In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra. A graded Lie superalgebra extends the notion of a graded Lie algebra in such a way that the Lie bracket is no longer assumed to be necessarily anticommutative. These arise in the study of derivations on graded algebras, in the deformation theory of Murray Gerstenhaber, Kunihiko Kodaira, and Donald C. Spencer, and in the theory of Lie derivatives. A supergraded Lie superalgebra is a further generalization of this notion to the category of superalgebras in which a graded Lie superalgebra is endowed with an additional super -gradation. These arise when one forms a graded Lie superalgebra in a classical (non-supersymmetric) setting, and then tensorizes to obtain the supersymmetric analog. Still greater generalizations are possible to Lie algebras over a class of braided monoidal categories equipped with a coproduct and some notion of a gradation compatible with the braiding in the category. For hints in this direction, see Lie superalgebra#Category-theoretic definition. Graded Lie algebras In its most basic form, a graded Lie algebra is an ordinary Lie algebra , together with a gradation of vector spaces such that the Lie bracket respects this gradation: The universal enveloping algebra of a graded Lie algebra inherits the grading. Examples sl(2) For example, the Lie algebra of trace-free 2 × 2 matrices is graded by the generators: These satisfy the relations , , and . Hence with , , and , the decomposition presents as a graded Lie algebra. Free Lie algebra The free Lie algebra on a set X naturally has a grading, given by the minimum number of terms needed to generate the group element. This arises for example as the associated graded Lie algebra to the lower central series of a free group. Generalizations If is any commutative monoid, then the notion of a -graded Lie algebra generalizes that of an ordinary (-) graded Lie algebra so that the defining relations hold with the integers replaced by . In particular, any semisimple Lie algebra is graded by the root spaces