Graded Lie algebra

Source: Wikipedia, the free encyclopedia.

In

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

.

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 -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.

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

lower central series of a free group
.

Generalizations

If is any

, 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 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

  1. for ,
  2. , and
  3. 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-

commutative algebras
to the graded category.

On Der(A), one can define a bracket via:

[d, δ ] = − (−1)ijδd, for d ∈ Deri(A) and δ ∈ Derj(A).

Equipped with this structure, Der(A) inherits the structure of a graded Lie superalgebra over k.

Further examples:

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:

  1. 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 map
    for the additive structure on the ring .

Notes

  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 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 -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.

References

  • .

See also