Automorphisms and representations of quasi Laurent polynomial algebras

Abstract

We study automorphisms and representations of quasi polynomial algebras (QPAs) and quasi Laurent polynomial algebras (QLPAs). For any QLPA defined by an arbitrary skew symmetric integral matrix, we explicitly describe its automorphism groups at generic $q$ and at roots of unity. Any QLPA is isomorphic to the tensor product of copies of the QLPA of degree $2$ at different powers of $q$ and the centre, thus the study of representations of QPAs and QLPAs largely reduces to that of ${\mathcal L}_q(2)$ and ${\mathcal A}_q(2)$, the QLPA and QPA of degree $2$. We study a category of ${\mathcal A}_q(2)$-modules which have finite covers by submodules with natural local finiteness properties and satisfy some condition under localisation, determining its blocks, classifying the simple objects and providing two explicitly constructions for the simples. One construction produces the simple ${\mathcal A}_q(2)$-modules from ${\mathcal L}_q(2)$-modules via monomorphisms composed of the natural embedding of ${\mathcal A}_q(2)$ in ${\mathcal L}_q(2)$ and automorphisms of ${\mathcal L}_q(2)$, and the other explores a class of holonomic ${\mathcal D}_q$-modules for the algebra ${\mathcal D}_q$ of $q$-differential operators.