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Integrable affine WZW highest-weight modules are graded and restricted

Statement

Let $g$ be finite-dimensional complex simple, use the invariant form $B$ for which long roots have square $2$ , and let $θ$ be the highest root. Let $k \in Z_{\geq 0}$ and let $λ$ be dominant integral with $⟨ λ, θ^{\lor} ⟩ \leq k$ . For the full untwisted affine algebra with $[x_{m}, y_{n}] = [x, y]_{m + n} + m δ_{m, - n} B (x, y) c, [d, x_{m}] = m x_{m},$ put $\hat{λ} ∣_{h} = λ$ , $\hat{λ} (c) = k$ , and $\hat{λ} (d) = 0$ . The simple highest-weight module $L_{k} (λ)$ exists and is integrable. Its degree-zero space for the grading $N = - d$ is the finite-dimensional simple $g$ -module $V_{λ}$ ; all degrees are nonnegative integers and finite-dimensional. In the sense of def-positive-energy-affine-module, the module is restricted: for every vector $v$ , $x_{m} v = 0$ for all sufficiently large $m > 0$ .

The grading asserted here is the affine degree grading. No relation to a Sugawara $L_{0}$ -grading or integral $L_{0}$ -spectrum is asserted.

Statement references

Restricted affine-current module

Facts & Assumptions

Given: $g, B, θ, k, λ$ as in the Statement, with the full affine Cartan and degree convention of Degree derivation and full untwisted affine algebra.

[A1]

Full AC is assumed by The Axiom of Choice, as required by the finite-dimensional highest-weight suppliers below.

[F1]

Untwisted affine central extension fixes the normalized bracket and central generator $c$ .

[F2]

Loop and affine GCM presentations are isomorphic identifies the affine node-zero coroot as $h_{0} = c - θ^{\lor}$ .

[F3]

Kac moody verma module has a unique simple quotient supplies a nonzero simple highest-weight module for every affine weight.

[F4]

Kac moody integral and dominant integral weights: dominant integral means every simple-coroot label lies in $Z_{\geq 0}$ . Integrable kac moody module defines integrability by local nilpotence of the Chevalley generators. Integrability criterion for simple highest weight kac moody modules identifies these label conditions with integrability of the simple highest-weight module.

[F5]

Dominant simple highest-weight modules are finite-dimensional: a simple finite-dimensional-semisimple-Lie-algebra highest-weight module at dominant integral $λ$ is finite-dimensional; its highest vector is nonzero.

[F6]

Highest-weight classification identifies the resulting finite-dimensional simple $g$ -module as $V_{λ}$ .

[F7]

PBW for countably presented Kac Moody Lie algebras gives ordered PBW monomials for the countably based affine Lie algebra; Universal property and pbw character of kac moody verma modules gives the negative-then-Borel spanning of the affine Verma module.

[F8]

Restricted affine-current module uses “restricted” to mean that sufficiently high positive current modes annihilate each fixed vector; it does not itself assert an energy grading.

Proof

1.1givenF1F2algebra

For each finite simple coroot $h_{i}$ , the affine weight has label $\hat{λ} (h_{i}) = λ (h_{i}) \in Z_{\geq 0}$ . By [F2], the remaining label is $\hat{λ} (h_{0}) = k - ⟨ λ, θ^{\lor} ⟩$ , also a nonnegative integer. Thus $\hat{λ}$ is dominant integral for the untwisted affine generalized Cartan matrix.

1.2F3F7given

The highest vector has $N = - d$ degree zero. The negative-then-Borel PBW spanning in [F7] shows that every vector is spanned by negative-root and negative-loop monomials on the highest vector. A negative loop mode $x_{- m}$ , $m > 0$ , raises $N$ by $m$ , while horizontal modes preserve it. Thus the module has no negative degrees and its degree-zero subspace is exactly the horizontal highest-weight module generated by the highest vector.

2.1F3F4step 1.1

By [F3] the simple highest-weight module $L (\hat{λ})$ exists and is nonzero. Applying [F4] to the labels computed in step 1.1 proves that this simple module is integrable. Denote it by $L_{k} (λ)$ .

2.2A1F3F5F6F7step 1.2

This degree-zero subspace is simple as a $g$ -module. Indeed, if $W$ were a proper nonzero $g$ -submodule, every positive loop mode would kill $W$ , because it lowers degree below zero. Order words by PBW with positive loops on the right, horizontal modes in the middle, and negative loops on the left. Any word applied to $W$ that has degree zero therefore reduces to horizontal modes on $W$ , hence stays in $W$ ; negative loops have strictly positive degree. The affine submodule generated by $W$ would have degree-zero part exactly $W$ , contradicting simplicity from [F3]. Its highest weight is $λ$ , so [F5] and [F6] identify it with finite-dimensional $V_{λ}$ .

3.1F7F8step 2.2algebra∎

At degree $N$ , PBW expresses vectors as sums of negative-loop monomials of total degree $N$ applied to $V_{λ}$ . There are finitely many partitions of $N$ , and each mode space $g \otimes t^{- m}$ is finite-dimensional; thus each such span is finite-dimensional. If $v$ has degrees at most $N$ , then $x_{m} v$ has degrees below zero whenever $m > N$ , so it is zero. This proves finite-dimensional graded pieces and the restrictedness defined in [F8].

Integrable affine WZW highest-weight modules are graded and restricted

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources