Reductive Lie Algebra
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a Lie algebra is reductive if its adjoint representation in completely reducible, whence the name. More concretely, a Lie algebra is reductive if is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: mathfrak{g} = mathfrak{s} oplus mathfrak{a}; there are alternative characterizations, given below.The most basic example is the Lie algebra mathfrak{gl}_n of n times n matrices with the commutator as Lie bracket, or more abstractly as the...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a Lie algebra is reductive if its adjoint representation in completely reducible, whence the name. More concretely, a Lie algebra is reductive if is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: mathfrak{g} = mathfrak{s} oplus mathfrak{a}; there are alternative characterizations, given below.The most basic example is the Lie algebra mathfrak{gl}_n of n times n matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an n-dimensional vector space, mathfrak{gl}(V). This is the Lie algebra of the general linear group GL(n), and is reductive as it decomposes as mathfrak{gl}_n = mathfrak{sl}_n oplus mathfrak{k}, corresponding to traceless matrices and scalar matrices.
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