## Spiders for rank 2 Lie algebras.(English)Zbl 0870.17005

Summary: 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 $$\text{sl}(2)$$ 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 $$A_2$$, $$B_2$$, and $$G_2$$. 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 François Jaeger.

### MathOverflow Questions:

$$\mathfrak{sl}_3$$ webs without faces having a multiple of 4 sides

### MSC:

 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B20 Simple, semisimple, reductive (super)algebras 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010)

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### References:

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