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    Lie Algebra

    Basic definition

    A Lie algebra is a nonassociative algebra corresponding to a Lie group.

    Detailed definition

    A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket. Elements f, g, and h of a Lie algebra satisfy [f, f] = 0 [f + g, h] = [f, h] + [g, h], and [f, [g, h]] + [g, [h, f]] + [h, [f, g]] = 0 (the Jacobi identity). The relation [f, f] = 0 implies [f, g] = - [g, f]. For characteristic not equal to two, these two relations are equivalent. The binary operation of a Lie algebra is the bracket [f g, h] = f[g, h] + [f, h] g.

    Related terms

    Ado's theorem | derivation algebra | Dynkin diagram | Iwasawa's theorem | Jacobi identities | Lie algebroid | Lie bracket | Lie group | Poincaré-Birkhoff-Witt theorem | Poisson bracket | reduced root system | root system | Weyl group

    Educational grade level

    graduate school level

    Associated person

    Marius Sophus Lie

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