Volume 21, issue 3 (2021)

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A combinatorial description of the centralizer algebras connected to the Links–Gould invariant

Cristina Ana-Maria Anghel

Algebraic & Geometric Topology 21 (2021) 1553–1593
Abstract

We study the tensor powers of the standard representation of the super-quantum algebra Uq(sl(2|1)), focusing on the rings of its algebra endomorphisms, called centralizer algebras and denoted by LGn. Their dimensions were conjectured by I Marin and E Wagner (Adv. Math. 248 (2013) 1332–1365). We prove this conjecture, describing the intertwiner spaces from a semisimple decomposition as sets consisting of certain paths in a planar lattice with integer coordinates. Using this model, we present a matrix unit basis for the centralizer algebra LGn, by means of closed curves in the plane, which are included in the lattice with integer coordinates.

Keywords
quantum algebra, representation theory, centralizer algebras
Mathematical Subject Classification
Primary: 57K10
Secondary: 16T20, 17B37, 20F36, 57K31
References
Publication
Received: 3 July 2020
Revised: 16 November 2020
Accepted: 27 December 2020
Published: 11 August 2021
Authors
Cristina Ana-Maria Anghel
Mathematical Institute
University of Oxford
Oxford
United Kingdom
https://www.maths.ox.ac.uk/people/cristina.palmer-anghel