ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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The typicality boundary in a gl(1|1) highest-weight module

Example

For every $(a, b) \in C^{2}$ , let $K (a, b)$ be the two-dimensional highest-weight supermodule from Two-dimensional highest-weight module data for gl(1|1), with highest vector $v$ even and $w = f v$ odd. It is simple exactly when the parameter is typical, namely when $a + b \neq 0$ .

Facts & Assumptions

Given: The gl(1|1) module $K (a, b) = C v \oplus C w$ , its $Z_{2}$ -grading, and the highest-weight convention from the fixed triangular decomposition.

[F1]

The gl(1|1) bracket data define a Lie superalgebra: the positive subalgebra is $C e$ and the Cartan subalgebra is $C h_{1} \oplus C h_{2}$ .

[F2]

Highest-weight vectors and modules for a Lie superalgebra: a highest-weight vector is a nonzero homogeneous weight vector annihilated by the selected positive subalgebra.

[F3]

Two-dimensional highest-weight module data for gl(1|1): $v$ has weight $(a, b)$ , $e v = 0$ , $f v = w$ , and $e w = (a + b) v$ .

[F4]

The two-dimensional gl(1|1) action is well-defined: these data make $K (a, b)$ a supermodule and $v$ a highest-weight vector.

[F5]

Typical and atypical highest weights for gl(1|1): this parameter is typical when $a + b \neq 0$ and atypical when $a + b = 0$ .

[F6]

Simple and semisimple Lie supermodules: a nonzero supermodule is simple when it has no proper nonzero graded submodule.

Verification

technique · direct

1.1F1F2F3F4F5F6given

By [F1], $n^{+} = C e$ and $h = C h_{1} \oplus C h_{2}$ . The vector $v$ is nonzero, even, has the displayed $h$ -weight, and is killed by $e$ , so [F2] and [F4] identify $K (a, b)$ as the stated highest-weight supermodule. Its only nonzero proper graded subspaces are $C v$ and $C w$ , since its even and odd parts are one-dimensional. Neither is invariant when $a + b \neq 0$ : $f v = w \notin C v$ , while $e w = (a + b) v \notin C w$ . By [F6], it is simple for a typical parameter.

2.1givenF3F4F5F6∎

If $a + b = 0$ , then $e w = 0$ , $f w = 0$ , and $h_{1}, h_{2}$ preserve $C w$ , so $C w$ is a nonzero proper graded submodule. By [F6], $K (a, b)$ is not simple; by [F5], the parameter is atypical.

Depends on

Used by

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Sources

Sigiswald Barbier, A minimal representation for the orthosymplectic Lie-superalgebra, §§2.4 and 2.6 (standard reference, not scraped)
Vera Serganova, Kac–Moody Superalgebras and Integrability, §13 (standard reference, not scraped)