As we saw in the first post, a representation of a finite group can be thought of simply as a module over a certain ring: the group ring. The analog for Lie algebras is the enveloping algebra. That’s the topic of this post.
Definition
The basic idea is as follows. Just as a representation of a finite group was a group-homomorphism
for a vector space, a representation of a Lie algebra
is a Lie-algebra homomorphism
. Now,
is the Lie algebra constructed from an associative algebra,
—just as
is the group constructed from
taking invertible elements.
So, in general, it follows that what we’re really looking at here is maps for an associative algebra
, such that
is a Lie-algebra homomorphism, i.e.
Call such a function an
-map. Note that
makes any
-module into a representation of
, so these are interesting objects to consider.
The universal enveloping algebra has the universal -map. In detail, if
is fixed, we can consider the category
of pairs
, where
is an associative algebra and
an
-map. A morphism between
and
is an algebra-homomorphism
such that
.
Definition 1 In this category, the universal enveloping algebra
is the initial object: thus for any
-map
, we have a unique algebra-homomorphism
making the usual diagram commutative.
By the uniqueness of universal objects, the enveloping algebra is unique if it exists.
Proposition 2 To give a
-representation is the same as to give a
-module. Alternatively, the categories of
-representations and
-modules are equivalent.
Proof: Indeed, a -module is a vector space
with an algebra homomorphism
. The set of such algebra-homomorphisms is functorially isomorphic to the set of
-mappings
, or
-representations.
Of course, we have to still show exists.
Tensor Algebras
Given a vector space , we can form its tensor powers
for all
(say
if we are working over the complex numbers).
We can thus form the graded vector space
which we can make into a noncommutative ring. Indeed, we have a bilinear map
and we can extend by linearity to the direct sum to get a graded ring. Basically, the structure is described simply: if , form a basis for
, then multiplication sends
The ordering of the ‘s does matter, which explains the noncommutativity; if we demand that the
‘s do commute, we end up with a polynomial algebra.
Anyway, the importance of the tensor algebra is the universal property:
Proposition 3 Given a linear map
for a ring
, there is a unique algebra-homomorphism
such that the map
is just
.
Proof: The proof follows by sending to
; this gives a ring-homomorphism
. The rest is straightforward to check.
The Enveloping Algebra
It is often useful to quotient out the tensor algebra by some relations to still get a useful object. For instance, if one abelianizes, one gets the symmetric algebra; if one quotients out by the ideal generated by
, one gets the exterior algebra.
In our case, we have a Lie algebra , and I claim that
Proposition 4
is isomorphic to
where
is the two-sided ideal generated by
Proof: We can check this by looking at universal properties. So, first, there’s a map (this is true for any vector space) and a quotient map
. Thus there’s a map
. Now
by the relations defining
. From this it follows that
is an
-map.
Conversely, if is an
-map, then there is a unique algebra-homomorphism
. By the definition of
-maps, it must factor through
, so it follows that
is universal.
So, next we should probably look at some of the properties of the enveloping algebra. For instance, it’s Noetherian. It also has a nice basis by the PBW theorem.
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