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A symmetrizable indefinite rank two gcm

Example

The symmetric GCM $A = (\begin{array}{cc} 2 & - 3 \\ - 3 & 2 \end{array})$ is indefinite. The nonzero vector $[e_{1}, e_{2}]$ has imaginary root $β = α_{1} + α_{2}$ with squared length $- 2$ for $D = I$ .

Facts & Assumptions

Given: The displayed rank-two matrix and D=I.

[F1]

The positive half is free modulo its positive Serre ideal. (Serre presentation of a kac moody algebra).

[F2]

The root metric has entries d_i a_ij. (Invariant bilinear form for a symmetrizable kac moody algebra).

[F3]

Imaginary means a root outside W Pi. (Real and imaginary kac moody roots).

[F4]

A positive vector with negative image characterizes indefinite type. (Finite affine indefinite trichotomy for indecomposable gcms).

Verification

1.1

The matrix meets every GCM condition, is connected and symmetric, has determinant $4 - 9 = - 5$ , and $A (1, 1)^{t} = (- 1, - 1)^{t}$ . Hence it is indefinite by F4.

F4given

1.2

The two positive Serre generators have degrees $(4, 1)$ and $(1, 4)$ , each of total height five. Every element of their generated positive ideal is a linear combination of these and positive adjoints, so has no component below height five. The free bracket $[e_{1}, e_{2}]$ is nonzero: its tensor image is the difference of the distinct words $e_{1} e_{2} - e_{2} e_{1}$ . F1 therefore ensures its degree-(1,1) class survives. It is a root vector of weight $β$ .

F1given

2.1

F2 gives $(β, β) = 2 + 2 - 3 - 3 = - 2$ . For each reflection, $(α_{i}, λ) = λ (h_{i})$ and $(α_{i}, α_{i}) = 2$ make expansion of $(λ - λ (h_{i}) α_{i})^{2}$ equal to $(λ, λ)$ . Thus every Weyl translate of a simple root has squared length 2. The root from step 1.2 has length −2, so cannot be such a translate and is imaginary by F3.

F2F3step 1.2∎

Sources

Source comparison: Kleshchev, §4.1 and Theorem 9.3.5, pp.50–57 and 125–126; local degree-(1,1) calculation.

Depends on

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Sources

Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §4.1 and Theorem 9.3.5, pp.50–57 and 125–126; local degree-(1,1) calculation (standard reference, not scraped)