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Spiders for rank 2 Lie algebras

Abstract

A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namelyA 2,B 2, andG 2. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger.

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Additional information

The author was supported by an NSF Postdoctoral Fellowship, grant #DMS-9107908.

Communicated by H. Araki

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Kuperberg, G. Spiders for rank 2 Lie algebras. Commun.Math. Phys. 180, 109–151 (1996). https://doi.org/10.1007/BF02101184

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Keywords

  • Neural Network
  • Statistical Physic
  • Complex System
  • Nonlinear Dynamics
  • Quantum Computing