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Lie Algebras

3 Poincaré-Birkhoff-Witt

We would in general like to understand the structure of U(g).

To do this, we will use a general tool due to George Bergman for finding canonical forms for elements of associative k-algebras.

Say that we have a set X, and we consider the tensor algebra TR(X) on X over a commutative ring R. Say we also have some relations σS of the form Wσ=fσ, where W is a word in X. For any other words A,B, we can consider rAσB, the R-linear endomorphism on TR(X) replacing AWσB with AfσB. We call applying this to an element of TR(X) a reduction. If all reductions on an element are trivial, then that element is irreducible. The submodule of irreducible elements is called TR(X)irr. We say that an element is reduction finite if for any infinite sequence of reductions, only finitely many act nontrivially. The reduction finite elements form an R-submodule of TR(X) called TR(X)fin. A sequence of reductions is final if it results in an irreducible element. An element a is reduction unique if it is reduction finite and any final sequence results in the same irreducible element. The unique result will be denoted r(a).

  • Lemma 3.1. The reduction unique elements form a submodule denoted TR(X)un, and r:TR(X)unTR(X)irr is R-linear.

  • Proof. Suppose a,bTR(X)un,kR. a+kb is reduction finite, and if we have two reduction sequences of it to an irreducible, we can consider doing the same sequence on a,b and extend them to one that makes each irreducible (note that this won’t change the result, only the sequence). Then by uniqueness for a,b we see that the two reductions are the same, and that r is R-linear.

  • Lemma 3.2. Let a,b,cTR(X) have the property that if A,B,C are nonzero monomials in a,b,c, then ABC is reduction unique. Then if rΣ(b) is the result of some finite reductions on b, then arΣ(b)c is reduction unique.

  • Proof. Note in particular that the hypotheses imply that if abc is reduction unique, and that if the conclusion holds, r(arΣ(b)c)=r(abc). By linearity we only need to show this when a,b,c are monomials, and when rΣ(b) is a single reduction rdσe. But ardσe(b)c=radσec(abc), so this follows since abc is reduction unique.

We say that a overlap ambiguity is a pair σ,σS and a triple A,B,C of nonempty words such that AB=Wσ,BC=Wσ (they overlap). It is resolvable if there is are sequences of reductions r,r such that rrσ(ABC)=rrσ(ABC). An inclusion ambiguity is the same, except when Wσ=B,Wσ=ABC, and has the same conditions for resolvability (except A,C can be empty).

Say that a compatible partial ordering on words of X is one such that ABCADCBD and such that the monomials of the fσ are smaller than the monomials of Wσ. Let I be the two-sided ideal generated by the relations in S, and IA be the R-submodule of TR(X) generated by BσC<A,σS (every monomial is smaller than A). An ambiguity is resolvable relative to if rσ(ABC)rσ(ABC)IABC. Note that this is an easy condition to check.

Finally we arrive at this theorem, which can be considered a Diamond Lemma for rings:

  • Theorem 3.3. Suppose we have S,X,,I as above where is compatible with S and satisfies the descending chain condition. Then the following are equivalent:

    • 1. Every ambiguity is resolvable.

    • 2. Every ambiguity is resolvable with respect to .

    • 3. Every element is reduction unique.

    • 4. The natural quotient identifies the submodule spanned by irreducible monomials with TR(X)/I, which is TR(X)irr.

  • Proof. Clearly (3)(1)(2). If (3) is true, r defines a projection onto TR(X)irr. The kernel is contained in I by definition of r and contains I, as for any AB, r(AσB)=0 by Lemma 3.2. Thus as an R-module, TR(X)=ITR(X)irr, giving (4). Conversely if (4) is true, Then since reductions are equal in the quotient, they must be unique.

    Thus it suffices to prove (2)(3), and by linearity, we need only show this for a monomial A. From the descending chain condition, we can assume that any smaller monomial is reduction unique. Now suppose that we have two reductions of a monomial A.

    If there is no ambiguity, they commute, so we can create two more reductions to an irreducible where the first two steps are the first step of these reductions but in different orders. They will give the same element, and by induction will show that the two original reductions are also the same.

    If there is ambiguity, then since it is resolvable relative to by induction the difference can be resolved to 0, so by linearity the two resolutions must agree.

  • Theorem 3.4 (Poincaré-Birkhoff-Witt). Let g be a Lie algebra over R where the underlying module is free. Choose a well-ordered basis of g, xα,αI. Then xa1e1xanen for xa1 in increasing order, over all possible ai and ei form a basis of U(g).

  • Proof. Construct an ordering on monomials in the basis where monomials of smaller degree are smaller, and if two monomials have the same degree, we compare them lexicographically using the order on our basis. This is a well-ordering on monomials, and a sufficient set of relations for U(g) can be written as xy=yx+[x,y] for x>y in the basis. The order is compatible with this, and we can check that every ambiguity is resolvable with respect to our order. Here, the only nontrivial kind of ambiguity that can occur is an overlap ambiguity, when we have something of the form A(yxz+[x,y]z)B=AxyzB=A(xzy+x[y,z])B. The difference is:

    A(yxz+[x,y]z(xzy+x[y,z]))B=A(yzx+y[x,z]+z[x,y]+[[x,y],z]

    (zxy+[x,z]y+[y,z]x+[x,[y,z]]))B=A(zyx+[y,z]x+y[x,z]+z[x,y]+[[x,y],z]

    (zyx+z[x,y]+y[x,z]+[[x,z],y]+[y,z]x+[x,[y,z]])B

    =A([[x,y],z][[x,z],y][x,[y,z]])B=0

    Where at the last step, we have used the Jacobi identity. This shows that the ambiguity is resolvable under the ordering, so that by Theorem 3.3 we are done.

Note that as a corollary, any Lie algebra g (not necessarily finite-dimensional) is faithfully represented in U(g).