$X_2$ series of universal quantum dimensions

Abstract

The antisymmetric square of the adjoint representation of any simple Lie algebra is equal to the sum of adjoint and $X_2$ representations. We present universal formulae for quantum dimensions of an arbitrary Cartan power of $X_2$. They are analyzed for singular cases and permuted universal Vogel's parameters. $X_2$ has been the only representation in the decomposition of the square of the adjoint with unknown universal series. Application to universal knot polynomials is discussed.

arXiv comment33 pages

References in corpus (13)

• Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
• On universal knot polynomials
• Universality in Chern-Simons theory
• Nonperturbative universal Chern-Simons theory
• Refined Chern-Simons versus Vogel universality
• On Universal Quantum Dimensions
• Universal volume of groups and anomaly of Vogel's symmetry
• Partition function of Chern-Simons theory as renormalized q-dimension
• Diophantine equations, Platonic solids, McKay correspondence, equivelar maps and Vogel's universality
• On the map of Vogel's plane
• Extending and quantising the Vogel plane
• Exact Chern-Simons / Topological String duality
• Lie algebras in symmetric monoidal categories

Cited by in corpus (2)

• On Universal Eigenvalues of Casimir Operator
• On universal quantum dimensions of certain two-parameter series of representations