Graded Lie algebra

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^[1] 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^[2] is a further generalization of this notion to the category of superalgebras in which a graded Lie superalgebra is endowed with an additional super $\mathbb {Z} /2\mathbb {Z}$ -gradation. These arise when one forms a graded Lie superalgebra in a classical (non-supersymmetric) setting, and then tensorizes to obtain the supersymmetric analog.^[3]

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.

^ The "super" prefix for this is not entirely standard, and some authors may opt to omit it entirely in favor of calling a graded Lie superalgebra just a graded Lie algebra. This dodge is not entirely without warrant, since graded Lie superalgebras may have nothing to do with the algebras of supersymmetry. They are only super insofar as they carry a $\mathbb {Z} /2\mathbb {Z}$ gradation. This gradation occurs naturally, and not because of any underlying superspaces. Thus in the sense of category theory, they are properly regarded as ordinary non-super objects.
^ In connection with supersymmetry, these are often called just graded Lie superalgebras, but this conflicts with the previous definition in this article.
^ Thus supergraded Lie superalgebras carry a pair of $\mathbb {Z} /2\mathbb {Z}$ -gradations: one of which is supersymmetric, and the other is classical. Pierre Deligne calls the supersymmetric one the super gradation, and the classical one the cohomological gradation. These two gradations must be compatible, and there is often disagreement as to how they should be regarded. See Deligne's discussion of this difficulty.

[1] The "super" prefix for this is not entirely standard, and some authors may opt to omit it entirely in favor of calling a graded Lie superalgebra just a graded Lie algebra. This dodge is not entirely without warrant, since graded Lie superalgebras may have nothing to do with the algebras of supersymmetry. They are only super insofar as they carry a $\mathbb {Z} /2\mathbb {Z}$ gradation. This gradation occurs naturally, and not because of any underlying superspaces. Thus in the sense of category theory, they are properly regarded as ordinary non-super objects.

[2] In connection with supersymmetry, these are often called just graded Lie superalgebras, but this conflicts with the previous definition in this article.

[3] Thus supergraded Lie superalgebras carry a pair of $\mathbb {Z} /2\mathbb {Z}$ -gradations: one of which is supersymmetric, and the other is classical. Pierre Deligne calls the supersymmetric one the super gradation, and the classical one the cohomological gradation. These two gradations must be compatible, and there is often disagreement as to how they should be regarded. See Deligne's discussion of this difficulty.

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