Quasi Frobenius Lie algebra - Alchetron, the free social encyclopedia

In mathematics, a quasi-Frobenius Lie algebra

( g , [ , ] , β )

over a field k is a Lie algebra

( g , [ , ] )

equipped with a nondegenerate skew-symmetric bilinear form

β : g × g → k , which is a Lie algebra 2-cocycle of g with values in k . In other words,

for all X , Y , Z in g .

If β is a coboundary, which means that there exists a linear form f : g → k such that

β ( X , Y ) = f ( [ X , Y ] ) ,

then

( g , [ , ] , β )

is called a Frobenius Lie algebra.

Equivalence with pre-Lie algebras with nondegenerate invariant skew-symmetric bilinear form

If ( g , [ , ] , β ) is a quasi-Frobenius Lie algebra, one can define on g another bilinear product ◃ by the formula

Then one has [ X , Y ] = X ◃ Y − Y ◃ X and

( g , ◃ )

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