Orthogonal Euler Pairs in Lie Theory

• Orthogonal Euler pairs are pairs of elements in a Lie algebra (or related structures) that satisfy an involutive condition, linking Euler-type objects with explicit symmetry properties.
• They are classified using Cartan involutions and restricted root systems, allowing for precise parametrizations in simple Lie algebras and beyond.
• Their applications span operator theory, orthogonal polynomial systems, combinatorial design, and AQFT, providing robust frameworks for spectral and geometric analyses.

An orthogonal Euler pair refers to two elements (“Euler elements”) in a finite-dimensional Lie algebra, or two mathematical structures (such as polynomials, operators, or combinatorial designs) that satisfy a precise algebraic or combinatorial orthogonality criterion intimately connected to Euler-type objects. The paradigm originates in Lie theory, where orthogonal Euler pairs $(h, k)$ are pairs of elements in a Lie algebra $\mathfrak{g}$ satisfying $e^{\pi i \operatorname{ad} h} k = -k$, and generalizes to diverse settings including operator theory, orthogonal polynomial systems, and combinatorial matrix design. The paper of orthogonal Euler pairs encompasses their classification in Lie algebras, spectral and metric characterizations in operator algebras, precise recurrence relations in the theory of orthogonal polynomials, and the construction of highly structured combinatorial objects.

1. Algebraic Definition and Lie-Theoretic Foundations

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. An Euler element is a nonzero, diagonalizable element with respect to the adjoint action whose spectrum is contained in $\{-1, 0, 1\}$. A pair $(h, k)$ of Euler elements is called an orthogonal Euler pair if

$e^{\pi i\,\operatorname{ad} h} k = -k.$

This involutive condition translates into the statement that the automorphism $\tau_h := e^{\pi i\,\operatorname{ad} h}$ acts as $-1$ on $k$. When $(h, k)$ and $(k, h)$ are both orthogonal in this sense (i.e., $\tau_h(k) = -k$ and $\tau_k(h) = -h$), $h$ and $k$, together with their Lie bracket $[h, k]$, generate a three-dimensional simple subalgebra isomorphic to $\mathfrak{sl}_2(\mathbb{R})$ (Morinelli et al., 14 Aug 2025).

This mechanism is a categorical generalization of the way the Cartan involution grades semisimple Lie algebras and provides a structural connection to symmetric spaces and modular data in Algebraic Quantum Field Theory (AQFT). The central role of these involutions in the theory is analogous to modular conjugations in the Tomita–Takesaki theory of von Neumann algebras.

2. Classification in Simple Lie Algebras

The classification of orthogonal Euler pairs in simple Lie algebras proceeds by analyzing the adjoint orbits of symmetric Euler elements. Given a fixed symmetric Euler element $h$ (meaning $-h$ is in its adjoint orbit), an explicit classification of all orthogonal Euler elements $k$ (up to conjugation) is achieved via restricted root system theory.

The process is as follows:

• Select a Cartan involution $\theta$ with $\theta(h) = -h$.
• Consider the maximal abelian subspace $\mathfrak{t}_-$ fixed by $-\theta$.
• Identify the subset of roots $\{\gamma_1, \ldots, \gamma_r\}$ in the restricted root system for which $\gamma_j(h) = 1$ and are strongly orthogonal.
• For each $\gamma_j$, a nonzero root vector $e_j$ yields $k_j = e_j - \theta(e_j)$.
• Construct

$k^j := k_1 + \dots + k_j - k_{j+1} - \dots - k_r$

for $j = 0, \ldots, r$.

The classification theorem (Morinelli et al., 14 Aug 2025) states: | Restricted Root System | $k^j$ Representatives | |------------------------|-----------------------------------------------| | $A_{r-1}$ | $k^0, \ldots, k^r$ | | $C_r$ | $k^r$ | | $D_r$ | $k^{r-1}, k^r$ |

Each $k^j$ (along with $h$) completes an orthogonal Euler pair, distinguishing orbits by the structure of the root system. This explicit parametrization connects the algebraic data of the Lie algebra to combinatorial and geometric invariants.

The fundamental group of the adjoint orbit $\mathcal{O}_h$ associated with $h$ is then classified: it is trivial for complex or non-split types, isomorphic to $\mathbb{Z}$ in the hermitian case, and to $\mathbb{Z}_2$ otherwise.

3. Orthogonal Euler Pairs in Operator Algebras and Functional Analysis

Orthogonal Euler pairs find a direct analogue in operator theory as orthonormal (or Pythagoras orthogonal) pairs of operators (Magajna, 2022). Given $A, B \in B(H)$ (bounded operators on a Hilbert space $H$), they form an orthogonal pair if there exists a linear isometry $f$ mapping $\operatorname{span}\{A, B\}$ to $\mathbb{C}^2$, with $A, B$ sent to classical orthonormal vectors. The necessary and sufficient conditions for this isometry descend to norm and spectral characterizations:

• For $A, B$ of norm $1$,

$$|A + X B|^2 = 1 + |X|^2 \quad \forall\, X \in \mathbb{C}$$

or, more operator-theoretically,

$(1 + |X|^2)I - (A + X B)^* (A + X B) \geq 0$

with positivity and singularity at all $X$ (Magajna, 2022).

For commuting normal operators, the joint spectrum must fill the unit half-ball. When generalized to finite sets, the existence of a complete isometry to column Hilbert space requires all normalized operators to attain their supremum on a single state vector.

4. Orthogonal Euler Pairs in Orthogonal Polynomial Systems

In the context of orthogonal polynomials, orthogonal Euler pairs arise as pairs of moment sequences and associated orthogonal polynomial families where the moments are Euler-type sequences. For example, the higher-order Euler polynomials $E_n^{(p)}(x)$, with generating function

$$\left(\frac{2}{e^z + 1}\right)^p e^{xz} = \sum_{n \geq 0} E_n^{(p)}(x) \frac{z^n}{n!},$$

act as “moments” for which a monic orthogonal polynomial system $\Omega_n^{(p)}(y)$ is constructed, explicitly identified as Meixner–Pollaczek polynomials (Jiu et al., 2017):

$$\Omega_{n+1}^{(p)}(y) = \left( y - x + \frac{p}{2} \right) \Omega_n^{(p)}(y) + \frac{n(n+p-1)}{4} \Omega_{n-1}^{(p)}(y).$$

Such pairs $(E_n^{(p)}(x), \Omega_n^{(p)}(y))$ are “orthogonal Euler pairs,” encapsulating both algebraic (recurrence relations, spectral) and combinatorial (lattice path) properties. Matrix representations, continued fractions, and explicit Hankel determinants are available for these sequences (Jiu et al., 2017, Dilcher et al., 2020).

More generally, for any two quasi-definite moment functionals $u_0, u_1$, if $u_0$ is classical, a rational modification produces $u_1$ so that their orthogonal polynomials are connected by an extended coherent (Euler-type) algebraic differential relation (Lee et al., 2023):

$$R_n(x) - d_{n-1} R_{n-1}(x) - e_{n-2} R_{n-2}(x) = Q_n(x) - \sigma_{n-1} Q_{n-1}(x) - \tau_{n-2} Q_{n-2}(x).$$

5. Orthogonality Criteria in Combinatorial and Geometric Structures

In combinatorial design and block matrix theory, orthogonal Euler pairs appear as structures with controlled intersections and block orthogonality. The construction of generalized Euler squares (GES) via finite fields yields block-orthogonal binary matrices $A(n, k, t)$ (Sasmal et al., 2019):

• Each column represents a $k$-tuple with one “1” per block.
• Block structure and GES properties guarantee orthogonality within blocks and low block coherence.
• For columns $v, w$ in the same block, $\langle v, w \rangle = 0$. The block coherence $\mu_B \leq t$ allows for applications in signal processing and sparse recovery.

In geometric representation theory, mutually unbiased bases (MUBs) in quantum information correspond to special orthogonal Euler pairs in $\mathfrak{sl}_n$, where systems of minimal projectors $\{p_i\}$, $\{q_j\}$ verify

$\operatorname{Tr}(p_i q_j) = \frac{1}{n}$

for all $i, j$ (Bondal et al., 2015). These orthogonality conditions match the algebraic and geometric framework developed for Euler pairs in Lie theory.

6. Applications in Algebraic Quantum Field Theory and Modular Duality

The systematic paper of orthogonal Euler pairs is foundational in the theory of Algebraic Quantum Field Theory (AQFT), especially in the modular localization program (Morinelli et al., 14 Aug 2025). The abstract notion of an Euler wedge, $(x, \sigma)$—with $\sigma = e^{\pi i x}$ in an appropriate group extension—enables the translation of causal complement and twisted duality phenomena in the spacetime net of wedges into algebraic terms. Every twisted complement in the abstract wedge space can be realized as a chain of successive complements induced by $3$-dimensional simple subalgebras generated via orthogonal Euler pairs.

Explicitly, the involutive operation

$W' = (-x, \sigma)$

encodes geometric duality (complementation), while the connectedness of twisted complements is governed by central elements produced by exponentiating commutators $[h, k]$ for orthogonal Euler pairs $(h, k)$, with

$\zeta_{h,k} := \exp(2\pi [h, k])$

generating the subgroup parametrizing all twisted complements.

This machinery not only elucidates the geometric and topological structure of covariant nets, but also enables the derivation of commutation and spin–statistics relations in relativistic quantum field models.

7. Related Orthogonality in Linear and Operator Algebra Contexts

In systems identification and control theory, orthogonal Euler pair parameterizations provide minimal, robust representations of linear system pairs $(A, C)$ via products of orthogonal matrices mimicking Euler-angle decompositions (Mullhaupt et al., 2018). The key condition $I - A^* A = C^* C$ ensures output-normality, with the parameterization of the stacked matrix $Q = [C\,;A]$ as a product of $n$ orthogonal blocks, each analogously to an “Euler rotation.” Such representations are optimal for well-conditioned identification algorithms and state update procedures in high-dimensional dynamical systems.

In the analysis and approximation of pairwise comparisons (PC) matrices, orthogonalization based on a generalized Frobenius inner product decomposes a PC matrix into consistent (“Eulerian”) and inconsistent (“orthogonal”) parts (Benitez et al., 18 Mar 2024). This process realizes an orthogonal decomposition in the sense that the consistent subspace is complemented by a (weighted) orthogonal complement in the space of skew-symmetric matrices, directly paralleling orthogonal Euler pair decompositions.

References and Further Reading

This summary draws primarily on (Morinelli et al., 14 Aug 2025) for the Lie algebraic and modular/AQFT structures; (Bondal et al., 2015) for the representation-theoretic and quantum information perspective; (Jiu et al., 2017, Dilcher et al., 2020), and (Lee et al., 2023) for orthogonal polynomials; (Magajna, 2022) for Pythagoras orthogonality in operator theory; (Sasmal et al., 2019) for combinatorial constructions; and (Benitez et al., 18 Mar 2024, Mullhaupt et al., 2018) for applications to matrix analysis and systems identification.

The structural insight underlying orthogonal Euler pairs—namely, the interplay of involutive symmetries, strict orthogonality, and the algebraic or combinatorial invariants they generate—continues to inform research across representation theory, functional analysis, combinatorics, and quantum mathematical physics.