Quaternionic Forms of Split Rank 4

• Quaternionic forms of split rank 4 are structures defined by split quaternionic algebras exhibiting signature (2,2) and deep symmetry properties.
• They underpin advanced representation theory by enabling the decomposition into discrete, continuous, and minimal series, with applications in automorphic and quantum frameworks.
• Integrating harmonic analysis, differential geometry, and mathematical physics, these forms provide a unifying framework for innovative analytical techniques and computational models.

Quaternionic forms of split rank 4 constitute a fundamental class of structures in representation theory, harmonic analysis, differential geometry, and mathematical physics, with deep connections to split quaternionic algebras, minimal and discrete series representations, exceptional Lie groups, automorphic forms, and even quantum field theory. The split rank 4 condition encodes a symmetry where the underlying space or group exhibits maximal noncompactness in four directions, generating a rich interplay between algebraic, analytic, and geometric properties.

1. Foundations: Split Quaternionic Algebras and Geometric Models

Split quaternions $\mathbb{H}_\mathbb{R}$ differ from Hamilton’s quaternions by having signature (2,2) rather than (4,0); their unit elements form the group $SL(2,\mathbb{R})$ as opposed to $SU(2)$ for classical quaternions. Analytically, this leads to a hyperboloid geometry rather than the compact 3-sphere, shifting the functional-analytic and representation-theoretic framework from compact to noncompact real forms (Frenkel et al., 2010).

Letting $$X = x^0 e_0 + x^1 \tilde{e}_1 + x^2 \tilde{e}_2 + x^3 e_3$$, the split-quaternionic quadratic form is

$N(X) = (x^0)^2 - (x^1)^2 - (x^2)^2 + (x^3)^2.$

This is central in the definition of norm and regularity conditions, and underlies both the group structure and analysis.

Quaternionic Grassmannians and homogeneous models, such as

$$Gr_k(\mathbb{H}^n) = Sp(n)/(Sp(k) \times Sp(n - k)),$$

and their complex analogues provide the algebraic-geometric setting—serving as classifying spaces for quaternionic vector bundles and as arenas where characteristic 4-forms (e.g., symplectic Pontrjagin forms) play a universal role (Datta, 2012).

2. Representation Theory: Discrete, Continuous, and Minimal Series

A defining principle is the decomposition of function spaces on split quaternionic domains into irreducible representations of Lie groups such as $SL(2,\mathbb{R})$ or their conformal extensions (e.g., $SL(4,\mathbb{R}) \cong SO(3,3)$) (Frenkel et al., 2010).

Discrete series emerge as polynomial (matrix coefficient) representations analogous to holomorphic discretizations, while the continuous series component encodes the minimal representation of the conformal group: $$\text{Minimal representation:} \quad K \simeq SO(3) \times SO(3), \quad \text{K-types given by polynomials in spherical harmonics and associated Legendre functions.}$$ The explicit separation of these spectral components, and their realization as modules for algebras such as $$\mathfrak{gl}(2, \mathbb{H}_\mathbb{C}) \simeq \mathfrak{sl}(4, \mathbb{C})$$, is achieved using split quaternionic matrix coefficient expansions and integral projectors (Libine, 2014).

The decomposition

$${\cal D}^h \oplus {\cal D}^a = {\cal D}^h_< \oplus {\cal D}^{--} \oplus {\cal D}^h_> \oplus {\cal D}^a_< \oplus {\cal D}^{++} \oplus {\cal D}^a_>,$$

where each summand is characterized by the range of norm powers and representation indices, is preserved by the natural $\mathfrak{sl}(4, \mathbb{C})$-action and supports explicit intertwining projectors via integral kernel methods (Libine, 2014).

3. Split Quaternionic Analysis: Dirac Operators, Integral Formulas, and Kernels

Split Dirac operators generalize monogenic function theory to the (2,2) setting (Libine, 2010): $$\nabla^+_{R} = e_0 \frac{\partial}{\partial x^0} - \tilde{e}_1 \frac{\partial}{\partial x^1} - \tilde{e}_2 \frac{\partial}{\partial x^2} + e_3 \frac{\partial}{\partial x^3}$$ define regular functions as solutions to $\nabla^+_R f = 0$. The ultrahyperbolic operator $\square_{2,2} = \nabla_R \nabla^+_R$ replaces the Laplacian.

Key tools are split analogues of the Cauchy–Fueter and Poisson integral formulas: $$f(W) = C \int_{\partial U} (Z-W)^{-1} D\mathcal{Z} \, f(Z),$$ with appropriately regularized kernels and contours (e.g., substitution $N(X - W) \mapsto N(X - W) \pm i\epsilon$), yielding projectors onto discrete and continuous representations and realising the separation of series (Frenkel et al., 2010, Libine, 2010).

These formulas encode the Plancherel measure, as the kernel expansions have coefficients involving hyperbolic functions (e.g., $\coth(\pi \operatorname{Im} \lambda)$, $\tanh(\pi \operatorname{Im} \lambda)$), providing new geometric interpretations of harmonic analysis on $SL(2,\mathbb{R})$.

4. Dual Pairs and Minimal Representations in Split Rank 4

Split rank 4 quaternionic forms appear as special real forms of exceptional Lie groups (e.g., $F_{4,4}$, $E_{6,4}$, $E_{7,4}$, $E_{8,4}$) whose geometric and representation-theoretic features are determined by quaternionic symmetries and the split nature of the real form (Bakic et al., 3 Aug 2025).

Minimal representations (e.g., $\sigma_Z$) of these groups are characterized by annihilator ideals (such as the Joseph ideal) and have K-type decompositions

$$\sigma_Z = \bigoplus_{n \geq 0} S^{n+1}(U_2) \otimes \pi_M(n \omega)$$

with explicit branching laws and dual pair correspondences—relating the restriction of quaternionic representations to products of classical and split rank one groups (dual pairs $G \times G'$, $G'$ of type $G_2$).

Short exact sequence computations and coordinate ring methods are critical in describing restrictions, and the associated theta correspondences produce explicit "dictionaries" between representation categories (Bakic et al., 3 Aug 2025).

5. Modular and Automorphic Forms, Fourier–Jacobi Expansions

Split rank 4 quaternionic forms govern the structure of vector-valued quaternionic modular forms, especially for groups like $SO^*(8)$, $Sp(n,\mathbb{H})$, and their congruence subgroups (Freitag et al., 2014, Johnson-Leung et al., 27 Jan 2024). These spaces admit highly structured Fourier–Jacobi expansions respecting Heisenberg parabolic subgroups and cubic norm-induced Heisenberg radicals (Narita, 12 Jan 2025):

• Scalar and vector-valued forms arise as sections over modular varieties tied to congruence subgroups of Hurwitz integers.
• Each nontrivial central Fourier coefficient is shown to belong solely to the continuous spectrum of the Jacobi group; this vanishing is tightly bound to the representation theory of the ambient split quaternionic group.
• The Köcher principle (automatic moderate growth at the archimedean place) for automorphic forms generating quaternionic discrete series is verified in split rank 4 (with exceptions for $G_2$ and $SO(4,N)$ types) (Narita, 12 Jan 2025).

These analytic properties encode foundational spectral decompositions, structure the theory of automorphic L-functions, and control the possible contributions of discrete spectrum in Fourier expansions.

6. Differential Geometry: Invariant Forms, Grassmannians, and Perturbed Structures

Quaternions and split quaternions organize the geometry of Grassmannians, symmetric pairs, and invariant forms in higher-dimensional settings. In particular:

• Quaternionic Grassmannians $Gr_k(\mathbb{H}^n)$ serve as classifying spaces for principal $Sp(k)$-bundles; the universal symplectic Pontrjagin form $$\sigma = p_1(\omega_0) = \operatorname{tr}(\Omega^2)$$ realizes any closed 4-form on a manifold $M$ in the appropriate cohomology class as a pullback from the Grassmannian (Datta, 2012).
• Invariant $4$-forms govern the existence and structure of $s$-representations in symmetric spaces: a real, complex, or quaternionic representation $m$ leads to a symmetric extension

$\tilde{h} = h \oplus sp(1)$

with the invariance condition

$$\operatorname{dim}(\Lambda^4_\mathbb{R} m)^{h \oplus sp(1)} = 1,$$

ensuring the existence of the symmetric extension and providing the algebraic underpinning for many exceptional Lie algebras (Moroianu et al., 2012).

In quaternionic Kähler geometry and its deformation theory, split rank 4 appears in the GL(8,$\mathbb{R}$)-orbit structure of closed $4$-forms, with parameterizations of nilpotent perturbations (e.g., $v: S^2 \to S^4$) reflecting the split structure (Conti et al., 2016).

7. Quantum Deformations, Lattice Theory, and Mathematical Physics

Quantizations of symmetry algebras in split rank 4, such as $$\mathfrak{o}^*(4) \cong \mathfrak{o}(2,1) \oplus \mathfrak{o}(3)$$, require a detailed classification of classical $r$-matrices and the construction of suitable universal quantum $R$-matrices: $$r_1(\gamma, \chi, \eta) = \gamma E_+ \wedge E_- + i\chi E_+ \wedge H + \eta E_+ \wedge H$$ subject to anti-Hermiticity under quaternionic conjugation (Borowiec et al., 2015, Borowiec et al., 2017). The corresponding quantum groups have applications in integrable models, noncommutative geometry, and quantum gravity.

In lattice theory, the automorphism group of high-symmetry lattices such as the Barnes–Wall lattice $BW_{16}$ can be constructed as subalgebras of high-rank tensor products of Hurwitz quaternionic integers, with explicit generators in $H^{\otimes 4}$ encoding the full automorphism structure (Ohta, 2022).

In mathematical physics, the kernels appearing in split quaternionic harmonic analysis correspond to massless propagators in four-dimensional field theory, directly tying harmonic analysis to Feynman diagrammatics and singularity theory (Frenkel et al., 2010).

Table: Key Structures and Their Roles in Split Rank 4 Quaternionic Forms

Structure	Description / Role	Source
Split Quaternions $\mathbb{H}_\mathbb{R}$	Basic algebraic setting, signature (2,2)	(Frenkel et al., 2010)
Minimal Representation	Realized in harmonic analysis, lowest GK dimension	(Frenkel et al., 2010)
Cayley Transform (Split)	Transfers analysis between $\mathbb{H}_\mathbb{R}$ and M (Minkowski)	(Frenkel et al., 2010)
Cauchy–Fueter/Poisson Formulas	Integral projectors, separation of series	(Libine, 2010)
S-representations and Symmetric Pairs	Extension of isotropy algebra via $sp(1)$	(Moroianu et al., 2012)
Automorphic Forms, Fourier–Jacobi	Modular forms, moderate growth, split spectra	(Narita, 12 Jan 2025)
Quantum Deformations	Classification of $r$-matrices, quantum groups	(Borowiec et al., 2015)
Grassmannians, Universal Forms	Classifying spaces, realization of 4-forms	(Datta, 2012)

8. Connections to Hyperbolic Geometry, Spin and Noncommutative Structures

Quaternionic forms of split rank 4 are integral in higher-dimensional hyperbolic and conformal geometry. Recent work establishes explicit correspondences between pairs of quaternions (“spinors”), flags in Minkowski space, and decorated horospheres in hyperbolic 4-space. Lambda lengths generalize to quaternionic values, with a noncommutative Ptolemy equation arising from quasi-Plücker relations, connecting geometric, algebraic, and topological structures (Mathews et al., 9 Dec 2024).

Such constructions extend lower-dimensional Teichmüller theory and cluster algebra principles into the quaternionic, noncommutative regime, offering new tools for topology, higher Teichmüller theory, and mathematical physics.

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In aggregate, quaternionic forms of split rank 4 provide a framework that unifies diverse structures—algebraic (split quaternion algebras, exceptional groups), analytic (harmonic and automorphic analysis, Fourier–Jacobi expansions), geometric (Grassmannians, symmetric pairs, hyperbolic and quaternionic Kähler geometry), and quantum (quantum groups, noncommutative geometry), with further ramifications in mathematical physics and number theory. The split rank 4 perspective ensures maximal interplay between compact and noncompact directions, enabling both detailed structural results and concrete computational/analytic techniques.