Irreducible Lie-Yamaguti algebras of generic type. (English) Zbl 1225.17001
A Lie-Yamaguti algebra \(V\) is a vector space provided with a bilinear product \(\cdot: V\times V\to V\) and a trilinear one \([\;,\;,\;]: V\times V \times V\to V\) such that:
\[ x\cdot x=0,\quad [x,x,y]=0,\quad \sum_{(x,y,z)}([x,y,z]+(x\cdot y)\cdot z)=0, \]
\[ \sum_{(x,y,z)}[x\cdot y,z,t]=0,\quad [x,y,u\cdot v]=[x,y,u]\cdot v+u\cdot [x,y,v], \]
\[ [x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]], \]
for any \(x,y,z,u,v,w\in V\) (the notation \(\sum_{(x,y,z)}\) meaning cyclic sum).
For any two elements \(x,y\in V\) the linear map \(D(x,y): V\to V\) such that \(D(x,y)z:=[x,y,z]\) is a derivation of \(V\) relative to its binary product and also of \(V\) relative to its ternary operation. The linear span \(D(V,V)\) of all derivations \(D(x,y)\) is a Lie subalgebra of \(\text{gl}(V)\) called the inner derivation algebra of \(V\). When \(V\) is an irreducible module for \(D(V,V)\) we say that \(V\) is an irreducible Lie-Yamaguti algebra.
In a previous paper by the same authors [J. Pure Appl. Algebra 213, No. 5, 795–808 (2009; Zbl 1223.17001)], it has been proved that the classification of irreducible Lie-Yamaguti algebras splits into three non-overlapping types:
Adjoint type: \(V\) is the adjoint module for \(D(V,V)\).
Non-simple type: \(D(V,V)\) is not simple.
Generic Type: both \(g(V)\) (the standard enveloping algebra) and \(D(V,V)\) are simple.
The Lie-Yamaguti algebras of the first two types were classified in a previous work through a generalized Tits construction of Lie algebras. It turns out that the ones of adjoint type are just the simple Lie algebras, and those of Non-simple type can be described through reductive decompositions modeled by the mentioned generalized Tits construction using quaternions, octonions and simple Jordan algebras.
The aim of this paper is the classification of the remaining type: the generic one. At the same time it is investigated the connection of these Lie-Yamaguti algebras with some well-known nonassociative algebraic systems. For instance most irreducible Lie-Yamaguti algebras of generic type appear inside simple Lie algebras as orthogonal complements of subalgebras of derivations of Lie and Jordan algebras and triple systems, Freudenthal and orthogonal triple systems or Jordan and anti-Jordan pairs.
\[ x\cdot x=0,\quad [x,x,y]=0,\quad \sum_{(x,y,z)}([x,y,z]+(x\cdot y)\cdot z)=0, \]
\[ \sum_{(x,y,z)}[x\cdot y,z,t]=0,\quad [x,y,u\cdot v]=[x,y,u]\cdot v+u\cdot [x,y,v], \]
\[ [x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]], \]
for any \(x,y,z,u,v,w\in V\) (the notation \(\sum_{(x,y,z)}\) meaning cyclic sum).
For any two elements \(x,y\in V\) the linear map \(D(x,y): V\to V\) such that \(D(x,y)z:=[x,y,z]\) is a derivation of \(V\) relative to its binary product and also of \(V\) relative to its ternary operation. The linear span \(D(V,V)\) of all derivations \(D(x,y)\) is a Lie subalgebra of \(\text{gl}(V)\) called the inner derivation algebra of \(V\). When \(V\) is an irreducible module for \(D(V,V)\) we say that \(V\) is an irreducible Lie-Yamaguti algebra.
In a previous paper by the same authors [J. Pure Appl. Algebra 213, No. 5, 795–808 (2009; Zbl 1223.17001)], it has been proved that the classification of irreducible Lie-Yamaguti algebras splits into three non-overlapping types:
Adjoint type: \(V\) is the adjoint module for \(D(V,V)\).
Non-simple type: \(D(V,V)\) is not simple.
Generic Type: both \(g(V)\) (the standard enveloping algebra) and \(D(V,V)\) are simple.
The Lie-Yamaguti algebras of the first two types were classified in a previous work through a generalized Tits construction of Lie algebras. It turns out that the ones of adjoint type are just the simple Lie algebras, and those of Non-simple type can be described through reductive decompositions modeled by the mentioned generalized Tits construction using quaternions, octonions and simple Jordan algebras.
The aim of this paper is the classification of the remaining type: the generic one. At the same time it is investigated the connection of these Lie-Yamaguti algebras with some well-known nonassociative algebraic systems. For instance most irreducible Lie-Yamaguti algebras of generic type appear inside simple Lie algebras as orthogonal complements of subalgebras of derivations of Lie and Jordan algebras and triple systems, Freudenthal and orthogonal triple systems or Jordan and anti-Jordan pairs.
Reviewer: Candido Martín González (Málaga)
MSC:
| 17A30 | Nonassociative algebras satisfying other identities |
| 17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
| 17A40 | Ternary compositions |
Keywords:
Lie-Yamaguti algebra; classification; Lie algebra; Lie triple system; Jordan systems; Freudenthal triple systemsCitations:
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