Invariant Projectors in Lie Theory

• Invariant projectors are canonical linear operators in representation theory that commute with the Lie algebra action and project onto specific invariant subspaces.
• They are constructed as spectral projectors using the split Casimir operator and expressed universally as rational functions via Vogel parameters.
• Their applications include decomposing tensor powers, constructing invariant R-matrices for Yang–Baxter solutions, and deriving quantum knot invariants.

An invariant projector is a canonical, representation-theoretic object: a linear operator on a tensor power of a group or algebra representation (e.g., $V^{\otimes n}$), which commutes with the diagonal action of the symmetry group or algebra, and projects onto a specific invariant (often irreducible) subspace. In the modern context of Lie theory and quantum algebra, invariant projectors are constructed as explicit rational functions—often polynomials—of central/mutually commuting operators, most notably the split (polarized) Casimir operators. The most advanced formulation, due to Vogel, expresses all data universally for the class of complex simple Lie algebras (classical and exceptional) as rational functions of three parameters $(\alpha,\beta,\gamma)$, called Vogel parameters. This framework allows for uniform formulae for invariant projectors in low tensor powers of the adjoint and defining representations, and generalizes further to superalgebras and quantum deformations.

1. Universal Construction of Invariant Projectors

Invariant projectors are built as spectral projectors of $G$-invariant (or $\mathfrak{g}$-invariant) self-adjoint operators that commute with the action on tensor powers of a representation. In the context of simple Lie algebras, the split (polarized) Casimir operator is the central tool: for the adjoint representation,

$$C = g^{ab}\,X_a \otimes X_b \in \operatorname{End}(\mathfrak{g} \otimes \mathfrak{g})$$

where $g^{ab}$ is the inverse Killing form and $X_a$ is a basis of $\mathfrak{g}$. For simple Lie algebras, universal factorized characteristic identities for the Casimir operator and its higher tensor analogs allow one to write the minimal polynomial in terms of Vogel parameters, with all roots corresponding to the irreducible subrepresentations. The general principle is

$$P_\lambda = \prod_{\mu \neq \lambda} \frac{C - \mu \mathbf{1}}{\lambda - \mu}$$

where $P_\lambda$ projects onto the invariant subspace with Casimir eigenvalue $\lambda$. This simple Lagrange interpolation construction, rendered universal in the Vogel parametrization, produces the complete set of invariant projectors for (adjoint) tensor squares and cubes, as well as for mixed tensor products $T \otimes Y_n$ ($T$ fundamental, $Y_n$ Cartan powers of adjoint) for all but the most exceptional cases (Isaev et al., 2022, Isaev et al., 2020, Isaev et al., 2021, Isaev, 4 Jan 2026).

2. Vogel Parametrization and Universality

Vogel parametrization assigns to each complex simple Lie algebra a triple of numbers $(\alpha,\beta,\gamma)$, defined up to scale and permutation, with $t = \alpha+\beta+\gamma$ (dual Coxeter number) fixing the scale. All group-theoretic quantities (representation dimensions, Casimir eigenvalues, etc.) become rational functions of these parameters. For example, the dimension of the adjoint is

$$\dim_\mathrm{ad} = \frac{(2t-\alpha)(2t-\beta)(2t-\gamma)}{\alpha\beta\gamma}$$

and all relevant subrepresentations in (anti)symmetric squares or cubes of adjoint (or fundamental) also admit rational formulas for their dimensions and Casimir spectra. This universality extends to projectors: $$P_i = \prod_{j\neq i} \frac{C - a_j \mathbf{1}}{a_i - a_j}$$ where the $a_j$ are the Vogel-universal eigenvalues of the split Casimir on the relevant representation. For higher tensors, characteristic identities in terms of Vogel data yield, in each Young symmetry sector, polynomials whose roots directly provide the spectral projectors onto irreducibles (Isaev et al., 2022, Isaev, 4 Jan 2026, Westbury, 2015).

Table 1. Typical Vogel parameters for classical and exceptional Lie algebras

Algebra	$\alpha$	$\beta$	$\gamma$	$t$
$A_n$ ($\mathfrak{sl}_{n+1}$)	$-2$	$2$	$n+1$	$n+1$
$B_n$ ($\mathfrak{so}_{2n+1}$)	$-2$	$4$	$2n-3$	$2n-1$
$C_n$ ($\mathfrak{sp}_{2n}$)	$-2$	$1$	$n+2$	$n+1$
$G_2$	$-2/3$	$10/3$	$8/3$	$16/3$
$F_4$	$-2$	$5$	$6$	$9$
$E_6$	$-2$	$6$	$8$	$12$
$E_8$	$-2$	$12$	$20$	$30$

(Westbury, 2015, Isaev, 4 Jan 2026)

3. Explicit Projector Formulas in Low Tensor Powers

In the adjoint tensor square, the split Casimir $C$ has eigenvalues $-1$, $-\tfrac{\alpha}{2t}$, $-\tfrac{\beta}{2t}$, $-\tfrac{\gamma}{2t}$, $-\tfrac12$, and $0$, corresponding to the singlet, three symmetric-square irreducibles $Y(\alpha)$, $Y(\beta)$, $Y(\gamma)$, the adjoint itself, and the antisymmetric-square component $X_2$. The explicit projectors are (Isaev et al., 2020, Westbury, 2015, Isaev et al., 2021): $$P_i = \prod_{j\neq i} \frac{C - a_j \mathbf{1}}{a_i - a_j}$$ Observables (traces, dimensions, etc.) constructed from these projectors are universal rational functions of $(\alpha,\beta,\gamma)$.

In the adjoint cube, the 3-split Casimir $C(3)$, projected into each Young symmetry sector, yields universal characteristic polynomials, whose roots (linear in $t$, $\alpha$, $\beta$, $\gamma$) define the sectors’ irreducibles and enable direct construction of the corresponding projectors (Isaev et al., 2022).

For mixed products such as $T \otimes Y_n$ (fundamental by Cartan power of adjoint), a universal four-factor minimal polynomial yields four explicit projectors—though, in classical cases, one or more summands may vanish, leaving three nontrivial irreducible projectors (Isaev, 4 Jan 2026).

4. Projectors for Lie Superalgebras and Quantum Deformations

The same construction extends to the Lie superalgebras $\mathfrak{sl}(M|N)$ and $\mathfrak{osp}(M|N)$, with Vogel parameter triples defined appropriately ($t$ becomes the dual Coxeter number for superalgebras). Universal characteristic identities for split Casimir operators in both defining and adjoint representations lead to the same Lagrange interpolation expressions for projectors, and dimensions of irreducibles are again rational in Vogel data (Isaev et al., 2022).

Quantum deformations replace the usual Casimir with suitable $q$-analogues, and all eigenvalue and projector formulas acquire $q$-polynomial forms, maintaining universality—e.g., quantum dimensions and spectral decomposition of the quantum $R$-matrix (Westbury, 2015).

5. Applications: Tensor Decompositions, Knot Invariants, Yang–Baxter Solutions

The uniform construction of invariant projectors is indispensable in explicitly decomposing tensor powers of $G$- or $\mathfrak{g}$-modules—crucial for:

• Computing universal formulas for the decomposition of tensor powers into irreducible $G$-modules, including all multiplicities.
• Constructing $G$- or $\mathfrak{g}$-invariant $R$-matrices as spectral decompositions in terms of projectors onto irreducible summands; these $R$-matrices are universal solutions to the Yang–Baxter equation in both defining and adjoint settings (Isaev et al., 2021, Isaev et al., 2022).
• Recovering and generalizing universal knot invariants (Chern–Simons, HOMFLY, Kauffman, and refined invariants), whose structure constants and eigenvalues are directly parameterized in the Vogel framework (Westbury, 2015, Krefl et al., 2013).
• Extending Vogel plane technology to symmetric spaces and their quantum/super counterparts, via projective and quantized extensions.

6. Universality Principle and Impact

The central achievement is the universality principle: all invariant projectors, spectra, traces, and (super)dimensions in low-degree tensor powers are not merely specific to an individual Lie algebra but become rational functions of $(\alpha,\beta,\gamma)$, with the actual algebra/class only reflected in the chosen triple. This elevates the decomposition of tensor products and the analysis of Casimir operators to a universal algebraic problem, simultaneously encoding the full classical and exceptional series (with rare exceptions, e.g., $E_8$ in certain mixed tensor products).

The construction is directly applicable to Deligne-type series, quantum knot invariants, dimension formulas in Chern–Simons theory, and solutions of Yang–Baxter equations. The canonical invariant projectors become building blocks for any representation-theoretic or quantum group computation requiring explicit understanding of subrepresentation structure (Isaev et al., 2022, Westbury, 2015, Krefl et al., 2013, Isaev et al., 2020, Isaev, 4 Jan 2026, Isaev et al., 2021, Isaev et al., 2022).

7. Limitations and Open Directions

While the framework is essentially complete for all classical simple Lie algebras and the vast majority of exceptional cases, there are notable boundaries:

• In the decomposition $T \otimes Y_n^\prime$ for $E_8$, Vogel-universal expressions for projectors fail (Isaev, 4 Jan 2026).
• In higher tensor degrees and for quantum/super-extensions, explicit universal projector formulas become algebraically complex but remain, in principle, reachable via the Lagrange interpolation/spectral projection method.

A plausible implication is that further extension of this technology may yield universal formulas for more general symmetric spaces, arbitrary degree Casimirs, and possibly invariants related to categorification and quantum topology.

---

References: (Isaev et al., 2022, Westbury, 2015, Isaev et al., 2020, Isaev et al., 2021, Krefl et al., 2013, Isaev, 4 Jan 2026, Isaev et al., 2022)