Each homogeneous layer of the polynomial vector-field Lie algebra W_n splits into two irreducible modules over sl_n: the divergence-free derivations and the multiples of the Euler vector field.
Journal of Algebra, (2021), v.576, 1-26
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Module structure of the Lie algebra $W_n(K)$ over $sl_n(K)$
Each homogeneous layer of the polynomial vector-field Lie algebra W_n splits into two irreducible modules over sl_n: the divergence-free derivations and the multiples of the Euler vector field.