Definable twist-and-tensor conjecture for modules of algebraic groups

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Let (G,V)(G,V) be a faithful, irreducible module of finite Morley rank, where G=G(K)G=\mathbb{G}(\mathbb{K}) is the group of K\mathbb{K}-points of a simple algebraic group. Definable twist-and-tensor conjecture. There are a compatible definable K\mathbb{K}-vector-space structure on VV, irreducible algebraic modules W1,…,WdW_1,\ldots,W_d, and (G⋉V)(G\ltimes V)-definable field automorphisms φ1,…,φd\varphi_1,\ldots,\varphi_d such that

V≃⨂i=1dWiφiV\simeq\bigotimes_{i=1}^d W_i^{\varphi_i}

as K[G]\mathbb{K}[G]-modules; equivalently, VV lies in the definable twist-and-tensor category. This is intended as a finite-Morley-rank analogue of Steinberg's tensor-product theorem, and the general classification of such modules remains open.

References

Primary source

Alexandre Borovik and Adrien Deloro, “Binding groups, permutations groups and modules of finite Morley rank”, arXiv:1909.02813 (2019).

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