ` λ8λ4λ1 f93 “ λ1µ6µ5 ` λ2λ6µ5 ` λ3µ6λ5 ` λ4λ6λ5 ` λ1µ8µ1 ` λ2λ8µ1 ` λ3µ8λ1 ` λ4λ8λ1 ` λ5µ1µ8 ` λ6λ1µ8 ` λ7µ1λ8 ` λ8λ1λ8 ` λ5µ3µ4 ` λ6λ3µ4 ` λ7µ3λ4 ` λ8λ3λ4 f94 “ λ1µ6µ6 ` λ2λ6µ6 ` λ3µ6λ6 ` λ4λ6λ6 ` λ1µ8µ2 ` λ2λ8µ2 ` λ3µ8λ2 ` λ4λ8λ2 ` λ5µ2µ8 ` λ6λ2µ8 ` λ7µ2λ8 ` λ8λ2λ8 ` λ5µ4µ4 ` λ6λ4µ4 ` λ7µ4λ4 ` λ8λ4λ4 f95 “ λ1µ6µ7 ` λ2λ6µ7 ` λ3µ6λ7 ` λ4λ6λ7 ` λ1µ8µ3 ` λ2λ8µ3 ` λ3µ8λ3 ` λ4λ8λ3 ` λ5µ1µ6 ` λ6λ1µ6 ` λ7µ1λ6 ` λ8λ1λ6 ` λ5µ3µ2 ` λ6λ3µ2 ` λ7µ3λ2 ` λ8λ3λ2 f96 “ λ1µ6µ8 ` λ2λ6µ8 ` λ3µ6λ8 ` λ4λ6λ8 ` λ1µ8µ4 ` λ2λ8µ4 ` λ3µ8λ4 ` λ4λ8λ4 ` λ5µ2µ6 ` λ6λ2µ6 ` λ7µ2λ6 ` λ8λ2λ6 ` λ5µ4µ2 ` λ6λ4µ2 ` λ7µ4λ2 ` λ8λ4λ2

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 33 / 45

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SLIDE 118

f97 “ ´µ1µ1 ´ µ2λ1 ` λ1µ2µ2 ` λ2λ2µ2 ` λ3µ2λ2 ` λ4λ2λ2 ` λ1µ4µ6 ` λ2λ4µ6 ` λ3µ4λ6 ` λ4λ4λ6 f98 “ ´µ1µ2 ´ µ2λ2 ` λ1µ1µ2 ` λ2λ1µ2 ` λ3µ1λ2 ` λ4λ1λ2 ` λ1µ3µ6 ` λ2λ3µ6 ` λ3µ3λ6 ` λ4λ3λ6 f99 “ ´µ1µ3 ´ µ2λ3 ` λ1µ2µ1 ` λ2λ2µ1 ` λ3µ2λ1 ` λ4λ2λ1 ` λ1µ4µ5 ` λ2λ4µ5 ` λ3µ4λ5 ` λ4λ4λ5 f100 “ ´µ1µ4 ´ µ2λ4 ` λ1µ1µ1 ` λ2λ1µ1 ` λ3µ1λ1 ` λ4λ1λ1 ` λ1µ3µ5 ` λ2λ3µ5 ` λ3µ3λ5 ` λ4λ3λ5 f101 “ ´µ1µ5 ´ µ2λ5 ` λ5µ2µ2 ` λ6λ2µ2 ` λ7µ2λ2 ` λ8λ2λ2 ` λ5µ4µ6 ` λ6λ4µ6 ` λ7µ4λ6 ` λ8λ4λ6 f102 “ ´µ1µ6 ´ µ2λ6 ` λ5µ1µ2 ` λ6λ1µ2 ` λ7µ1λ2 ` λ8λ1λ2 ` λ5µ3µ6 ` λ6λ3µ6 ` λ7µ3λ6 ` λ8λ3λ6 f103 “ ´µ1µ7 ´ µ2λ7 ` λ5µ2µ1 ` λ6λ2µ1 ` λ7µ2λ1 ` λ8λ2λ1 ` λ5µ4µ5 ` λ6λ4µ5 ` λ7µ4λ5 ` λ8λ4λ5 f104 “ ´µ1µ8 ´ µ2λ8 ` λ5µ1µ1 ` λ6λ1µ1 ` λ7µ1λ1 ` λ8λ1λ1 ` λ5µ3µ5 ` λ6λ3µ5 ` λ7µ3λ5 ` λ8λ3λ5 f105 “ ´µ3µ1 ´ µ4λ1 ` λ1µ2µ4 ` λ2λ2µ4 ` λ3µ2λ4 ` λ4λ2λ4 ` λ1µ4µ8 ` λ2λ4µ8 ` λ3µ4λ8 ` λ4λ4λ8 f106 “ ´µ3µ2 ´ µ4λ2 ` λ1µ1µ4 ` λ2λ1µ4 ` λ3µ1λ4 ` λ4λ1λ4 ` λ1µ3µ8 ` λ2λ3µ8 ` λ3µ3λ8 ` λ4λ3λ8 f107 “ ´µ3µ3 ´ µ4λ3 ` λ1µ2µ3 ` λ2λ2µ3 ` λ3µ2λ3 ` λ4λ2λ3 ` λ1µ4µ7 ` λ2λ4µ7 ` λ3µ4λ7 ` λ4λ4λ7 f108 “ ´µ3µ4 ´ µ4λ4 ` λ1µ1µ3 ` λ2λ1µ3 ` λ3µ1λ3 ` λ4λ1λ3 ` λ1µ3µ7 ` λ2λ3µ7 ` λ3µ3λ7 ` λ4λ3λ7 f109 “ ´µ3µ5 ´ µ4λ5 ` λ5µ2µ4 ` λ6λ2µ4 ` λ7µ2λ4 ` λ8λ2λ4 ` λ5µ4µ8 ` λ6λ4µ8 ` λ7µ4λ8 ` λ8λ4λ8 f110 “ ´µ3µ6 ´ µ4λ6 ` λ5µ1µ4 ` λ6λ1µ4 ` λ7µ1λ4 ` λ8λ1λ4 ` λ5µ3µ8 ` λ6λ3µ8 ` λ7µ3λ8 ` λ8λ3λ8 f111 “ ´µ3µ7 ´ µ4λ7 ` λ5µ2µ3 ` λ6λ2µ3 ` λ7µ2λ3 ` λ8λ2λ3 ` λ5µ4µ7 ` λ6λ4µ7 ` λ7µ4λ7 ` λ8λ4λ7 f112 “ ´µ3µ8 ´ µ4λ8 ` λ5µ1µ3 ` λ6λ1µ3 ` λ7µ1λ3 ` λ8λ1λ3 ` λ5µ3µ7 ` λ6λ3µ7 ` λ7µ3λ7 ` λ8λ3λ7

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 34 / 45

slide-119
SLIDE 119

f113 “ ´µ5µ1 ´ µ6λ1 ` λ1µ6µ2 ` λ2λ6µ2 ` λ3µ6λ2 ` λ4λ6λ2 ` λ1µ8µ6 ` λ2λ8µ6 ` λ3µ8λ6 ` λ4λ8λ6 f114 “ ´µ5µ2 ´ µ6λ2 ` λ1µ5µ2 ` λ2λ5µ2 ` λ3µ5λ2 ` λ4λ5λ2 ` λ1µ7µ6 ` λ2λ7µ6 ` λ3µ7λ6 ` λ4λ7λ6 f115 “ ´µ5µ3 ´ µ6λ3 ` λ1µ6µ1 ` λ2λ6µ1 ` λ3µ6λ1 ` λ4λ6λ1 ` λ1µ8µ5 ` λ2λ8µ5 ` λ3µ8λ5 ` λ4λ8λ5 f116 “ ´µ5µ4 ´ µ6λ4 ` λ1µ5µ1 ` λ2λ5µ1 ` λ3µ5λ1 ` λ4λ5λ1 ` λ1µ7µ5 ` λ2λ7µ5 ` λ3µ7λ5 ` λ4λ7λ5 f117 “ ´µ5µ5 ´ µ6λ5 ` λ5µ6µ2 ` λ6λ6µ2 ` λ7µ6λ2 ` λ8λ6λ2 ` λ5µ8µ6 ` λ6λ8µ6 ` λ7µ8λ6 ` λ8λ8λ6 f118 “ ´µ5µ6 ´ µ6λ6 ` λ5µ5µ2 ` λ6λ5µ2 ` λ7µ5λ2 ` λ8λ5λ2 ` λ5µ7µ6 ` λ6λ7µ6 ` λ7µ7λ6 ` λ8λ7λ6 f119 “ ´µ5µ7 ´ µ6λ7 ` λ5µ6µ1 ` λ6λ6µ1 ` λ7µ6λ1 ` λ8λ6λ1 ` λ5µ8µ5 ` λ6λ8µ5 ` λ7µ8λ5 ` λ8λ8λ5 f120 “ ´µ5µ8 ´ µ6λ8 ` λ5µ5µ1 ` λ6λ5µ1 ` λ7µ5λ1 ` λ8λ5λ1 ` λ5µ7µ5 ` λ6λ7µ5 ` λ7µ7λ5 ` λ8λ7λ5 f121 “ ´µ7µ1 ´ µ8λ1 ` λ1µ6µ4 ` λ2λ6µ4 ` λ3µ6λ4 ` λ4λ6λ4 ` λ1µ8µ8 ` λ2λ8µ8 ` λ3µ8λ8 ` λ4λ8λ8 f122 “ ´µ7µ2 ´ µ8λ2 ` λ1µ5µ4 ` λ2λ5µ4 ` λ3µ5λ4 ` λ4λ5λ4 ` λ1µ7µ8 ` λ2λ7µ8 ` λ3µ7λ8 ` λ4λ7λ8 f123 “ ´µ7µ3 ´ µ8λ3 ` λ1µ6µ3 ` λ2λ6µ3 ` λ3µ6λ3 ` λ4λ6λ3 ` λ1µ8µ7 ` λ2λ8µ7 ` λ3µ8λ7 ` λ4λ8λ7 f124 “ ´µ7µ4 ´ µ8λ4 ` λ1µ5µ3 ` λ2λ5µ3 ` λ3µ5λ3 ` λ4λ5λ3 ` λ1µ7µ7 ` λ2λ7µ7 ` λ3µ7λ7 ` λ4λ7λ7 f125 “ ´µ7µ5 ´ µ8λ5 ` λ5µ6µ4 ` λ6λ6µ4 ` λ7µ6λ4 ` λ8λ6λ4 ` λ5µ8µ8 ` λ6λ8µ8 ` λ7µ8λ8 ` λ8λ8λ8 f126 “ ´µ7µ6 ´ µ8λ6 ` λ5µ5µ4 ` λ6λ5µ4 ` λ7µ5λ4 ` λ8λ5λ4 ` λ5µ7µ8 ` λ6λ7µ8 ` λ7µ7λ8 ` λ8λ7λ8 f127 “ ´µ7µ7 ´ µ8λ7 ` λ5µ6µ3 ` λ6λ6µ3 ` λ7µ6λ3 ` λ8λ6λ3 ` λ5µ8µ7 ` λ6λ8µ7 ` λ7µ8λ7 ` λ8λ8λ7 f128 “ ´µ7µ8 ´ µ8λ8 ` λ5µ5µ3 ` λ6λ5µ3 ` λ7µ5λ3 ` λ8λ5λ3 ` λ5µ7µ7 ` λ6λ7µ7 ` λ7µ7λ7 ` λ8λ7λ7

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 35 / 45

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SLIDE 120

Non-commutative and non-anticommutative

We have 128 different polynomials fi P Zrλ1, . . . , λ8, µ1, . . . , µ8s that the coefficients λ1, . . . , λ8, µ1, . . . , µ8 have to satisfy.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 36 / 45

slide-121
SLIDE 121

Non-commutative and non-anticommutative

We have 128 different polynomials fi P Zrλ1, . . . , λ8, µ1, . . . , µ8s that the coefficients λ1, . . . , λ8, µ1, . . . , µ8 have to satisfy. Thanks to the computer algebra package SINGULAR we know that there exist some gi P Qrλ1, . . . , λ8, µ1, . . . , µ8s such that 1 “ f1g1 ` ¨ ¨ ¨ ` f128g128

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 36 / 45

slide-122
SLIDE 122

Non-commutative and non-anticommutative

We have 128 different polynomials fi P Zrλ1, . . . , λ8, µ1, . . . , µ8s that the coefficients λ1, . . . , λ8, µ1, . . . , µ8 have to satisfy. Thanks to the computer algebra package SINGULAR we know that there exist some gi P Qrλ1, . . . , λ8, µ1, . . . , µ8s such that 1 “ f1g1 ` ¨ ¨ ¨ ` f128g128 If X “ px1, . . . , x8, y1, . . . , y8q P K16 is a solution of our polynomials fi, then 1 “ f1pXqg1pXq ` ¨ ¨ ¨ ` f128pXqg128pXq “ 0

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 36 / 45

slide-123
SLIDE 123

Non-commutative and non-anticommutative

We have 128 different polynomials fi P Zrλ1, . . . , λ8, µ1, . . . , µ8s that the coefficients λ1, . . . , λ8, µ1, . . . , µ8 have to satisfy. Thanks to the computer algebra package SINGULAR we know that there exist some gi P Qrλ1, . . . , λ8, µ1, . . . , µ8s such that 1 “ f1g1 ` ¨ ¨ ¨ ` f128g128 If X “ px1, . . . , x8, y1, . . . , y8q P K16 is a solution of our polynomials fi, then 1 “ f1pXqg1pXq ` ¨ ¨ ¨ ` f128pXqg128pXq “ 0

Theorem

Let K be a field of characteristic zero. If V is a (LACC) variety of non-associative algebras without any identity of order 2, then V is abelian.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 36 / 45

slide-124
SLIDE 124

Prime characteristic

If char ‰ 2, all results previously done work without any problem, but we should do a computation for all primes p.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 37 / 45

slide-125
SLIDE 125

Prime characteristic

If char ‰ 2, all results previously done work without any problem, but we should do a computation for all primes p. On the other hand, we know that 1 “ f1g1 ` ¨ ¨ ¨ ` f128g128 where fi P Zrλ1, . . . , µ8s and gi P Qrλ1, . . . , µ8s.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 37 / 45

slide-126
SLIDE 126

Prime characteristic

If char ‰ 2, all results previously done work without any problem, but we should do a computation for all primes p. On the other hand, we know that 1 “ f1g1 ` ¨ ¨ ¨ ` f128g128 where fi P Zrλ1, . . . , µ8s and gi P Qrλ1, . . . , µ8s. Therefore, there exists m such that m “ f1g 1

1 ` ¨ ¨ ¨ ` f128g 1 128

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 37 / 45

slide-127
SLIDE 127

Prime characteristic

If char ‰ 2, all results previously done work without any problem, but we should do a computation for all primes p. On the other hand, we know that 1 “ f1g1 ` ¨ ¨ ¨ ` f128g128 where fi P Zrλ1, . . . , µ8s and gi P Qrλ1, . . . , µ8s. Therefore, there exists m such that m “ f1g 1

1 ` ¨ ¨ ¨ ` f128g 1 128

Then it is just enough to compute a Gröbner-Shirshov basis in characteristic the prime divisors of this n.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 37 / 45

slide-128
SLIDE 128

Prime characteristic

In our case, m “87023663716111410986717845740599868868355247730561 80191922975621327712489528654597642065042289512 p97 decimal digitsq

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 38 / 45

slide-129
SLIDE 129

Prime characteristic

In our case, m “87023663716111410986717845740599868868355247730561 80191922975621327712489528654597642065042289512 p97 decimal digitsq which factors intro primes as m “ 23 ˆ 32 ˆ 7 ˆ 1049 ˆ 14479 ˆ 12133021861 ˆ 16113739806343 ˆ 6887165068164869 ˆ 18481555969123738547989 ˆ 45682348205218520398213951.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 38 / 45

slide-130
SLIDE 130

Prime characteristic

In our case, m “87023663716111410986717845740599868868355247730561 80191922975621327712489528654597642065042289512 p97 decimal digitsq which factors intro primes as m “ 23 ˆ 32 ˆ 7 ˆ 1049 ˆ 14479 ˆ 12133021861 ˆ 16113739806343 ˆ 6887165068164869 ˆ 18481555969123738547989 ˆ 45682348205218520398213951. the maximum characteristic that SINGULAR admits is 231 ´ 1 “ 2147483647

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 38 / 45

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SLIDE 131

Prime characteristic

Computing a Gröbner basis with a different monomial ordering (swapping µ7 with µ8) gave us a different linear combination: m1 “ f1g 1

1 ` ¨ ¨ ¨ ` f128g 1 128

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 39 / 45

slide-132
SLIDE 132

Prime characteristic

Computing a Gröbner basis with a different monomial ordering (swapping µ7 with µ8) gave us a different linear combination: m1 “ f1g 1

1 ` ¨ ¨ ¨ ` f128g 1 128

where m1 “259428036880596829755696425536436202871595677 3862814343963891594296990709562550944826763410 which factors into primes as m1 “ 2 ˆ 3 ˆ 5 ˆ 7 ˆ 13 ˆ 67 ˆ 8878743659183 ˆ 980432662345198665679 ˆ 1629334190706617301312819947580437423912985358007543.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 39 / 45

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SLIDE 133

Prime characteristic

The good news is that gcdpm, m1q “ 2 ˆ 3 ˆ 7 so we only needed to compute a Gröbner basis on these three characteristics.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 40 / 45

slide-134
SLIDE 134

Prime characteristic

The good news is that gcdpm, m1q “ 2 ˆ 3 ˆ 7 so we only needed to compute a Gröbner basis on these three characteristics.

Theorem

Let K be a field of characteristic zero. If V is a (LACC) variety of non-associative algebras without any identity of order 2, then V is abelian.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 40 / 45

slide-135
SLIDE 135

Characteristic 2

If char K “ 2, the identity xx “ 0 implies the identity xy “ ´yx, but not the

  • ther way around.
  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 41 / 45

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SLIDE 136

Characteristic 2

If char K “ 2, the identity xx “ 0 implies the identity xy “ ´yx, but not the

  • ther way around.

Then, we can define the variety of quasi-Lie algebras denoted by qLieK, which satisfies the Jacobi identity and xy “ ´yx.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 41 / 45

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SLIDE 137

Characteristic 2

If char K “ 2, the identity xx “ 0 implies the identity xy “ ´yx, but not the

  • ther way around.

Then, we can define the variety of quasi-Lie algebras denoted by qLieK, which satisfies the Jacobi identity and xy “ ´yx.

Theorem

Both varieties LieK and qLieK are (LACC).

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 41 / 45

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SLIDE 138

n-ary operations

Theorem

If V is an algebraically coherent variety of n-algebras, with n ‰ 2, then it is abelian.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 42 / 45

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SLIDE 139

n-ary operations

Theorem

If V is an algebraically coherent variety of n-algebras, with n ‰ 2, then it is abelian.

Corollary

If V is a (LACC) variety of n-algebras, with n ‰ 2, then it is abelian.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 42 / 45

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SLIDE 140

Main theorem

Summarising:

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 43 / 45

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SLIDE 141

Main theorem

Summarising:

Theorem

Let V be a non-abelian (LACC) variety of non-associative n-algebras

  • ver an infinite field K. Then,
  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 43 / 45

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SLIDE 142

Main theorem

Summarising:

Theorem

Let V be a non-abelian (LACC) variety of non-associative n-algebras

  • ver an infinite field K. Then,

If char K ‰ 2, then V “ LieK,

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 43 / 45

slide-143
SLIDE 143

Main theorem

Summarising:

Theorem

Let V be a non-abelian (LACC) variety of non-associative n-algebras

  • ver an infinite field K. Then,

If char K ‰ 2, then V “ LieK, If char K “ 2, then V “ LieK or V “ qLieK.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 43 / 45

slide-144
SLIDE 144

Main theorem

Summarising:

Theorem

Let V be a non-abelian (LACC) variety of non-associative n-algebras

  • ver an infinite field K. Then,

If char K ‰ 2, then V “ LieK, If char K “ 2, then V “ LieK or V “ qLieK.

  • X. G-M, T. Van der Linden

A characterisation of Lie algebras amongst anti-commutative algebras. Journal of Pure and Applied Algebra, 223(11), 4857–4870 (2019).

  • X. G-M, T. Van der Linden

A characterisation of Lie algebras via algebraic exponentiation. Advances in Mathematics, 341, 92–117, (2019).

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 43 / 45

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SLIDE 145

Another characterisation?

Definition (Borceux-Janelidze-Kelly, 2005)

Let L and M be two Lie algebras. An action of L on M is a Lie algebras homomorphism L Ñ DerKpMq

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 44 / 45

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SLIDE 146

Another characterisation?

Definition (Borceux-Janelidze-Kelly, 2005)

Let L and M be two Lie algebras. An action of L on M is a Lie algebras homomorphism L Ñ DerKpMq

Definition

A category is action representable if for any objects B and X, there exists an

  • bject rXs such that

ActpB, Xq – HompB, rXsq

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 44 / 45

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SLIDE 147

Representability of actions

Theorem

Let V be a non-abelian action representable variety of non-associative algebras

  • ver an infinite field K. Then,

If char K ‰ 2, then V “ LieK,

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 45 / 45

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SLIDE 148

Representability of actions

Theorem

Let V be a non-abelian action representable variety of non-associative algebras

  • ver an infinite field K. Then,

If char K ‰ 2, then V “ LieK, If char K “ 2, then V “ LieK or V “ qLieK.

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 45 / 45

slide-149
SLIDE 149

Representability of actions

Theorem

Let V be a non-abelian action representable variety of non-associative algebras

  • ver an infinite field K. Then,

If char K ‰ 2, then V “ LieK, If char K “ 2, then V “ LieK or V “ qLieK.

Remark (Borceux-Janelidze-Kelly 2005, Gray 2012)

The variety of Boolean Rings over Z2 is action representable but not (LACC).

  • X. García-Martínez

(UVigo) Characterising Lie algebras Ottawa, August, 2019 45 / 45