Vogel universality and beyond
A. P. Isaev (BLTPh, JINR, Dubna)

TL;DR
This paper develops universal characteristic identities and explicit projectors for simple Lie algebras, enabling a unified description of their representation decompositions using Vogel parameters, except for e8_8.
Contribution
It introduces universal characteristic identities and projectors for simple Lie algebra representations, extending the Vogel parametrization to a broad class of decompositions.
Findings
Derived explicit formulas for invariant projectors in Lie algebra representations.
Established universal identities valid for all simple Lie algebras except e8_8.
Calculated dimensions of Casimir subrepresentations in tensor product decompositions.
Abstract
For simple Lie algebras we construct characteristic identities for split (polarized) Casimir operators in representations and , where -- defining (minimal fundamental for exceptional Lie algebras) representation, -- n-Cartan powers of the adjoint representations and Y_n' -- special representations appeared in the Clebsch-Gordan decomposition of symmetric part of . By means of these characteristic identities, we derive (for all simple Lie algebras, except ) explicit formulae for invariant projectors onto irreducible subrepresentations arose in the decomposition of . These projectors and characteristic identities are written in the universal form for all simple Lie algebras (except ) in terms of Vogel parameters. Universal formulas for the dimensions of the Casimir…
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Taxonomy
TopicsAdvanced Operator Algebra Research · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology
