Universal enveloping algebras for Malcev color algebras
aa r X i v : . [ m a t h . R A ] S e p Universal enveloping algebras forMalcev color algebras
Daniel de la Concepción ∗ Universidad de la Rioja
Departamento de Matemáticas y Computación
Abstract
In this paper we give a construction of the universal enveloping algebra ofa Malcev algebra in categories of group algebra comodules with a symmetrygiven by a bicharacter of the group. A particular example of such categoriesis the category of super vector spaces.
There is a very well known relation between connected Hopf algebras and Liealgebras; as it is shown in [5]. In fact this categories are equivalent in the caseof having a field of characteristic zero. This relation has been extended toconnected cocommutative and coassociative bialgebras and Sabinin algebrasin [6].It has been shown that this relation between connected Hopf algebras andLie algebras is deeper, as it stands when moving to categories other than
Vec .Specifically, the monoidal categories that have a forgetful functor to
Vec anda braiding. A survey can be read in [4], where the most common examples areexplained. This relations makes us wonder if this results can be extended towider subclasses of Sabinin algebras in this types of categories.One easy and general subclass of this categories mentioned in the previousparagraph, are the categories of group comodules with a symmetry given by agroup bicharacter. Results about this structures in relation with Lie algebrascan be read in [9], [8] and [1], originally.In this paper we generalize this relation with the easiest examples: Malcevalgebras on K [ G ]-comodule categories; using already known category theoretictools. ∗ FPU12 Grant from the Ministerio de Educación, Cultura y Deporte in Spain; and associatedto the research project MTM2013-45588-C3-3-P Category theory
In this first section, we recall some category theory and define the categoricalobjects that will appear through the paper.Also, everywhere in this papaer, unless otherwise stated, K is a field ofcharacteristic zero and any group G is abelian. Firstly, we define some categorical structures through commutative diagrams;that in the next sections will be used.
Definition 2.1.
A category C is called monoidal if it is equiped with: • A bi-functor ⊗ : C × C → C called tensor product. • An object I called the unit element. • A natural isomorphism α A,B,C : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ) calledassociator. • Two natural isomorphisms λ A : I ⊗ A → A and ρ A : A ⊗ I → A , respec-tively called left and right unitor.such that the diagrams 1 and 2 are commutative. The first examples of monoidal categories are those with finite products orcoproducts:
Example 2.2.
Let C be a category with finite products. Then ( C , × , T, α, ρ, λ ) with T as the terminal object of the category and × as the categorical productis a monoidal category given the following: • α A,B,C as the morphisms obtained by recombining the projections from ( A × B ) × C to match A × ( B × C ) . • T is the unit element. • λ A : A × T → A and ρ A : T × A → A as the projections. Example 2.3.
Let C be a category with finite coproducts. Then ( C , ⊕ , ⊥ , α, ρ, λ ) with ⊥ as the initial object of the category and ⊕ as the categorical coproductis a monoidal category given the following: • α A,B,C as the morphisms obtained by recombining the injections to ( A ⊕ B ) ⊕ C to match A ⊕ ( B ⊕ C ) . • ⊥ as the unit element. • λ A and ρ A as the morphisms obtained by the coproduct from the identityon A and the only morphism f A : ⊥ → A . ⊗ ( Y ⊗ ( Z ⊗ T )) X ⊗ (( Y ⊗ Z ) ⊗ T ))( X ⊗ ( Y ⊗ Z )) ⊗ T (( X ⊗ Y ) ⊗ Z ) ⊗ T ( X ⊗ Y ) ⊗ ( Z ⊗ T )1 ⊗ αα α ⊗ αα Figure 1: Pentagon diagram X ⊗ ( I ⊗ Y )( X ⊗ I ) ⊗ Y X ⊗ Yαρ ⊗ id id ⊗ λ Figure 2: Triangle diagram
The next step is to consider isomorphisms between A ⊗ B and B ⊗ A , thistype of categories are those denoted as braided categories. Definition 2.4.
A natural isomorphism c A,B : A ⊗ B → B ⊗ A for ( C , ⊗ , I , α, ρ, λ ) is called a braiding in case the hexagon diagrams 3 and 4 commute. In thiscase ( C , ⊗ , I , c, α, ρ, λ ) is called a braided monoidal category.In case c A,B ◦ c B,A = id B ⊗ A , the braiding is called symmetric and theresulting category is a symmetric monoidal category. It is obvious to wonder if there are a special type of functors between ( B ⊗ C ) ⊗ AA ⊗ ( B ⊗ C )( A ⊗ B ) ⊗ C ( B ⊗ A ) ⊗ C B ⊗ ( A ⊗ C ) B ⊗ ( C ⊗ A ) c αα α id ⊗ cc ⊗ id Figure 3: First hexagon diagram3 ⊗ ( A ⊗ B )( A ⊗ B ) ⊗ CA ⊗ ( B ⊗ C ) A ⊗ ( C ⊗ B ) ( C ⊗ A ) ⊗ B ( A ⊗ C ) ⊗ Bc αα α c ⊗ idid ⊗ c Figure 4: Second hexagon diagram
F A ⊗ ( F B ⊗ F C )( F A ⊗ F B ) ⊗ F CF ( A • B ) ⊗ F CF (( A • B ) • C ) F ( A • ( B • C )) F A ⊗ F ( B • C ) α id ⊗ φφ ⊗ id F α φφ Figure 5: First monoidal functor diagram monoidal categories; i.e, functors that ‘preserve’ the monoidal structure insome sense.
Definition 2.5.
A functor F : ( C , • , I C ) → ( D , ⊗ , I D ) between two monoidalcategories is called monoidal if it is equiped with: • φ : ( F A ) ⊗ ( F B ) → F ( A • B ) a natural transformation. • r : I D → F I C another natural tranformation.such that the diagrams 5, 6 and 7 commute.In case there is a braiding, the functor F is called braided if also the diagram8 commutes. Definition 2.6.
The dual concept of a monoidal functor is called a comonoidalfunctor. In other words, a comonoidal functor is a functor F : ( C , • , I C ) → ( D , ⊗ , I D ) between two monoidal cateogries equiped with: • φ : F ( A • B ) → F ( A ) ⊗ F ( B ) a natural transformation. • r : F I C → I D a natural transformation.such that the dual diagrams of 5, 6 and 7 commute.In case there is a braiding, the functor F is called braided if also the diagramdual to 8 commutes. I C ⊗ F B I D ⊗ F BF B F ( I C • B ) r ⊗ id φF λλ Figure 6: Second monoidal functor diagram
F A ⊗ F I C F A ⊗ I D F A F ( A • I C ) id ⊗ r φF ρρ Figure 7: Third monoidal functor diagram
F B ⊗ F AF A ⊗ F BF ( A • B ) F ( B • A ) c φF cφ Figure 8: Braided monoidal functor diagram5 ( A, B ) C ( B, D ) ⊗ C ( A, B )( C ( C, D ) ⊗ C ( B, C )) ⊗ C ( A, B ) C ( C, D ) ⊗ ( C ( B, C ) ⊗ C ( A, B )) C ( C, D ) ⊗ C ( A, C ) ◦◦ ⊗ id α id ⊗ ◦◦ Figure 9: First enriched category diagram C ( B, B ) ⊗ C ( A, B ) I ⊗ C ( A, B ) C ( A, B ) u ⊗ id ◦ λ Figure 10: Second enriched category diagram
Sometimes, the more natural framework is one where the morphisms
Hom ( A, B )is an object in some other category and the composition morphism is an arrowin that other category. For example, in
Vec , Hom ( V, W ) is a vector space andthe composition of two such morphisms is bilinear. The general definition ofthis situation is the following:
Definition 2.7.
Given ( M , ⊗ , I , α, ρ, λ ) a monoidal category. C is called anenriched category over M if there is: • A family of objects
Obj ( C ) • ∀ A, B ∈ Obj ( C ) , an object in M , Hom C ( A, B ) = C ( A, B ) ∈ M• ∀ A, B, C ∈ Obj ( C ) , a composition map ◦ : C ( B, C ) ⊗ C ( A, B ) → C ( A, C ) ,an arrow in M• ∀ A ∈ Obj ( C ) , a unity map u A : I → C ( A, A ) , an arrow in M And all of them make the diagrams 9, 10 and 11 commute. In case M isthe category of abelian groups, we call C a preadditive category. In case M isthe category of vector spaces over K with natural tensor product, we call C a K -linear category. In general, C is called an M -category. There exist several examples of enriched categories:
Example 2.8.
A natural first example is the
Set -categories. This are just ( A, B ) ⊗ C ( B, B ) C ( A, B ) ⊗ I C ( A, B ) id ⊗ uρ ◦ Figure 11: Third enriched category diagram categories since rewriting the definition of enriched category with M = Set the definition of a category is reached.
Example 2.9.
Another usefull example is a K -linear category: the categoryof vector spaces over K ; i.e., Vec . • The set of linear maps between two vector spaces is a vector space. • The composition of linear maps defines a bilinear map; hence it definesa linear map ◦ : Hom ( B, C ) ⊗ Hom ( A, B ) → Hom ( A, C ) . • There is a linear map u A : K → End ( A ) such that u A (1) = id ( A ) .The previous arrows make the diagrams 9, 10 and 11 commutative. Enriched categories also have enriched functors and enriched natural trans-formations; which are those that respect the category structure of M . Forinstance, a K -linear functor is one that acts on morphisms as F ( λf + µg ) = λF ( f ) + µF ( g ); meaning that F X,Y : Hom ( X, Y ) → Hom ( F X, F Y ) is a linearmap.
Definition 2.10.
An enriched functor between two M -categories, C and C is defined as: • A function F : Obj ( C ) → Obj ( C ) on the objects. • A collection of M -morphisms F x,y : Hom C ( x, y ) → Hom C ( F x, F y ) such that the diagrams 12 and 13 are commutative. Definition 2.11.
An enriched natural transformation between two M -functors, F, G : C → C is defined as a colletion of M -morphisms µ c : I → Hom C ( F c, Gc ) ;such that the diagram 14 is commutative. The product of any two categories can be defined as
Obj ( C × C ) = Obj ( C ) × Obj ( C ) and Hom (( A , A ) , ( B , B )) = Hom ( A , A ) × Hom ( B , B ).This definition is used to define, for example, tensor products as functors froma product category C × C . The problem is that it doesn’t make sence for M -categories; since the morphisms no longer are defined as sets.What would make sense is to define Hom (( A , A ) , ( B , B )) = Hom ( A , A ) ⊗ Hom ( B , B ); since the objects can be tensored: om C ( a, c ) Hom C ( b, c ) ⊗ Hom C ( a, b ) Hom C ( F b, F c ) ⊗ Hom C ( F a, F b ) Hom C ( F a, F b ) F ◦ F ⊗ F ◦ Figure 12: First enriched functor diagram
Hom C ( F A, F A ) Hom C ( A, A ) I u u F A,A
Figure 13: Second enriched functor diagram
Hom C ( F b, Gb ) ⊗ Hom C ( F a, F b ) I ⊗ Hom C ( a, b ) Hom C ( a, b ) Hom C ( a, b ) ⊗ I Hom C ( Ga, Gb ) ⊗ Hom C ( F a, Ga ) Hom C ( F a, Gb ) ∼ = ∼ = G ⊗ µ a ◦ ◦ µ b ⊗ F Figure 14: Enriched natural transformation diagram8 efinition 2.12.
Let C and C be M -categories; with M symmetric monoidal.We then define C ⊗ M C as the M -category with: • Objects as
Obj ( C ⊗ M C ) = Obj ( C ) × Obj ( C ) • Morphisms as
Hom (( A , A ) , ( B , B )) = Hom ( A , B ) ⊗ Hom ( A , B ) • Composition • : Hom ( B, C ) ⊗ Hom ( A, B ) → Hom ( A, C ) , as • = ( ◦ ⊗◦ ) ◦ ( id ⊗ c ⊗ id ) ; where ◦ i : Hom ( B i , C i ) ⊗ Hom ( A i , B i ) → Hom ( A i , C i ) is the composition map of C i . We are considering A = ( A , A ) , B =( B , B ) and C = ( C , C ) . With this last defintion we have all the tools to define monoidal and sym-metric monoidal M -categories in a natural way. Just rewriting the definitionof monoidal and symmetric monoidal categories using enriched functors andenriched natural transformations instead of regular ones; and tensor of M -categories instead of product of categories.The last concept on monoidal categories we are going to consider is closedmonoidal strutures: Definition 2.13.
Let ( C , ⊗ , I , α, λ, µ ) be a monoidal category, then it is closedin case each functor − ⊗ A : C → C has an adjoint functor [ A, − ] : C → C forevery A ∈ Obj ( C ) ; i.e., C ( B ⊗ A, C ) ∼ = C ( B, [ A, C ]) . The object [ A, C ] is calledthe inner morphisms between A and C . In this subsection we define the categories of G -graded vector spaces; which isthe category of K [ G ]-comodules; and give them several symmetric structures. Definition 2.14. A G -graded vector space is defined as a vector space V suchthat V = L g ∈ G V g ; where G could be non-abelian. Lemma 2.15. Vec G K = ( V ec G K , ⊗ , K , α, λ, µ ) is a monoidal category; where G could be non-abelian.The objects in this category are G -graded vector spaces and for any V and W , G -graded vector spaces, the arrows are defined as Hom ( V, W ) = { f : V → W | f is linear and ∀ g ∈ G, f ( V g ) ⊆ W g } The other elements in the tuple are defined as: ⊗ : V ec G K × V ec G K → V ec G K ( V, W ) V ⊗ Wρ V : V ⊗ K → Vv ⊗ µ µv V : K ⊗ V → Vµ ⊗ v µvα V,U,W : ( V ⊗ U ) ⊗ W → V ⊗ ( U ⊗ W )( v ⊗ u ) ⊗ w v ⊗ ( u ⊗ w ) Proof.
It is clear that
V ec G K is a category, α makes the pentagon diagram 1commutative, and ρ , λ and α make the triangle diagram 2 commutative; andthat the three are isomorphisms for all objects.What is left to prove is that − ⊗ − is a functor.If we define | v ⊗ w | = | v || w | , it follows that V ⊗ W = L g ∈ G ( V ⊗ W ) g where( V ⊗ W ) g = L h ∈ G V h ⊗ W h − g . In conclusion ⊗ : V ec G K × V ec G K → V ec G K iswell defined for objects.The tensor on morphisms is defined as ⊗ ( f, g )( v ⊗ w ) = ( f ⊗ g )( v ⊗ w ) =( f v ) ⊗ ( gw ). This is well defined and ⊗ ( id V , id W ) = id V ⊗ W . Also ( f ◦ g ) ⊗ ( h ◦ t ) = ( f ⊗ h ) ◦ ( g ⊗ t ). Hence ⊗ is a functor. Observation 2.16.
The category of G -graded vector spaces is just the categoryof comodules of the group algebra K [ G ] ; the coaction is defined as v → v ⊗ g for v ∈ V g . The monoidal structure of the category has been stablished, but the numberof symmetric structures that this particular categories admit is big; and so weneed to concern ourselves now with some specific symmetries. In particularwe will deal with symmetries related to the group structure of G . Definition 2.17.
A skew-symmetric bicharacter is a mapping χ : G × G → K × such that: • χ ( ab, c ) = χ ( a, c ) χ ( b, c ) and χ ( a, bc ) = χ ( a, b ) χ ( a, c ) • χ ( a, b ) χ ( b, a ) = 1 We will define as a color the pair ( G, χ ) as above. Lemma 2.18.
Let ( G, χ ) be a color, then χ ( g, g ) = ± . Definition 2.19. G + = { g ∈ G | χ ( g, g ) = 1 } are called the even G -elementsand G − = { g ∈ G | χ ( g, g ) = − } are called the odd G -elements. For a G -graded vector space V = ( L g ∈ G + V g ) ⊕ ( L g ∈ G − V g ) = V + ⊕ V − . V − is called the odd part and V + is called the even part. Lemma 2.20.
Given a color C = ( G, χ ) , the natural isomorphism defined onhomogeneous elements as c CV,W : V h ⊗ W g → W g ⊗ V h v ⊗ w χ ( h, g ) w ⊗ v gives a symmetric braiding to the monoidal cathegory Vec G K . n this last result, it is mandatory that G be abelian; since otherwise c C isnot a morphism of graded spaces.Most of the time, by abuse of notation we shall write χ ( v, w ) instead of χ ( | v | , | w | ) for homogenous elements v and w . Definition 2.21.
For a color ( G, χ ) , denote by Vec K ( G , χ ) the symmetricmonoidal category given by the monoidal category Vec G K and the braiding givenby χ .This family of categories are the color categories. Examples of the previous categories are: • Vector spaces: the color given by G = { } and χ (0 ,
0) = 1. • G -graded vector spaces: the color given by χ ( g, h ) = 1. • Super vector spaces: the color given by G = Z and χ ( a, b ) = ( − ab .The key property of this categories, that will be used through the paper,is that we can consider this category as a K -linear category: Lemma 2.22.
The tensor of G -graded vector spaces and all the natural trans-formations that define the symmetric monoidal structure of Vec K ( G , χ ) aremultilinear; and hence they define linear mappings on the tensor products.Such structures make ( V ec G K , b ⊗ , K , ˆ α, ˆ λ, ˆ ρ, c c C ) into a K -linear category. A notation that we shall use extensively is given by the following:Since the category
Vec K ( G , χ ) is symmetric, it implies that there is a leftaction of S n onto V ⊗ n by automorphisms. This action, using the definition ofthe symmetry c CA,B ( v ⊗ w ) = χ ( | v | , | w | )( w ⊗ v ) on homogeneous elements, isgiven by σ : V ⊗ n → V ⊗ n such that σ · x = σ ( x ⊗ · · · ⊗ x n ) = k ( x σ − ⊗ · · · ⊗ x σ − n ) on homogenous elements; where k ∈ K × depends on χ, x and σ . Definition 2.23.
Usually we will denote the non zero element k by χ ( x, σ ) or χ ( x, x σ − ⊗ · · · ⊗ x σ − n ) Definition 2.24.
The linear character of a group G is a group homomorphism χ : G → K × . This set of group homomorphisms form a group by pointwisemultiplication denoted by G ∗ or b G , and called the dual group. In case the group G is finite and abelian, the bicharacter can be takenfrom a much bigger selection, as the dual group G ∗ of linear characters of G isisomorphic to G . This is no longer true for non abelian groups. For example S has only two non-isomorphic linear characters; and the linear charactersand in close relation with the bicharacters. emma 2.25. For a finite group G , Bicharacters ( G ) ∼ = Hom ( Ab ( G ) , G ∗ ) ;where Ab ( G ) = G/ [ G, G ] .Proof. Let χ : G × G → K be a bicharacter. Then for all g ∈ G , χ ( g, − ) : G → K × is a linear character. We have then that χ ρ : G → G ∗ is a grouphomomorphism; where χ ρ ( g )( h ) = χ ( g, h ).Consider now a group homomorphism ρ : G → G ∗ , and g, h ∈ G . Then ρ χ : G × G → K × is a bicharacter; where ρ χ ( g, h ) = ρ ( g )( h ).In conclusion Bicharacters ( G ) ∼ = Hom ( G, G ∗ ). Since G ∗ is an abeliangroup, it follows that Bicharacters ( G ) ∼ = Hom ( G/ [ G, G ] , G ∗ ).In case K = C and the group is topological, one can consider only continu-ous characters and bicharacters χ such that im ( χ ) ⊆ S . Corollary 2.26.
In case G is finite, Bicharacters ( G ) ∼ = End ( G ∗ ) ; and incase G is abelian Bicharacter ( G ) ∼ = Hom ( G, G ∗ ) . Reconsidering our example, S ∗ ∼ = Z and End ( Z ) ∼ = Z ; concluding that S has only two bicharacters.Another example can be Z with the discrete topology and field C , where Bicharacter ( Z ) ∼ = S ; since χ ( n, m ) = χ (1 , nm and χ (1 , ∈ S .If one considers Z in the more general setting, then Bicharacter ( Z ) ∼ = K × following the same argument as before.If we restrict then to skew-symmetric bicharacters of finite abelian groups,then we have the following results: Theorem 2.27.
The only skew symmetric bicharacter of the group Z k +1 isthe trivial one: χ ( a, b ) = 1 for any a, b ∈ Z k +1 Proof. χ (0 ,
0) = χ (2 k + 1 , k + 1) = χ (1 , (2 k +1) = χ (1 , k +4 k +1 = χ (1 , χ (1 , k + k ) ) = χ (1 , χ (1 , ∈ {− , } as shown in2.18.Since χ ( a, b ) = χ (1 , ab for a, b = 0 it follows that then χ ( a, b ) = 1 sincewe know already that χ (0 , a ) = χ ( a,
0) = 1.
Theorem 2.28.
The only skew symmetric bicharactesr of the group Z k is thetrivial one and even-odd one: χ ( a, b ) = 1 if ab is even and χ ( a, b ) = − if ab is odd.Proof. χ (0 ,
0) = χ (2 k, k ) = χ (1 , (2 k ) = χ (1 , k Then the element χ (1 ,
1) can be chosen either as 1 or as −
1. If χ (1 ,
1) = 1we get the trivial bicharacter and if χ (1 ,
1) = − .4 Categorical tools The main non-trivial example of color category is that of super vector spaces; or
SVec K . For this particular category there have been developed some functorialtools that helped on the study of superalgebras, or algebras on SVec K . In thissubsection we study a generalization of this tools.Firstly, there is a concept of Grassmann superalgebra as the free associtive,commutative and unital algebra on SVec K generated by an infinite countabledimensional vector space of odd elements.The generalization of this concept is given by: Definition 2.29.
Consider V a G -graded vector space, then T ( V ) is also G -graded. T ( V ) with the natural product is an associative algebra in the colorcategory of any color ( G, χ ) .Consider then the ideal generated by I ( χ ) = h u ⊗ v − χ ( g, h ) v ⊗ u | v ∈ V g , u ∈ V h i .Then G ( V, χ ) := T ( V ) /I ( χ ) is commutative in Vec K ( G , χ op ) ; and it is G -graded since I ( χ ) is also graded.Specify V = K < { x ig | g = 1; i ∈ N } ∪ { } > and define Gr Gχ = G ( V, χ ) .This is denoted as the Grassmann color algebra of ( G, χ ) . Examples of the previous constructions are: • In case of vector spaces G ( V, χ ) = S ( V ); the symmetric algebra. • In case of super vector spaces G ( V, χ ) = S ( V ) ⊗ G ( V ); where G ( V ) isthe Grassmann super algebra generated by V .Notice that in the case of super vector spaces χ ( a, b ) = χ ( b, a ) and so χ op = χ . This is the reason why the Grassmann superalgebra is commutativein SVec K .The next step is to define a generalization of the Grassmann envelope. Toreach this concept we first need this three other functors: Definition 2.30. ( − ) : Vec ( G , χ ) → Vec defined as ( V ) is the vector spaceof homogenous elements in V with degree e ∈ G ; the neutral element.Given an arrow f ∈ Hom ( V, W ) in Vec ( G , χ ) ; f ∈ Hom ( V , W ) isdefined as f = f | V . Definition 2.31.
F orget : Vec ( G , χ ) → Vec ( G , ) defined as the forgetfulfunctor that forgets the symmetry; but no the grading. Definition 2.32. Gr Gχ : Vec ( G , ) → Vec ( G , χ ) defined as Gr Gχ ( V ) = Gr Gχ ⊗ V . Given an arrow f ∈ Hom ( V, W ) in Vec ( G , ) ; Gr Gχ ( f ) ∈ Hom ( Gr Gχ ( V ) , Gr Gχ ( V )) s defined as Gr Gχ ( f ) = id ( Gr Gχ ) ⊗ f Proposition 2.33.
The functors ( − ) , F orget and Gr Gχ are monoidal func-tors.Proof. It is widely known that ( − ) : Vec ( G , χ ) → Vec is an actual functorand is an easy exercise to check that i : V ⊗ W → ( V ⊗ W ) is a naturaltransformation between the functors ( − ) ⊗ ( − ) and ( − ⊗ − ) ; where i isdefined as the inclusion since ( V ⊗ W ) = L g ∈ G V g ⊗ W g − . Hence we only needto prove that it is monoidal with the natural transformation i and morphism φ = id ( K ) : K → K .This means that the diagrams 5, 6 and 7 must commute. This is also atrivial exercise.Trivially, F orget : Vec ( G , χ ) → Vec ( G , ) is a functor. It is monoidalsince F orget ( V ⊗ W ) = F orget ( V ) ⊗ F orget ( W ) and F orget ( K ) = K .To prove that Gr Gχ : Vec ( G , ) → Vec ( G , χ ) is monoidal we need to findan appropiate i : Gr Gχ ( V ) ⊗ Gr Gχ ( W ) → Gr Gχ ( V ⊗ W ) and φ : K → Gr Gχ ( K )that make the diagrams 5, 6 and 7 commute.This mappings are given by i ( g ⊗ v, h ⊗ w ) = ( gh ) ⊗ ( v ⊗ w ) and φ ( a ) =1 ⊗ a . Definition 2.34.
Define then
Env Gχ = ( − ) ◦ Gr Gχ ◦ F orget : Vec ( G , χ ) K → Vec K . Proposition 2.35.
The functor
Env Gχ is symmetric monoidal with associatednatural transformations i : Env Gχ ( V ) ⊗ Env Gχ ( W ) → Env Gχ ( V ⊗ W ) given by i ( g ⊗ v, h ⊗ w ) = gh ⊗ ( v ⊗ w ) and φ : K → Env Gχ ( K ) given by φ ( a ) = 1 ⊗ a .Proof. By definition,
Env Gχ is a monoidal functor since it is composition ofmonoidal functors. It rests to show that it is in fact symmetric.Let t = ( g ⊗ v ) ⊗ ( h ⊗ w ) ∈ Env Gχ ( V ) ⊗ Env Gχ ( W ); then t = i ( t ) = gh ⊗ ( v ⊗ w ) ∈ Env Gχ ( V ⊗ W ). Hence Env Gχ ( c )( t ) = gh ⊗ ( χ ( | v | , | w | )( w ⊗ v )) = gh ⊗ ( χ ( g − , h − )( w ⊗ v )) = χ ( g, h ) gh ⊗ ( w ⊗ v ).On the other hand, t = c ( t ) = ( h ⊗ w ) ⊗ ( g ⊗ v ) and i ( t ) = hg ⊗ ( w ⊗ v ) = χ ( g, h ) gh ⊗ ( w ⊗ v ).In conclusion the diagram 8 commutes.The functor Env χG reduces to the grassmann envelope in the case of SVec and is symmetric monoidal, hence algebras in
Vec ( G , χ ) are transported toalgebras in Vec .The trick of this functor is translating the grading and the symmetry fromthe categorical structure to the algebraic structure, so the categorial structuretransforms to
Vec without forgeting the group G and its bicharacter χ thatcan be recovered from the algebraic structure. Algebraic Definitions
In this section, we define what it is meant as an algebra in some category
Vec K ( G , χ ) and some definitions related to them. An algebraic structure in some category is defined as expected:
Definition 3.1.
In a category C with a tensor product ⊗ , an algebra is definedas an object A and a family of morphisms { f i : A ⊗ n i → A } . The most interesting categories are those which are symmetric monoidalwhich let us define identities with the use of the action of the symmetric groups.In fact, our previously defined monoidal functor
Env Gχ lets us give a definitionof some algebra belonging to a variety: Definition 3.2.
Given a class of algebras D in Vec , we will define the class ofalgebras D G,χ as the algebras ( B, { f i : B ⊗ χ n i → B } ) in the category Vec K ( G , χ ) such that ( Env Gχ ( B ) , { Env Gχ ( f i ) ◦ i n i : Env Gχ ( B ) ⊗ n i → Env Gχ ( B ) } is an algebrain the class D .In this case i n i : Env Gχ ( B ) ⊗ n i → Env Gχ ( B ⊗ χ n i ) is the natural transforma-tion resulting of applying i : Env Gχ ( A ) ⊗ Env Gχ ( B ) → Env Gχ ( A ⊗ χ B ) severaltimes; for A = B ⊗ q where q ∈ N . Lemma 3.3.
Let D , C be two classes of algebras in the category of vector spacesand D G,χ , C G,χ the classes defined previously in the category
Vec K ( G , χ ) . Let R G,χ : C G,χ → D
G,χ be a linear functor defined for every color. If there is alinear natural transformation N G,χ : R { } , ◦ Env χG → Env χG ◦ R G,χ defined forevery color, then any polynomial equation fulfilled by R { } , ( Env Gχ ( M )) maygive at least one equation that is fulfilled by R G,χ ( M ) for every algebra M in C G,χ ; in case ∀ y ∈ R G,χ ( M ) , ∃ g ∈ Gr Gχ such that g ⊗ y ∈ im ( N G,χ ) .Proof. Let θ ( x , . . . , x n ) = 0 be some polynomial equation fullfiled by R ( Env χG ( M )).Then N M : R ◦ Env χG ( M ) → Env χG ◦ R G,χ ( M ) is a morphism of algebras givenby the natural transformation.It follows then that N M ◦ θ ( x , . . . , x n ) = 0. Since N M is morphism and θ is polynomial, there exists ¯ θ such that ¯ θ ( N M ( x ) , . . . , N M ( x n )) = 0.We can choose elements x i such that N M ( x i ) = g i ⊗ y i for any y i ∈ R G,χ ( M ),by the last hypothesis.Expanding the equation,¯ θ ( N M ( x ) , . . . , N M ( x n )) =¯ θ ( g ⊗ y , . . . , g n ⊗ y n ) = X i h i ⊗ p i = 0 here p i are ponomials in R G,χ ( M ) valued in ( y , . . . , y n ) and the set { h i } islinearly independent. Then p i = 0 for every i by linear independency. Hencethe polynomials p i = 0 are polynomial equations for R G,χ ( M ). Observation 3.4.
Iif the polynomial equations are n -linear; i.e. θ ( x , . . . , λx i , . . . , x n ) = λθ ( x , . . . , x i , . . . , x n ) , the result of the process gives one polynomial equation. In almost all examplesin this paper, this is the case. An example of this construction:
Proposition 3.5.
Consider U the enveloping algebra of a color Lie algebra;and U the enveloping algebra of a Lie algebra, then there is a natural trans-formation n : U ◦ Env Gχ → Env Gχ ◦ U such that ∀ y ∈ U( L ) , ∃ g ∈ Gr χ with g ⊗ y ∈ n L ( U ( Env Gχ ( L ))) for any L Lie color algebra.Proof.
Let L be any Lie color algebra, define n L : U ( Env Gχ ( L )) → Env Gχ ( U ( L ))as n L ( g ⊗ a ) = g ⊗ a and extend as an algebra morphism: n L (( g ⊗ a ) . . . ( g n ⊗ a n )) = ( g . . . g n ) ⊗ ( a . . . a n ).With this definition, n : U ◦ Env Gχ → Env Gχ ◦ U is a natural transformationand ∀ y ∈ U( L ) , ∃ g ∈ Gr χ with g ⊗ y ∈ n ( U ( Env Gχ ( L ))) for any L Lie coloralgebra; as we wanted.Since this functor,
Env Gχ , is not comonoidal, we cannot recover directlythe coproduct for U ( L ); where L is a Lie color algebra; as we can recover theproduct from n and U ( Env Gχ ( L )).The coproduct of U ( L ); for a color algebra L , must be a map ∆ χ : U ( L ) → U ( L ) ⊗ U ( L ) such that the diagram 15 commutes. Since Env Gχ is an injectivelinear functor, if such a map exists, it is unique. For it to exists, it musthappend that i ◦ ( n L ⊗ n L ) ◦ ∆(( g ⊗ a ) . . . ( g n ⊗ a n )) = ( g . . . g n ) ⊗ A , wherethen ∆ χ ( a . . . a n ) = A ∈ U ( L ) ⊗ U ( L ).It can be easily shown that everything holds and ∆ χ exists; since i ◦ ( n L ⊗ n L ) ◦ ∆ is an algebra morphism and it holds for primitives; which generate thewhole algebra. In this subsection we deal with the definition of Malcev algebra in a colorcategory. To get such an algebra, we only need to apply our definition to thevariety of Malcev algebras; and the result is:
Definition 3.6.
For a color ( G, χ ) , define color-alternative algebras as theclass of color-algebras A = L g ∈ G A g such that: ∀ a, b, c ∈ A homogenous ele-ments; ( a, b, c ) = − χ ( a, b )( b, a, c ) and ( a, b, c ) = − χ ( b, c )( a, c, b ) where ( a, b, c ) is the associator. Env Gχ ( L )) U( Env Gχ ( L )) ⊗ U( Env Gχ ( L )) Env Gχ (U( L )) Env Gχ (U( L )) ⊗ Env Gχ (U( L )) Env Gχ (U( L ) ⊗ U( L ))∆ n L n L ⊗ n L iEnv Gχ (∆ χ )Figure 15: Coproduct definition diagramm Notice that it may happen that ( a, a, b ) = 0 in case χ ( a, a ) = − . Let (
A, a · b ) be a color-alternative algebra and consider the new product[ a, b ] χ = ab − χ ( a, b ) ba then [ a, b ] χ = − χ ( a, b )[ b, a ] χ and[[ x, z ] χ , [ y, w ] χ ] χ χ ( y, z ) =[[[ x, y ] χ , z ] χ , w ] χ + χ ( x, y ) χ ( x, z ) χ ( x, w )[[[ y, z ] χ , w ] χ , x ] χ + χ ( y, z ) χ ( x, z ) χ ( y, w ) χ ( x, w )[[[ z, w ] χ , x ] χ , y ] χ + χ ( z, w ) χ ( y, w ) χ ( x, w )[[[ w, x ] χ , y ] χ , z ] χ which reduce to the linearization of the Malcev identity or Malcev super-identity chosing the proper color ( G, χ ) = ( Z , χ s ), hence the name linearizedMalcev color-identity.It is important to work with linearized identities because it may happendthat χ ( x, x ) = − J ( x, y, z ) , y ];which permutes y with itself. Lemma 3.7.
The algebras that fulfill the linearized Malcev color-identity andthe antisymmetry color-identity, are the Malcev color-algebras.Proof.
Direct application of the definition.From any algebra we can obtain a color-alternative algebra:
Definition 3.8.
Let A = L g ∈ G A g be a color-algebra and define N color ( A ) = L g ∈ G { x ∈ A g | ( x, y, z ) = − χ ( x, y )( y, x, z ) and ( y, z, x ) = − χ ( z, x )( y, x, z ) , ∀ y, z ∈ S h ∈ G A h } . This subspace is called the color-alternative nucleous of A . Lemma 3.9. N color ( A ) − is a Malcev color-algebra. roof. It is known that N alt ( Env Gχ ( A )) − is a Malcev algebra.What we need to prove is that the linearized Malcev color identity is fulfilledby this construction of that Env Gχ ( N color ( A ) − ) is a Malcev algebra.Define the following map: f : N alt ( Env Gχ ( A )) → Env Gχ ( A ) as the inclusion;and consider g ⊗ a ∈ im ( f ).Then g ⊗ a ∈ N alt ( Env Gχ ( A )) and ∀ h ⊗ b, q ⊗ c ; ( g ⊗ a, h ⊗ b, q ⊗ c ) = − ( h ⊗ b, g ⊗ a, q ⊗ c ). In conclusion, ghq ⊗ ( a, b, c ) = − hgq ⊗ ( b, a, c ) ghq ⊗ ( a, b, c ) = ghq ⊗ ( − χ ( a, b )( b, a, c ))It follows then that im ( f ) = Env Gχ ( N color ( A )).In conclusion, f is an isomorphism from N alt ( Env Gχ ( A )) to Env Gχ ( N color ( A ));hence Env Gχ ( N color ( A )) is a Malcev algebra. In this section K { V } indicates the free algebra generated by V .From a color-Malcev algebra M , construct the quotient algebra U ( M ) = K { M } /I for I the ideal generated by h x ⊗ y − χ ( x, y ) y ⊗ x − [ x, y ] χ , ( x, a, b ) + χ ( x, a )( a, x, b ) , ( a, x, b ) + χ ( x, b )( a, b, x ) | x, y ∈ M ; a, b ∈ K { V } homogeneouselements i . Let i : M → U ( M ) be i ( x ) = x + I . We shall prove next that U ( M )is the functor adjoint to N color ( − ) − applied to M . Theorem 3.10.
Let ( M, [ x, y ]) be a Malcev color-algebra. Let also A be analgebra and f : M → N color ( A ) − a Malcev color-algebra morphism. Thenexists a unique U ( f ) : U ( M ) → A algebra morphism such that U ( f ) ◦ i = f .Proof. Define F : K { M } → A as an algebra morphism and F | M = f ; whichcan be done in a unique way. In conclusion we can define e F : U ( M ) → A as e F ( x + I ) = F ( x ) for x ∈ K { M } , because I ⊆ ker ( F ) by the definition of im ( f ) ⊆ N color ( A ) − as a Malcev color-algebra and f as a Malcev color-algebramorphism. e F ◦ i = f by definition and concluding: U ( f ) = e F .Uniqueness follows from the fact that M generates U ( M ) as an algebra,and hence U ( f ) is completely defined by its image on M . Corollary 3.11.
Hom alg ( U ( M ) , A ) ∼ = Hom malcev ( M, N color ( A ) − )This construction as the adjoint functor through a quotion of algebrasdoesn’t let us check wether i ( M ) ∩ I = { } or not. The last part of thispaper solves this problem. In the case of Malcev algebras, this problem hasbeen solved in [3] and U ( M ) has a bialgebra structure called Hopf-Moufangalgebra.This class of bialgebras have been defined over the category of vector spaces.First we will define H -bialgebra and afterwards Hopf-Moufang algebras as asubclass of this algebras. ⊗ H [ H, H ] ⊗ H HH ⊗ [ H, H ] L ⊗ id evid ⊗ R ev ◦ cµ Figure 16: Adjoint maps
Theorem 3.12.
In a braided closed monoidal K -linear category C , let ( H, ∆ , ǫ, µ ) be a coassociative, counital bialgebra; then C ( H, [ H, H ]) is an associative unitalalgebra given the product a ∗ b = •◦ ( a ⊗ b ) ◦ ∆ ; where • : [ H, H ] ⊗ [ H, H ] → [ H, H ] is the composition map.Proof. Let a, b, c : H → [ H, H ], then we need to show that ( a ∗ b ) ∗ c = a ∗ ( b ∗ c )and to find an unit. ( a ∗ b ) ∗ c = ( • ◦ ( a ⊗ b ) ◦ ∆) ∗ c = • ◦ (( • ◦ ( a ⊗ b ) ◦ ∆) ⊗ c ) ◦ ∆ = • ◦ ( • ⊗ id ) ◦ (( a ⊗ b ) ⊗ c ) ◦ (∆ ⊗ id ) ◦ ∆ a ∗ ( b ∗ c ) = a ∗ ( • ◦ ( b ⊗ c ) ◦ ∆) = • ◦ ( a ⊗ ( • ◦ ( b ⊗ c ) ◦ ∆)) ◦ ∆ = • ◦ ( id ⊗ • ) ◦ ( a ⊗ ( b ⊗ c )) ◦ ( id ⊗ ∆) ◦ ∆The result follows from the associativity of composition, tensor productsand the coassociativity of the coproduct. T = λ ◦ ( ǫ ⊗ id ) : H ⊗ H → H and by the adjoint functors − ⊗ H and[ H, − ] there is a unique map u : H → [ H, H ] such that T = ev ◦ ( u ⊗ id ). Bythe definition of counit T ◦ ∆ = id : H → H .Similarly to the associativity computations, one realizes that u ∗ a = a ∗ u = a by use of the definition of the composition • : [ H, H ] ⊗ [ H, H ] → [ H, H ] andthe uniqueness of maps by the universal properties of adjoint functors.
Definition 3.13.
In a symmetric closed monoidal K -linear category, an alge-bra ( H, µ ) has two maps, called adjoint maps L : H → [ H, H ] and R : H → [ H, H ] that make the diagram 16 commutes. ⊗ HH ⊗ H ⊗ H ⊗ H ⊗ H ⊗ H ⊗ HH ⊗ H ⊗ Hid ⊗ c ⊗ idid ⊗ c ⊗ id ∆ ⊗ id ⊗ ∆ ⊗ id ⊗ id ⊗ ⊗ µ id ⊗ µµµ ⊗ id ⊗ µ ⊗ idµ Figure 17: Moufang relations’ diagram of a bialgebra L is denoted left adjoint and R , right adjoint. Their existence is explainedby the universal property of the adjoint functors. Since the category is sym-metric, R also exists by the use of the isomorphism between H ⊗ [ H, H ] and [ H, H ] ⊗ H ; as the diagram 16 shows. Definition 3.14.
In a braided closed monoidal K -linear category C a counitalcoassociative bialgebra ( H, ∆ , · , ǫ ) is called an H -bialgebra if its adjoint maps L and R have bilateral inverses in the algebra C ( H, [ H, H ]) defined in 3.12. The simplest example of an H -bialgebra is a Hopf algebra, in fact Hopfalgebras can be defined as associative unital H -bialgebras where there exists S : H → H such that L − = L ◦ S and R − = R ◦ S . Definition 3.15.
In a symmetric closed monoidal K -linear category C , a Hopf-Moufang algebra is a unital H - bialgebra ( H, µ, ∆ , ǫ, u ) in C such that the di-agram 17 commutes and there is S : H → H such that L − = L ◦ S and R − = R ◦ S .In the case C = Vec ( G , χ ) , the diagram 17 translates into the equation X χ ( | u (2) | , | v | )(( u (1) v ) u (2) ) w = X χ ( | u (2) | , | v | ) u (1) ( v ( u (2) w )) The relation between connected Hopf algebras and Lie algebras as equivalentcategories is well known. This subsection particularizes this relation to thosealgebras with triality.
Definition 3.16.
In a symmetric monoidal K -linear category C , a Lie algebrawith triality is a Lie algebra ( L, µ ) in C endowed with two automorphisms σ : L → L and σ : L → L such that h σ , σ i ∼ = S and that ( id + σ + σ ) ◦ ( σ − id ) = 0 . efinition 3.17. In a symmetric monoidal K -linear category C , a Hopf algebrawith triality is a Hopf algebra ( H, µ, ∆ , ǫ, u ) in C endowed with two automor-phisms σ : H → H and σ : H → H such that h σ , σ i ∼ = S and that X P ( x (1) ) σ ( P ( x (2) )) σ ( P ( x (3) )) = ǫ ( x )1 where P ( x ) = P σ ( x (1) ) S ( x (2) ) . The first step to construct the enveloping algebra of a Malcev color algebrawill be to construct a Lie color algebra with triality from the Malcev algebra.
Lemma 3.18.
Let A be some color-algebra. Consider then L a : A → A as L a ( x ) = ax and R b : A → A as R b ( x ) = χ ( b, x ) xb .Leat a, b ∈ N color ( A ) , then they fulfill the following equations:1. L ab = L a L b + [ R a , L b ] χ and L ba = L b L a + [ L b , R a ] χ [ R a , L b ] χ = [ L a , R b ] χ [ L a , L b ] χ = L [ a,b ] χ − R a , L b ] χ and [ R a , R b ] χ = − R [ a,b ] χ − R a , L b ] χ Proof.
The proof is just a colored version of the results on [3] and [2].In light of the previous result we can define a new Lie color-algebra:
Definition 3.19.
Let M be a Malcev color-algebra and consider the followingcolor vector space V = ⊕ g ∈ G hL a , R a | a ∈ M g i . Consider the L ( M ) as the Liecolor-algebra generated by this vector space and with the following restrictionsfor a, b ∈ M and k ∈ K : • L a + b = L a + L b and L ka = k L a . • R a + b = R a + R b and R ka = k R a . • [ R a , L b ] χ = [ L a , R b ] χ • [ L a , L b ] χ = L [ a,b ] χ − R a , L b ] χ • [ R a , R b ] χ = −R [ a,b ] χ − R a , L b ] χ The main feature of this construction is that this Lie algebra has a triality,as we shall prove, and contains the original Malcev color-algebra.In the article [2], where the universal enveloping algebra of a Malcev su-peralgebra is built, the constructions and results follow those of the Malcevalgebra case, done in [3].It is clear that this constructions are functorial and that they do not changewhen considering categories more complex than that of vector spaces.
Theorem 3.20.
Let M be a Malcev color-algebra, then L ( M ) fulfills the fol-lowing equations: [ T a , T b ] χ = 13 ad [ a,b ] χ + 23 D a,b • [ ad a , T b ] χ = T [ a,b ] χ • [ ad a , ad b ] χ = − ad [ a,b ] χ + 2 D a,b • [ D a,b , ad c ] χ = ad D a,b ( c ) • [ D a,b , T c ] χ = − T D a,b ( c ) • [ D a,b , D c,d ] χ = D D a,b ( c ) ,d + χ ( a, c ) χ ( b, c ) D c,D a,b ( d ) and hence has a Z graduation L ( M ) = L ( M ) − ⊕ L ( M ) + given by L ( M ) − = h T a | a ∈ M i and L ( M ) + = h ad a , D a,b | a, b ∈ M i ; where T a = L a + R a ad a = L a − R a and D a,b = ad [ a,b ] − L a , R b ] χ .Proof. Define F : L ( Env Gχ ( M )) → Env Gχ ( L ( M )) as F ( T g ⊗ a ) = g ⊗ T a F ( ad g ⊗ a ) = g ⊗ ad a F ( D g ⊗ a,h ⊗ b ) = gh ⊗ D a,b and extend by linearity and multiplicativity.The map F is a well defined Lie morphism:[ F ( L g ⊗ a ) , F ( R h ⊗ b )] χ = [ g ⊗ L a , h ⊗ R b ] χ = gh ⊗ [ L a , R b ] χ == gh ⊗ [ R a , L b ] χ = F ([ R g ⊗ a , L h ⊗ b ] χ )[ F ( L g ⊗ a ) , F ( L h ⊗ b )] χ = [ g ⊗ L a , h ⊗ L b ] χ = gh ⊗ ( L [ a,b ] χ − R a , L b ] χ ) = F ( L gh ⊗ [ a,b ] χ − R g ⊗ a , L h ⊗ b ] χ ) = F ([ L g ⊗ a , L h ⊗ b ] χ )[ F ( R g ⊗ a ) , F ( R h ⊗ b )] χ = [ g ⊗ R a , h ⊗ R b ] χ = gh ⊗ ( −R [ a,b ] χ − R a , L b ] χ ) = F ( −R gh ⊗ [ a,b ] χ − R g ⊗ a , L h ⊗ b ] χ ) = F ([ R g ⊗ a , R h ⊗ b ] χ )Applying the result 3.3 with R = L and N = F we get all the equationsin the theorem since they appear for L ( M ) in the category of vector spaces in[3]. Theorem 3.21.
Let M be a Malcev color-algebra and construct L ( M ) as ex-plained in 3.19. Then there are automorphisms σ , σ : L ( M ) → L ( M ) definedon a set of generators as σ ( L a ) = −R a ( R a ) = −L a σ ( L a ) = R a σ ( R a ) = −L a − R a that make L ( M ) a Lie color-algebra with triality.Proof. Trivially σ , σ : L ( M ) → L ( M ) exist and define automorphisms, sincethey satisfy the equations on the definition of L ( M ).Also, the group generated by these automorphisms is S . So it sufficiesto check if ( id + σ + σ )( id − σ ) = 0. A simple calculation shows that( id + σ + σ )( x ) = 0 for x = L a and x = R a .Since ( id − σ )( L a ) = L a + R a = ( id − σ )( R a ) it follows that ( id + σ + σ )( id − σ ) = 0 for L a and R a . Hence ( id + σ + σ )( id − σ ) = 0 for T a and ad a . It rests to check if it vanishes on D a,b or equvalently on [ L a , R b ] χ .( id − σ )([ L a , R b ] χ ) = [ L a , R b ] χ − [ R a , L b ] χ = 0. The next step is to show how the enveloping algebra of a Lie algebra withtriality is a Hopf algebra with triality. The key is our proof resides in thegrassmann envelope and the known result in the case of vector spaces.
Proposition 3.22.
The enveloping algebra of a Lie color algebra with trialityis a Hopf color algebra with triality.Proof.
Let L be a Lie color algebra with triality σ , σ . Then Env Gχ ( L ) isa Lie algebra and ρ = Env Gχ ( σ ) , ρ = Env Gχ ( σ ) are a triality for this Liealgebra: ( id + ρ + ρ )( id − ρ ) = Env Gχ (( id + σ + σ )( id − σ )) = 0It follows that U( Env Gχ ( L )) is a Hopf algebra with triality as it is shownin [7][Theorem 4.4]. The natural transformation n : U ◦ Env Gχ → Env Gχ ◦ Ustablishes that U( L ) fulfills the identities of a Hopf color algebra with triality. We remeber that for a Malcev color algebra M , the Lie color algebra L ( M )has a Z graduation and L ( M ) − has a very special property: Proposition 3.23.
The map φ : L ( M ) − → M where T a a is a linearisomorphism. Env Gχ ( H ) P Env Gχ ( H ) ⊗ P Env Gχ ( H ) P Env Gχ ( H ⊗ H ) b ∆ iEnv Gχ (∆ χ )Figure 18: Coproduct for the Grassmann envelope of H This last step consists in giving a functorial construction of a Hopf-Moufangcolor algebra from a Hopf color algebra with triality and to show how thisfunctor applied to U( L ( M )) gives as a result the universal enveloping algebraof M ; the Malcev color algebra. Definition 3.24.
Given a Hopf color algebra H with triality σ , σ , we define MH ( H ) = { P ( x ) | x ∈ H } ; where P ( x ) = P σ ( x (1) ) S ( x (2) ) . Proposition 3.25. MH ( H ) with product u ∗ v = P χ ( | u (2) | , | v | ) σ ( S ( u (1) )) vσ ( S ( u (2) )) on homogenous elements; together with the unit, counit and coproduct inheretedby H is a bialgebra.Proof. Consider the associative algebra
Env Gχ ( H ). Firstly, ρ = Env Gχ ( σ )and ρ = Env Gχ ( σ ) are automorphisms by the properties of functors.On the algebra Env Gχ ( H ) the coproduct is applied as Env Gχ (∆)( g ⊗ x ) = g ⊗ ∆( x ); also the counit is Env Gχ ( ǫ )( g ⊗ x ) = g ⊗ ǫ ( x ).Define P Env Gχ ( H ) as the subalgebra of Env Gχ ( H ) generated by Env Gχ ( P rimH )and the unit element. First, we see that
Env Gχ (∆)( P Env Gχ ( H )) ⊆ P Env Gχ ( H ) ⊗ P Env Gχ ( H ); and that i ( P Env Gχ ( H ) ⊗ P Env Gχ ( H )) ⊆ P Env Gχ ( H ⊗ H ). Evenmore, ρ and ρ can be considered automorphisms of the algebra P Env Gχ ( H ).To give P Env Gχ ( H ) a bialgebra structure, we need a coproduct and counit.The natural structure would come considering the diagramm 18.The problem in this setting is the fact that there is not a unique b ∆ thatmakes the diagramm 18 commutative; there are several such maps. They ariseby the fact that this algebra is not a UFD and could be g ⊗ xy = ( g ⊗ x )( g ⊗ y ) = ( h ⊗ x )( h ⊗ y ); where h = g and h = g . Nevertheless, we considerone of those coproducts.As counit, we consider ǫ ( g ⊗ x ) = g ⊗ ǫ ( x ). Note that if x ∈ K , then | g | = 0;and in case | g | 6 = 0, then ǫ ( x ) = 0.By construction, P Env Gχ ( H ) is then a bialgebra; with two special automor-phisms. And the following equations hold: ( g ⊗ x ) (1) Env Gχ ( S )(( g ⊗ x ) (2) ) = X g (1) g (2) ⊗ x (1) S ( x (2) ) = g ⊗ X χ ( x, x (1) x (2) ) x (1) S ( x (2) ) = g ⊗ ǫ ( x )Hence P Env Gχ ( H ) is a Hopf algebra, with two special automorphisms. X P (( g ⊗ x ) (1) ) ρ P (( g ⊗ x ) (2) ) ρ P (( g ⊗ x ) (3) ) = X g (1) g (2) g (3) ⊗ P ( x (1) ) ρ P ( x (2) ) ρ P ( x (3) ) = g ⊗ X χ ( x, x (1) x (2) x (3) ) P ( x (1) ) ρ P ( x (2) ) ρ P ( x (3) ) = g ⊗ ǫ ( x )In conclusion, P Env Gχ ( H ) is a Hopf algebra with triality.In this algebra MH ( P Env Gχ ( H )) can be defined following [7][Theorem 3.3]and it is a Hopf-Moufang algebra with product:( g ⊗ u ) ∗ ( h ⊗ v ) = X σ ( S (( g ⊗ u ) (1) ))( h ⊗ v ) σ ( S (( g ⊗ u ) (2) ))But then, the natural transformation n : MH ( P Env Gχ ( H )) → Env Gχ ( MH ( H ))results in:(( g . . . g n ) ⊗ ( u . . . u n )) ∗ (( h . . . h m ) ⊗ ( v . . . v m ) = ( gh ) ⊗ ( u ∗ v )= X ( g (1) hg (2) ) ⊗ σ ( S ( u (1) )) vσ ( S ( u (2) ))= X ( g (1) g (2) h ) ⊗ χ ( | u (2) | , | v | ) σ ( S ( u (1) )) vσ ( S ( u (2) ))( gh ) ⊗ ( u ∗ v − X χ ( | u (2) | , | v | ) χ ( u, u (1) u (2) ) σ ( S ( u (1) )) vσ ( S ( u (2) ))) = 0The conclusion that follows, rewriting the notation of the cocommutativecoproduct for the color category, is that defining u ∗ v as in the hypothesis MH ( H ) becomes a Moufang-Hopf color algebra; with the unit, counit andcoproduct inherited from H . Corollary 3.26. MH (U( L ( M ))) is a Moufang-Hopf algebra. This algebra is our candidate for the universal enveloping algebra of theMalcev color algebra M . The following results all lead to proof that fact.First, we find a basis of MH (U( L ( M ))) as a vector space. emma 3.27. Let L be a Lie color algebra and λ : S → Aut ( L ) an action of S as automorphisms of L ; with λ ((12)) = σ .Then MH (U( L )) = { P ( x ) | x ∈ U( L ) } = span h a n • ( a n − • · · · • ( a • a )) | a i ∈ E ( − σ ) , n ∈ N i ; where P ( x ) = P σ ( x (1) ) S ( x (2) ) and a • b = ab + χ ( | a | , | b | ) ba on homogeneous elements.Proof. We will procede by induction on the degree filtration on U ( L ). Thebase case is a ∈ L . P ( a ) = σ ( a ) − a ∈ E ( − σ )Assume know that it is true up to some natural number k ∈ N ; and consider a . . . a k +1 ∈ U ( L ). We may assume that a i ∈ E ( − σ ) ∪ E (1; σ ); since σ = Id . By induction, we only need to consider the cases where a i ∈ E ( − , σ ) for i ≤ k . Hence we are reduced to two cases: σ ( a k +1 ) = a k +1 or σ ( a k +1 ) = − a k +1 .In the first case P ( xa k +1 ) = X χ ( | x (2) | , | a k +1 | ) σ ( x (1) ) σ ( a k +1 ) S ( x (2) ) − X χ ( | x (2) | , | a k +1 | ) σ ( x (1) ) a k +1 S ( x (2) ) = 0and we are done.In the latter case P ( a x ) = X σ ( a ) σ ( x (1) ) S ( x (2) ) − X χ ( | a | , | x | ) σ ( x (1) ) S ( x (2) ) a = − a • P ( x )and we are done.The previous bases can be constructed using the product ∗ defined on MH (U( L ( M ))); instead of that of U( L ( M )); as the following lemma shows. Lemma 3.28.
In the previous settings for the Lie color algebra L ( M ) , T a ∗ u + χ ( | a | , | u | ) u ∗ T a = T a u + χ ( | a | , | u | ) uT a for any u ∈ MH (U( L ( M ))) Proof.
By [7][Lemma 5.2], ( g ⊗ T a ) ∗ ( h ⊗ u ) + ( h ⊗ u ) ∗ ( g ⊗ T a ) = ( g ⊗ T a )( h ⊗ u ) + ( h ⊗ u )( g ⊗ T a ); in Env Gχ (U( L ( M ))) and Env Gχ ( MH (U( L ( M ))))( g ⊗ T a ) ∗ ( h ⊗ u ) + ( h ⊗ u ) ∗ ( g ⊗ T a ) = ( gh ) ⊗ ( T a ∗ u ) + ( hg ) ⊗ ( u ∗ T a ) =( gh ) ⊗ ( T a ∗ u + χ ( | a | , | u | ) u ∗ T a )( g ⊗ T a )( h ⊗ u ) + ( h ⊗ u )( g ⊗ T a ) = ( gh ) ⊗ ( T a u ) + ( hg ) ⊗ ( u ∗ T a ) =( gh ) ⊗ ( T a u + χ ( | a | , | u | ) uT a )In conclusion, it follows that T a ∗ u + χ ( | a | , | u | ) u ∗ T a = T a u + χ ( | a | , | u | ) uT a ,applying the natural transformation n : U ◦ Env Gχ → Env Gχ ◦ U. Main result
All the previous results can be added to prove the equivalence between Malcevcolor algebras and connected color Hopf-Moufang algebras.
Theorem 4.1. MH (U( L ( M ))) ∼ = U( M ) and P rim ( U ( M )) ∼ = M .Proof. P ( T a ) = X σ ( T a ) (1) S ( T a ) (2) = σ ( T a ) − T a = − T a Hence T a ∈ MH (U( L ( M ))) and is a primitive element of MH (U( L ( M )));so they are part of the alternative nucleous.In Env Gχ ( MH (U( L ( M )))), from [7][Theorem 5.3] ( g ⊗ T a ) ∗ ( h ⊗ T b ) − ( h ⊗ T b ) ∗ ( g ⊗ T a ) = − [ g ⊗ T a , h ⊗ T b ]; which transforms itself to ( gh ) ⊗ ( T a ∗ T b − χ ( | a | , | b | ) T b ∗ T a ) = ( gh ) ⊗ ( − [ T a , T b ] χ ) when applied the natural transformation n : MH ◦ U ◦ Env Gχ → Env Gχ ◦ MH ◦ U.In conclusion T a ∗ T b − χ ( | a | , | b | ) T b ∗ T a = − [ T a , T b ] χ and hence the map a
7→ − T a , by the universal property of U, extends to an algebra morphism φ : U( M ) → MH (U( L ( M )))By the lemma 3.27, MH (U( L ( M ))) is spanned by { T a n • ( . . . ( T a • T a )) | a i ∈ M ; n ∈ N } from wich follows that φ is surjective using also lemma 3.28.Also, M ∩ ker ( φ ◦ i ) = { } and hence we have the PBW theorem forU( M ); where i : M → U( M ) is the map i ( m ) = m ; it follows then that P rim ( U ( M )) ∼ = M . Given { a i } i , an ordered basis of homogenous elements for M , the basis for U( M ) can be thought as { a i n • ( · · · • ( a i • a i )) | i j ≤ i j +1 if a i j is an even element and i j < i j +1 if a i j is an odd element } .When φ is applied to the basis of U( M ), the image of the elements resultin the set { ( − n T a in • ( · · · • ( T a i • T a i )) } , using again the lemma 3.28; whichis linearly independent in U( L ( M )). In conclusion, φ is injective.There it follows that MH (U( L ( M ))) ∼ = U( M ) through φ . References [1] Y. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, and M. V. Zaicev.
Infinite-dimensional Lie superalgebras . de Gruyter Expositions in Math-ematics, 1992.[2] E. Barreiro. A universal enveloping algebra of Malcev superalgebras.
Por-tugal. Math. , 68(3), 2011.[3] J.M. Pérez Izquierdo and I.Shestakov. An envelope for Malcev algebras.
J.Algebra , 272(1), February 2004.[4] V. K. Kharchenko and I. P. Shestakov. Generalizations of Lie algebras.
Advances in Applied Clifford Algebras , 22(3), 2012.
5] John W. Milnor and John C. Moore. On the structure of Hopf algebras.
The annals of mathematics , 81(2), March 1965.[6] J.M. Pérez-Izquierdo. Algebras, hyperalgebras, nonassociative bialgebrasand loops.
Advances in Mathematics , 208(2):834 – 876, 2007.[7] J.M. Pérez-Izquierdo and S. Madariaga G. Benkart. Hopf algebras withtriality.
Trans. Amer. Math. Soc. , 365:1001 – 1023, 2013.[8] V. Rittenberg and D. Wyler. Generalized lie superalgebras.
NuclearPhysics , B139:189–202, 1979.[9] M. Scheunert. Generalized Lie algebras.
J. Math. Phys. , 20:712–720, 1979., 20:712–720, 1979.