The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra.
In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups.
This relationship generalizes to the idea of Tannaka–Krein duality between compact topological groups and their representations.
Consider, for example, the Lie algebra sl(2,C), spanned by the matrices which satisfy the commutation relations
In general, elements of the universal enveloping algebra are linear combinations of products of the generators in all possible orders.
Using the defining relations of the universal enveloping algebra, we can always re-order those products in a particular order, say with all the factors of
Doing this sort of thing repeatedly eventually converts any element into a linear combination of terms in ascending order.
The Poincaré–Birkhoff–Witt theorem, discussed below, asserts that these elements are linearly independent and thus form a basis for the universal enveloping algebra.
The formal construction of the universal enveloping algebra takes the above ideas, and wraps them in notation and terminology that makes it more convenient to work with.
Here, a very explicit approach is adopted, to minimize any possible confusion about the meanings of expressions.
Let us define and It is straightforward to verify that the above definition is bilinear and skew-symmetric; one can also show that it obeys the Jacobi identity.
(The Gerstenhaber algebra should not be confused with the Poisson superalgebra; both invoke anticommutation, but in different ways.)
Likewise, the Poincaré–Birkhoff–Witt theorem, below, constructs a basis for an enveloping algebra; it just won't be universal.
This can be done in either one of two different ways: either by reference to an explicit vector basis on the Lie algebra, or in a coordinate-free fashion.
[6] The proof of the theorem involves noting that, if one starts with out-of-order basis elements, these can always be swapped by using the commutator (together with the structure constants).
The hard part of the proof is establishing that the final result is unique and independent of the order in which the swaps were performed.
One can also state the theorem in a coordinate-free fashion, avoiding the use of total orders and basis elements.
This is perhaps not immediately obvious: to get this result, one must repeatedly apply the commutation relations, and turn the crank.
can be expressed (non-uniquely) as a linear combination of products of left-invariant vector fields.
That is, what the PBW theorem obscures (the commutation relations) the algebra of symbols restores into the spotlight.
See, however, the discussion of the bialgebra structure in the article on tensor algebras for a review of some of the finer points of doing so: in particular, the shuffle product employed there corresponds to the Wigner-Racah coefficients, i.e. the 6j and 9j-symbols, etc.
Construction of representations typically proceeds by building the Verma modules of the highest weights.
(See, for example, the realization of the universal enveloping algebra as left-invariant differential operators on the associated group, as discussed above.)
For a finite-dimensional semisimple Lie algebra, the Casimir operators form a distinguished basis from the center
Recall that the adjoint representation is given directly by the structure constants, and so an explicit indexed form of the above equations can be given, in terms of the Lie algebra basis; this is originally a theorem of Israel Gel'fand.
That is, and explicit computation shows that after making use of the structure constants A key observation during the construction of
If the action of the algebra is isometric, as would be the case for Riemannian or pseudo-Riemannian manifolds endowed with a metric and the symmetry groups SO(N) and SO (P, Q), respectively, one can then contract upper and lower indices (with the metric tensor) to obtain more interesting structures.
Quartic Casimir operators allow one to square the stress–energy tensor, giving rise to the Yang-Mills action.
However, the Lie superalgebras are able to evade the premises of the Coleman–Mandula theorem, and can be used to mix together space and internal symmetries.
Given a Lie group G, one can construct the vector space C(G) of continuous complex-valued functions on G, and turn it into a C*-algebra.