of its adjoint representation. Graded Lie superalgebras A graded Lie superalgebra over a field k (not of characteristic 2) consists of a graded vector space E over k, along with a bilinear bracket operation such that the following axioms are satisfied. [-, -] respects the gradation of E: (Symmetry) For all x in Ei and y in Ej, (Jacobi identity) For all x in Ei, y in Ej, and z in Ek, (If k has characteristic 3, then the Jacobi identity must be supplemented with the condition for all x in Eodd.) Note, for instance, that when E carries the trivial gradation, a graded Lie superalgebra over k is just an ordinary Lie algebra. When the gradation of E is concentrated in even degrees, one recovers the definition of a (Z-)graded Lie algebra. Examples and Applications The most basic example of a graded Lie superalgebra occurs in the study of derivations of graded algebras. If A is a graded k-algebra with gradation then a graded k-derivation d on A of degree l is defined by for , , and for . The space of all graded derivations of degree l is denoted by , and the direct sum of these spaces, carries the structure of an A-module. This generalizes the notion of a derivation of commutative algebras to the graded category. On Der(A), one can define a bracket via: [d, δ&hairsp;] = dδ − (−1)ijδd, for d ∈ Deri&hairsp;(A) and δ ∈ Derj&hairsp;(A). Equipped with this structure, Der(A) inherits the structure of a graded Lie superalgebra over k. Further examples: The Frölicher–Nijenhuis bracket is an example of a graded Lie algebra arising naturally in the study of connections in differential geometry. The Nijenhuis–Richardson bracket arises in connection with the deformations of Lie algebras. Generalizations The notion of a graded Lie superalgebra can be generalized so that their grading is not just the integers. Specifically, a signed semiring consists of a pair , where is a semiring and is a homomorphism of additive groups. Then a graded Lie supalgebra over a signed semiring consists of a vector space E graded with respect to the additive structure on , and a bilinear bracket [-, -] which respects the grading on E and in addition satisfies: for all homogeneous elements x and y, and Further examples: A Lie superalgebra is a graded Lie superalgebra over the signed semiring , where is the identity