Algorithms for representations of quiver Yangian algebras

TL;DR

This paper reviews algorithms for constructing crystal representations of quiver Yangians, which are linked to BPS states in string theory, and demonstrates their application to specific algebra cases.

Contribution

It introduces detailed algorithms for crystal representation construction of quiver Yangians, emphasizing simplicity and potential generalizations to other algebraic structures.

Findings

Algorithms are based on linear algebra and symbolic computation.

Applications demonstrated on specific Yangian algebras.

Potential for generalization to toroidal and elliptic algebras.

Abstract

In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the fact that this subject was originally at the crossroads of algebraic geometry with effective supersymmetric field theories, equivariant toric action simplifies applied calculations drastically. So the sole pre-requisite for this algorithm's implementation is linear algebra. It can be easily taught to a machine with the help of any symbolic calculation system. Moreover, these algorithms may be generalized to toroidal and elliptic algebras and exploited in various numerical experiments with those algebras. We illustrate the work of the algorithms in applications to simple cases of $\mathsf{Y}(\mathfrak{sl}_2)$, $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1})$ and $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$.