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
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem