Spiders for rank 2 Lie algebras. Kuperberg, G. Communications in Mathematical Physics, 180(1):109–151, September, 1996. arXiv: q-alg/9712003![Spiders for rank 2 Lie algebras [link]](https://bibbase.org/img/filetypes/link.svg "link")
Paper doi abstract bibtex 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, namely A2, B2, and G2. 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.
@article{kuperberg_spiders_1996,
title = {Spiders for rank 2 {Lie} algebras},
volume = {180},
issn = {0010-3616, 1432-0916},
url = {http://arxiv.org/abs/q-alg/9712003},
doi = {10.1007/BF02101184},
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, namely A2, B2, and G2. 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.},
language = {en},
number = {1},
urldate = {2020-12-14},
journal = {Communications in Mathematical Physics},
author = {Kuperberg, Greg},
month = sep,
year = {1996},
note = {arXiv: q-alg/9712003},
keywords = {Mathematics - Combinatorics, Mathematics - Quantum Algebra},
pages = {109--151},
}
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