Unitarizability of weight modules over noncommutative Kleinian fiber products

TL;DR

This paper constructs and classifies pseudo-unitarizable representations of noncommutative fiber products associated with higher spin configurations, providing explicit inner products and conditions for unitarizability based on combinatorial data.

Contribution

It introduces a family of representations for noncommutative fiber products, explicitly describes their indefinite inner products, and characterizes unitarizability conditions using combinatorial configurations.

Findings

Constructed a one-parameter family of pseudo-unitarizable modules.

Provided an explicit combinatorial formula for the indefinite inner product signature.

Derived necessary and sufficient conditions for unitarizability of modules.

Abstract

For any $(m,n)$-periodic higher spin six-vertex configuration $\mathscr{L}$, we construct a one-parameter family $\Delta_\xi$ of pseudo-unitarizable representations of the corresponding noncommutative fiber product $\mathcal{A}(\mathscr{L})$ by difference operators acting on the space of sections of a complex line bundle $L_\xi$ over the face lattice $F$. The indefinite inner product is given explicitly in terms of a combinatorial sign function defined on $F$. We prove that each simple integral weight $\mathcal{A}(\mathscr{L})$-module (previously classified by the author, see arXiv:1612.08125) occurs as a submodule in one of these representation spaces. Lastly we give a combinatorial description of the signature of the unique (up to nonzero real multiples) indefinite inner product on any simple integral weight module, in terms of certain eight-vertex configurations canonically attached to $\mathscr{L}$. In particular we obtain necessary and sufficient conditions for such a module to be unitarizable.