REVIEW 9 cited by
Casimir eigenvalues for universal Lie algebra
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
Signed reviews
abstract
For two different natural definitions of Casimir operators for simple Lie algebras we show that their eigenvalues in the adjoint representation can be expressed polynomially in the universal Vogel's parameters $\alpha, \beta, \gamma$ and give explicit formulae for the generating functions of these eigenvalues.
Forward citations
Cited by 9 Pith papers
-
$N \leftrightarrow -N$ duality of SU(N) for stable sequences of representations
For stable sequences D(λ,τ) of SU(N) representations, dimensions satisfy dim(D(λ,τ),N) = (-1)^{Area(λ)+Area(τ)} dim(D(τ,λ),-N) and second-order Casimir eigenvalues satisfy C(D(λ,τ),N) = -C(D(λ^T,τ^T),-N).
-
Torus knots in adjoint representation and Vogel's universality
Universal adjoint invariants for T[4,n] torus knots are constructed via Vogel's universality, completing the T[4,n] case after previous T[2,n] and T[3,n] results.
-
Macdonald deformation of Vogel's universality and link hyperpolynomials
For the adjoint square in ADE Lie algebras, products of Macdonald dimensions with deformed Littlewood-Richardson coefficients are universal, yielding universal formulas for T[2,2n] link hyperpolynomials.
-
Construction of Lie algebra weight system kernel via Vogel algebra
Using Vogel's Lambda algebra, the authors construct and explicitly list the first Jacobi diagrams in the kernel of the sl_n weight system, up to order 10 for primitive diagrams.
-
On universal quantum dimensions of certain two-parameter series of representations
One rational expression in Vogel parameters computes quantum dimensions of Cartan powers of adjoint and X2 representations for many simple Lie algebras, though it vanishes in some cases where the representation exists.
-
Diagrammatic technique for Vogel's universality
Vogel's diagrammatic Lambda-algebra enables truly universal computations of Lie-theoretic quantities, demonstrated via multiple examples.
-
On Refined Vogel's universality
For the simply laced simple Lie algebras (A_n, D_n, E_6, E_7, E_8), the adjoint Macdonald dimension is captured by one universal rational function of Vogel's parameters.
-
Vogel's universality and Macdonald dimensions
The paper gives a single rational formula for adjoint Macdonald dimensions that unifies the simply laced Lie algebras A_n, D_n, E6, E7, E8, plus explicit mixed-root-system dimension formulas.
-
On Universal Eigenvalues of Casimir Operator
The second Casimir eigenvalue on powers of the X2 representation tensored with the adjoint is given by one universal formula for all simple Lie algebras, respecting SO/Sp duality.
Discussion (0). Continue with ORCID to comment.