In the sign convention used here, a graded Lie algebra over a field of characteristic zero is a vector space with a bilinear map satisfyingfor , and homogeneous . Thus two odd-degree elements have a symmetric graded Lie bracket, whereas an even-degree element has the ordinary antisymmetry rule. For a graded associative algebra, supplies an example; expanding its products proves the graded Jacobi identity. Reducing degrees modulo two gives a Lie superalgebra. A Gerstenhaber algebra uses this sign rule after shifting degrees by one.
A different common convention uses an ordinary Lie algebra with a grading satisfying and no degree signs. The loop algebra is an example of that ordinary graded convention. The degree convention must therefore be specified before applying a graded Lie bracket formula.
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