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Let L be a Lie algebra with basis $B=\{x_1,...,x_{10}\}$, Is there any property to describe the following lie bracket: for example how I can decompose $[x_1 x_2 x_3 , x_5]=$? Here $x_1 x_2 x_3$ is the formal product of the elements of Lie algebra.

mweiss
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Nil
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    What is a formal product? The commutator in the universal enveloping algebra? – darij grinberg Jul 20 '14 at 14:59
  • @ darij grinberg, yes , universal enveloping algebra is defined as the span of formal products of elements of Lie algebra. – Nil Jul 20 '14 at 15:01
  • As in @darijgrinberg's comment, there is some danger in attempting to describe the universal enveloping algebra as "formal products". Sure, there's the Poincare-Birkhoff-Witt theorem, but that oughtn't be relevant too often. Use the mapping-property characterization. – paul garrett Jul 20 '14 at 17:15

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If the Lie algebra derive from an associative Algebra, with the definition $$ [u,v]=uv-vu, $$ then you can easily prove that $$ [uv,w]=u[v,w]+[u,w]v. $$ For a situation like this: $$ [x_1^ex_2^fx_3^g,x_5] $$ with $e$, $f$, $g$ positive integers, you should have \begin{align} [x_1^ex_2^fx_3^g,x_5]&=\sum_{h=0}^{e-1} x_1^h [x_1,x_5]x_1^{e-h-1}x_2^fx_3^g\\ &+\sum_{h=1}^{f-1} x_1^e x_2^h [x_2,x_5]x_2^{f-h-1}x_3^g\\ &+\sum_{h=1}^{g-1} x_1^e x_2^fx_3^h[x_3,x_5]x_3^{g-h-1} \end{align}

Vincenzo Tibullo
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  • thank you for your answer, if we consider x_1 and x_2 and x_3 with powers, like this: x_1^(e ) x_2^f x_3^g then how is the result of Lie bracket? – Nil Jul 20 '14 at 15:31