Lie algebras from other algebras
1 Lie algebras from associative algebras
is a Lie algebra.
Next, so is alternating. Finally for the Jacobi identity we compute directly.
We notice this produces a functor from the category of associative algebras to the category of Lie algebras. However, to every commutative algebra , is a trivial Lie algebra, and so this functor is not faithful. More generally, the center of an arbitrary associative algebra is lost to the Lie algebra structure .
We do observe some relationships between the algebraic structure of and that of .
If then .
We observe that a submodule of is a submodule of as the two are identitcal as modules. It remains to show . So given and , then and as we conclude . ∎
2 Associative envelopes
Given a Lie algebra it is often desirable to reverse the process described above, that is, to provide an associative algebra for which . In general this is impossible as we will now explain.
Let be a vector space and the endomorphism algebra on . Then we give the name to the Lie algebra (noting that is associative under the composition of functions operation.) Then we can also define a subalgebra as the set of linear transformations with trace 0.
Now we claim that is not equal to for any associative (unital) algebra . For it is easy to see has a basis of three elements:
Therefore would also be 3-dimensional. We also know that is a simple Lie algebra, that is, it has no proper ideals. Therefore by Theorem 2, can have no ideals either, so must be simple. However the finite dimensional simple rings over are isomorphic to matrix rings (by the Wedderburn-Artin theorem) and thus cannot have dimension 3.
This forces the weaker question as to whether a Lie algebra can be embedded in for some associative algebra . We call such embeddings associative envelopes of the Lie aglebra. The existence of associative envelopes of arbitrary Lie algebras is answered by a corollary to the Poincare-Birkhoff-Witt theorem.
Every Lie algebra embeds in the universal enveloping algebra , where is an associative algebra.
Finite dimensional analogues also exist, some of which are simpler to observe. For instance, a Lie aglebra can be represented in by the adjoint representation. The representation is not faithful unless the center of is trivial. However, for semi-simple Lie algebras, the adjoint representation thus suffices as an associative envelope.
2.1 Lie algebra from non-associative algebras
If is not an associative algebra to begin with then we may still determine the commutator bracket is bilinear and alternating. However, the Jacobi identity is in question. If we define the associator bracket as then we can write the computation for the Jacobi identity as:
That is, in a non-associative algebra the corresponding Jacobi identity is the possibly non-trivial sum over all permutations of associators. We consider a few non-associative examples.
If is a commutative non-associative algebra (perhaps a Jordan algebra) then
so the Jacobi identity holds. However, if is commutative then to begin with so the associated Lie algebra product is trivial.
We can also consider beginning with a Lie algebra and producing . To avoid confusing the bracket of and that of we let the multiplication of be denoted by juxtaposition, , . Recall that in a Lie algebra of characteristic 0 or odd then so that in . So we have simply scaled the original product of by . To see the Jacobi identity still holds we note
So once again the associators cancel.
|Title||Lie algebras from other algebras|
|Date of creation||2013-03-22 16:37:33|
|Last modified on||2013-03-22 16:37:33|
|Last modified by||Algeboy (12884)|