The \mathfrak {sl}_{3}-web algebra |
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Authors: | M Mackaay W Pan D Tubbenhauer |
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Institution: | 1. CAMGSD, Instituto Superior Técnico, Lisbon, Portugal 2. Departamento de Matemática, FCT, Universidade do Algarve, Faro, Portugal 3. Courant Research Center “Higher Order Structures”, University of G?ttingen, G?ttingen, Germany 4. Saint Mary’s College of California, Moraga, CA, USA 5. Mathematisches Institut, Georg-August-Universit?t G?ttingen, G?ttingen, Germany
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Abstract: | In this paper we use Kuperberg’s $\mathfrak {sl}_3$ -webs and Khovanov’s $\mathfrak {sl}_3$ -foams to define a new algebra $K^S$ , which we call the $\mathfrak {sl}_3$ -web algebra. It is the $\mathfrak {sl}_3$ analogue of Khovanov’s arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$ -skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus }_0(\mathcal {W}^S)_{\mathbb {Q}(q)}$ , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that $K^S$ is a graded cellular algebra. |
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