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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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