Cubic Norm Pairs: Structures and Applications

• Cubic norm pairs are dual modules endowed with cubic norms, quadratic adjoints, and trace forms, establishing core multilinear identities and compatibility conditions.
• They extend classical Jordan algebra theory and structurable algebras by providing explicit coordinatizations of root-graded Lie algebras, including types G₂ and F₄.
• Cubic norm pairs underpin advanced constructions in algebraic groups and operator Kantor pairs, with applications spanning nonassociative spectral theory and arithmetic geometry.

A cubic norm pair is a dual pair of modules equipped with rich multilinear structure: cubic norms, quadratic adjoint maps, trace pairings, and compatibility identities that generalize both cubic norm structures of Jordan algebra theory and hermitian cubic norm structures relevant to structurable algebras and their attendant Lie theory. The theory of cubic norm pairs unifies and extends constructions arising in algebraic groups of types $G_2$ and $F_4$, structurable algebras of skew-dimension one, and operator Kantor pairs, and is central to the explicit coordinatization of root-graded Lie algebras over arbitrary commutative rings (Smet, 10 Jan 2025, Medts et al., 5 Feb 2026, Medts, 2017).

1. Foundations and Definition

Let $k$ be a commutative ring. A cubic norm pair consists of a pair of $k$-modules $(J, J')$ together with maps

• $N: J \to k$ (cubic norm), $N': J' \to k$,
• $\#: J \to J'$ and $\#': J' \to J$ (quadratic adjoints),
• $T: J \times J' \to k$, $T': J' \times J \to k$ (bilinear trace forms),
• bilinear products $\times: J \times J \to J'$, $\times': J' \times J' \to J$ (polarizations of $\#$ and $\#'$),

satisfying a set of axioms (using polarizations and multilinearizations of $N$ and the adjoints) that encode the interaction of norms, adjoints, and traces (Medts et al., 5 Feb 2026, Smet, 10 Jan 2025): $\begin{aligned} &(1)\ T(a, b^\#) = N^{(1,2)}(a, b) \ &(2)\ (a^\#)^{\#'} = N(a)\, a \ &(3)\ (a^\# \times' c) \times a = N(a) c + T(a, c) a^\# \ &(4)\ N(Q_a(c)) = N(a)^2 N'(c), \quad Q_a(c) = T(a, c)a - a^\# \times' c \ &\text{and the dual identities interchanging %%%%16%%%% with %%%%17%%%%.} \end{aligned}$

If there is a distinguished $1 \in J$ with $N(1) = 1$, the pair is called unital. In this case, one recovers the usual cubic norm structure as a special case via the identification $J \simeq J'$ (Smet, 10 Jan 2025).

2. Structural Identities and Jordan Pair Correspondence

The axioms of a cubic norm pair imply a system of polynomial and multilinear identities, including:

• $(x + y)^\# = x^\# + x \times y + y^\#$
• $N(x + y) = N(x) + T'(x^\#, y) + T(x, y^\#) + N(y)$
• $x^{\#\#} = N(x)x$
• $T(x, x^\#) = 3T(x)$

The construction of U-operators,

$U_a(b') = T(a, b') a - a^\# \times b',$

equips $(J, J')$ with the structure of a quadratic Jordan pair. In fact, giving a cubic norm pair is equivalent to giving a quadratic Jordan pair together with a divided power representation (Faulkner–Loos theory) with appropriate binomial identities (Medts et al., 5 Feb 2026, Smet, 10 Jan 2025).

3. Canonical Examples and Specializations

Classical and new examples of cubic norm pairs include:

• For $J = M_3(k)$, the determinant gives the cubic norm, the adjugate map is the adjoint, and the trace form is the usual matrix trace.
• The Albert algebra, $J = \mathrm{Her}_3(C)$ (hermitian $3 \times 3$ matrices over a composition algebra $C$), with cubic norm as the determinant and adjoint as the polarized adjugate (Medts et al., 5 Feb 2026).
• Hermitian cubic norm structures correspond to the situation where $J$ is a module over a quadratic étale extension $K/R$ with a $\bar{~}$-semilinear adjoint, a $K$-valued cubic norm, and a $\sigma$-hermitian trace, satisfying analogous (but twisted) identities (Medts, 2017, Smet, 10 Jan 2025). Ordinary cubic norm pairs are recovered as the split ($K \cong R \times R$) case.

Degenerate cases (such as $J = R_1$, $J' = R_2$ with $N = 0$) and new non-classical examples via descent and base change are included within this formalism (Smet, 10 Jan 2025).

4. Associated Structurable Algebras and Lie Theory

Given a cubic norm pair $(J, J')$, one constructs a structurable algebra as the $2 \times 2$ matrix algebra: $$\mathcal{A} = \begin{pmatrix} k & J \ J' & k \end{pmatrix}$$ with multiplication and involution defined using $T$, $\times$, and their duals (Smet, 10 Jan 2025): $$\begin{pmatrix} a & b \ c & d \end{pmatrix} \cdot \begin{pmatrix} e & f \ g & h \end{pmatrix} = \begin{pmatrix} ae + T(b, g) & a f + b h + c \times g \ c e + d g + b \times f & d h + T(c, f) \end{pmatrix}$$ and

$$\overline{ \begin{pmatrix} a & b \ c & d \end{pmatrix} } = \begin{pmatrix} d & b \ c & a \end{pmatrix} .$$

This algebra is structurable, with skew part corresponding to the diagonal subspace. The Kantor triple product $\{x, y, z\} = (x\bar y)z + (z\bar y)x - (z\bar x)y$ yields a Kantor triple system, from which the Tits–Kantor–Koecher (TKK) Lie algebra is constructed. In the hermitian cubic norm structure case, the algebra has the form $K \oplus J$ with product $$(k, j)\cdot(k', j') = (kk' + T(j, j'),\; k j' + \overline{k'} j + j \times j')$$ and involution $(\bar{k}, j)$ (Smet, 10 Jan 2025, Medts, 2017).

The associated $5$-graded Lie algebra encapsulates the algebraic and geometric automorphisms of the structurable algebra and is a key tool in nonassociative and Lie theory (Smet, 10 Jan 2025, Medts et al., 5 Feb 2026).

5. Root-Graded Lie Algebras and Algebraic Groups

The theory of cubic norm pairs provides explicit coordinatizations of root-graded Lie algebras:

• For any CNP $(J, J')$, the TKK construction yields a Lie algebra of type $G_2$ graded by long and short roots, with explicit descriptions of root spaces and dimensions (long roots are 1-dimensional, short roots correspond to copies of $J$ or $J'$) (Medts et al., 5 Feb 2026).
• When $(J, J)$ arises from a cubic Jordan matrix algebra, a refinement to an $F_4$-grading exists, matching the 52 root spaces of $F_4$ and their structure via the Tits index (map $F_4 \to G_2$) and Freudenthal–Springer–Tits formulas.

The exponential maps $$\exp_\alpha(a) = \sum_{i=0}^4 \frac{1}{i!} \mathrm{ad}_a^i$$ define root subgroups $U_\alpha$ that assemble to form crystallographic root-graded groups and Chevalley–Demazure type algebraic groups over arbitrary $k$, providing functoriality and uniformity for split and twisted forms (Medts et al., 5 Feb 2026, Smet, 10 Jan 2025).

The automorphism group of the associated Lie algebra contains the Chevalley–Demazure group of type $G_2$ or $F_4$, depending on the example.

6. Operator Kantor Pairs and Classification

Cubic norm pairs are in bijection with certain operator Kantor pairs, functorial extensions of the classical Kantor pair theory suited for root-graded and Moufang-identity settings. Specifically, operator Kantor pairs whose TKK–Lie algebra admits a split $G_2$-grading (with long-root subpair isomorphic to the elementary $3 \times 3$ pair) are specified precisely by cubic norm pairs (Smet, 10 Jan 2025).

A crucial classification result is that hermitian cubic norm structures are exactly the skew-dimension one forms of unital cubic norm structures, obtainable via Galois descent, and that every central simple skew-dimension one structurable algebra arises from exactly one hermitian cubic norm structure (Medts, 2017, Smet, 10 Jan 2025).

7. Connections, Isotopies, and Applications

The theory of cubic norm pairs interplays naturally with classical isotopy (non-linear automorphisms), structure groups (autotopy groups), and descent forms (Medts, 2017, Smet, 10 Jan 2025). Notably:

• The Cayley–Dickson process for certain structurable algebras is equivalent to the base change from a cubic norm structure to its hermitian twist over a quadratic étale extension.
• Cubic norm pairs provide a foundation for higher-order and nonassociative spectral theory, encoding “spectral radius” and nilpotence criteria for higher tensors (see Gelfand limit in third-order spectral theory) (Qi et al., 2019).
• Explicit correspondence exists with norm form equations in arithmetic geometry, as seen in rational solutions to $f(t) = N_{K/\mathbb{Q}}(x)$ for cubic $f$ and field norms $N$ (Irving, 2013).

This web of connections makes cubic norm pairs central objects in the structure theory of Lie algebras, algebraic groups, Jordan and structurable algebra theory, and arithmetic geometry.

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References:

(Smet, 10 Jan 2025): "Cubic norm pairs and hermitian cubic norm structures" (Medts et al., 5 Feb 2026): "From cubic norm pairs to $G_2$- and $F_4$-graded groups and Lie algebras" (Medts, 2017): "Structurable algebras of skew-dimension one and hermitian cubic norm structures" (Qi et al., 2019): "Tensor Norm, Cubic Power and Gelfand Limit" (Irving, 2013): "Cubic polynomials represented by norm forms"