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Root normalization and dual data for principal W levels

Definition

Fix a finite-dimensional complex simple Lie algebra $g$ , Cartan subalgebra $h$ , positive roots $Φ^{+}$ , and its unique highest root $θ$ from lem-the-affine-simple-root-alpha-zero-is-delta-minus-the-highest-root. Let $Δ = {α_{1}, \dots, α_{r}}$ be the simple roots and $ρ = \frac{1}{2} \sum_{α \in Φ^{+}} α$ . Normalize the invariant form so long roots have squared length $2$ . The Coxeter number is $h_{g} = 1 + ht (θ)$ , where $ht (\sum_{i} n_{i} α_{i}) = \sum_{i} n_{i}$ . The dual Coxeter number is $h_{g}^{\lor} = 1 + ρ (θ^{\lor})$ , with the normalized-form identification supplied by lem-qftx-basic-form-scale.

For $i \neq j$ , put $a_{i j} = α_{j} (α_{i}^{\lor})$ . Define the lacing number $r^{\lor} = \max ({1} \cup {∣ a_{i j} ∣ : i \neq j}) .$ This agrees with the maximal edge multiplicity in the Dynkin diagram: by thm-rank-two-root-system-classification, every nonzero off-diagonal pair has absolute Cartan integers $(1, 1)$ , $(1, 2)$ , or $(1, 3)$ .

The dual root datum is $Φ^{\lor} = {α^{\lor} : α \in Φ}$ with positive roots $(Φ^{+})^{\lor}$ and transposed Cartan matrix. When a finite-dimensional simple Lie algebra $\tilde{g}$ realizing that dual datum is supplied, use its normalized form to define $h_{\tilde{g}}^{\lor}$ by def-dual-coxeter-number-for-normalized-form. No existence or uniqueness proof for this realization is included in this definition.

Definition references

Rank-two root-system classification

Dual Coxeter number for the normalized invariant form

The affine simple root alpha zero is delta minus the highest root

The basic form is the Killing form divided by twice the dual Coxeter number

Facts & Assumptions

Given: The simple Lie algebra, chosen Cartan and positive system, invariant form, and—when the second lacing branch is used—a simple Lie algebra $\tilde{g}$ realizing the dual root datum.

[F1]

Finite semisimple Cartan, root and string structure: the Cartan root set is finite and crystallographic, its root inner product is positive definite, and its coroots are defined.

[F2]

Finite Weyl positive roots and simple reflections: simple roots form a basis, every root has integral coordinates of one sign, and simple coroots form the coroot basis.

[F3]

The affine simple root alpha zero is delta minus the highest root: the finite root system has a unique highest root $θ$ , which is long.

[F4]

The basic form is the Killing form divided by twice the dual Coxeter number: if $K$ is the Killing form and $B_{0}$ is the positive invariant form normalized so every long root has squared length $2$ , then $K = 2 h_{r o o t}^{\lor} B_{0}$ , where $h_{r o o t}^{\lor} : = 1 + ρ (θ^{\lor})$ . Dual Coxeter number for the normalized invariant form (Definition and Proof step 1.2): for a positive invariant form $B$ with this same long-root normalization, its adjoint Casimir $Ω_{B}$ satisfies $B (Ω_{B} x, y) = K (x, y)$ ; it defines $h_{B}^{\lor} = γ_{B} / 2$ when $Ω_{B} = γ_{B} I$ .

[F5]

Rank-two root-system classification: for two nonorthogonal simple roots the nonzero Cartan integers have absolute values $1$ and $1$ , $1$ and $2$ , or $1$ and $3$ .

[F6]

Serre presentation theorem: assuming Choice, the Serre algebra of a based finite reduced crystallographic root system is finite dimensional semisimple, is simple for an irreducible system, and realizes that exact Cartan/root datum. Its converse identifies any realization with that Serre algebra.

[A1]

The Axiom of Choice is used only for the optional unconditional construction of the dual realizing algebra through F6. The Definition itself remains conditional on supplied dual data and needs no such construction when those data are already supplied.

Proof

1.1givenF1F4algebra

The finite root datum and the unique highest long root make the displayed heights and $h_{r o o t}^{\lor} : = 1 + ρ (θ^{\lor})$ well-defined. Let $K$ be the Killing form and $B_{0}$ the positive invariant form normalized so every long root has squared length $2$ . The form $B$ in Dual Coxeter number for the normalized invariant form is exactly $B_{0}$ : both are positive scalar multiples of $K$ and impose the same normalization, which uniquely fixes that scalar. Write $h_{B_{0}}^{\lor} : = γ_{B_{0}} / 2$ for the normalized-form dual Coxeter number.

1.2F1F2F5algebra

The crystallographic Cartan integers and F5 give $r^{\lor} \in {1, 2, 3}$ , with the rank-one convention1, and identify it with the maximal edge multiplicity.

1.3F1F2F6A1algebra

The dual set is itself finite reduced crystallographic: $s_{α} (β^{\lor}) = (s_{α} β)^{\lor}$ proves reflection closure, and the Cartan integers transpose. Its simple coroot basis and positive signs follow from F2. The underlying connected diagram is unchanged, hence it is irreducible. Under A1, F6 therefore constructs a finite-dimensional SIMPLE algebra realizing precisely this dual datum. Any other supplied realization is isomorphic to that Serre algebra by the converse part of F6.

2.1F4F6A1step 1.3algebra∎

By F4, $B_{0} (Ω_{B_{0}} x, y) = K (x, y) = 2 h_{r o o t}^{\lor} B_{0} (x, y)$ for every $x, y \in g$ . Since $B_{0}$ is nondegenerate, $B_{0} ((Ω_{B_{0}} - 2 h_{r o o t}^{\lor} I) x, y) = 0$ for every $y$ implies $Ω_{B_{0}} = 2 h_{r o o t}^{\lor} I$ . Hence $γ_{B_{0}} = 2 h_{r o o t}^{\lor}$ and $h_{B_{0}}^{\lor} = γ_{B_{0}} / 2 = h_{r o o t}^{\lor}$ , which establishes the claimed identification of the root expression with the normalized-form dual Coxeter number. For a supplied dual realization $\tilde{g}$ , Dual Coxeter number for the normalized invariant form defines $h_{\tilde{g}}^{\lor}$ from its own normalized adjoint Casimir; isomorphisms of realizations preserve that normalized form and Casimir, so this value is independent of the realization. Thus all supplied-data expressions are well-defined, and under Choice the optional realizing algebra is available.

Proof status

The Definition is well-defined for its supplied data. Under the explicitly recorded Choice assumption, the published Serre realization theorem also supplies the dual simple algebra; no missing existence or normalization premise remains.

Root normalization and dual data for principal W levels

Definition

Definition references

Facts & Assumptions

Proof

Proof status

Depends on

Dependency tree · 2 levels

Sources