Definition:Lie Algebra

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Definition

Let $L$ be an algebra over a ring.


Then $L$ is a Lie algebra if and only if it satisfies the Lie algebra axioms:

\((\text L 0)\)   $:$   Closure      \(\ds \forall a, b \in L:\) \(\ds \sqbrk {a, b} \in L \)      
\((\text L 1)\)   $:$   Alternativity      \(\ds \forall a \in L:\) \(\ds \sqbrk {a, a} = 0 \)      
\((\text L 2)\)   $:$   Jacobi Identity      \(\ds \forall a, b, c \in L:\) \(\ds \sqbrk {a, \sqbrk {b, c} } + \sqbrk {b, \sqbrk {c, a} } + \sqbrk {c, \sqbrk {a, b} } = 0 \)      

where $\sqbrk {\, \cdot, \cdot \,}$ is the bilinear mapping on $L$.


Bracket

The bilinear mapping $\sqbrk {\, \cdot, \cdot \,}$ on $L$ which defines the underlying algebra of $L$ is referred to as the bracket operator.


Also see

  • Results about Lie algebras can be found here.


Source of Name

This entry was named for Marius Sophus Lie.


Sources