Quadratic Rank-Two Jacobi Algebra

• Quadratic Rank Two Jacobi Algebra is a quadratic algebra generated by five operators, capturing both differential and multiplication operations for bivariate Jacobi polynomials.
• It integrates multiple rank-one Jacobi and Racah subalgebras, enabling structured basis changes and overlap coefficients via a pentagonal organization.
• Its dual realization showcases bispectrality by linking variable-space differential equations with degree-space recurrence relations in superintegrable models.

Searching arXiv for the core papers on the rank-two/quadratic Jacobi algebra and closely related contexts. The quadratic rank two Jacobi algebra is the quadratic algebra generated by the bispectral operators of the two-variable Jacobi polynomials orthogonal on the triangle. In its standard presentation, it is generated by five operators,

$L,\qquad L_1,\qquad L_3,\qquad X_1,\qquad X_3,$

where $L,L_1,L_3$ are differential operators and $X_1=x$, $X_3=1-x-y$ are multiplication operators. It is called quadratic because its nested commutators close on polynomial expressions of degree at most two in the generators, and rank two because it governs a genuinely bivariate Jacobi system rather than the classical one-variable case. In recent work, the same algebra has been identified both as the structural algebra behind the two-variable Jacobi polynomials on the triangle and as the dynamical algebra of the generic quadratic superintegrable model on the two-sphere (Crampe et al., 10 Jul 2025, Crampé et al., 9 Sep 2025, Crampé et al., 28 Apr 2026).

1. Rank-one antecedent and passage to rank two

The immediate antecedent of the rank-two theory is the rank-one quadratic Jacobi algebra associated with the hypergeometric operator

$$L=x(1-x)\partial_x^2+\bigl(a+1-(a+b+2)x\bigr)\partial_x,$$

together with the multiplication operator $X$, $Xf(x)=x f(x)$. In that setting, the generators

$K_1=L,\qquad K_2=X,\qquad K_3=[K_1,K_2]$

satisfy quadratic relations, encode simultaneously the differential equation and three-term recurrence of the Jacobi polynomials, and admit a central Casimir-like element. The same work shows that tridiagonalization of the hypergeometric operator produces the quadratic Racah–Wilson algebra; Wilson polynomials arise in the infinite-dimensional spectral problem, Racah polynomials in finite-dimensional truncations, and Hahn polynomials in a degenerate first-order limit (Genest et al., 2015).

This rank-one picture is the conceptual template for the rank-two construction. In the bivariate case, the algebra no longer closes on a single differential operator and one multiplication operator. Instead, closure requires five generators and reveals, inside one structure, several rank-one Jacobi subalgebras together with a rank-one Racah subalgebra. This is not merely an analogy: the later rank-two papers make those subalgebras explicit and use them to organize basis changes, overlap coefficients, and symmetry transformations (Crampe et al., 10 Jul 2025).

2. Definition from the bivariate Jacobi system

The defining polynomial family is the two-variable Jacobi system on the triangle,

$$J_{n,k}^{(a,b,c)}(x,y) = J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right), \qquad n\ge k\ge 0,$$

orthogonal on

$0\le x,\qquad 0\le y,\qquad x+y\le 1.$

These polynomials satisfy two fundamental differential eigenvalue equations,

$L,L_1,L_3$0

$L,L_1,L_3$1

with

$L,L_1,L_3$2

$L,L_1,L_3$3

The recurrence operators are simply multiplication by the coordinate functions

$L,L_1,L_3$4

Closure of the commutator algebra generated by $L,L_1,L_3$5 forces the introduction of a further second-order operator,

$L,L_1,L_3$6

The total operator decomposes as

$L,L_1,L_3$7

and one may also introduce

$L,L_1,L_3$8

The quadratic rank two Jacobi algebra is then defined as the algebra generated by

$L,L_1,L_3$9

under commutators, with all triple commutators expressible as quadratic polynomials in these generators (Crampe et al., 10 Jul 2025).

3. Quadratic closure and defining relations

The algebra is characterized by a mixture of commuting pairs, first commutators, and quadratic triple-commutator identities. Its basic commuting relations are

$X_1=x$0

The primary nontrivial commutators are

$X_1=x$1

with explicit first-order operators

$X_1=x$2

$X_1=x$3

$X_1=x$4

$X_1=x$5

Quadraticity appears at the next level. For example,

$X_1=x$6

and

$X_1=x$7

The appendix formulation gives the same phenomenon in nested-commutator form: $X_1=x$8

$X_1=x$9

with analogous formulas for the pairs $X_3=1-x-y$0, $X_3=1-x-y$1, $X_3=1-x-y$2, and $X_3=1-x-y$3 (Crampe et al., 10 Jul 2025, Crampé et al., 9 Sep 2025).

The pair $X_3=1-x-y$4 furnishes the Racah-type sector. Its defining relations include

$X_3=1-x-y$5

$X_3=1-x-y$6

with the omitted terms given explicitly in the defining presentation. Because the central element in this sector is $X_3=1-x-y$7, the resulting structure is a centrally extended rank-one Racah algebra rather than an isolated Racah algebra. This is the algebraic source of the Racah connection coefficients that appear in the polynomial theory.

4. Subalgebras, pentagonal organization, and overlap interpretation

A central structural result is that the algebra contains several rank-one Jacobi algebras and one rank-one Racah algebra. The rank-one Jacobi sectors arise from the pairs $X_3=1-x-y$8, $X_3=1-x-y$9, $$L=x(1-x)\partial_x^2+\bigl(a+1-(a+b+2)x\bigr)\partial_x,$$0, and $$L=x(1-x)\partial_x^2+\bigl(a+1-(a+b+2)x\bigr)\partial_x,$$1, each with an appropriate central parameter or shifted normalization. The pair $$L=x(1-x)\partial_x^2+\bigl(a+1-(a+b+2)x\bigr)\partial_x,$$2 realizes the centrally extended rank-one Racah algebra. This makes the rank-two algebra a composite object in which two nested Jacobi mechanisms are coupled by a Racah mechanism (Crampe et al., 10 Jul 2025).

The same structure is organized in later work through a pentagonal graph of five maximal abelian subalgebras,

$$L=x(1-x)\partial_x^2+\bigl(a+1-(a+b+2)x\bigr)\partial_x,$$3

After introducing

$$L=x(1-x)\partial_x^2+\bigl(a+1-(a+b+2)x\bigr)\partial_x,$$4

one obtains an equivalent cyclic picture involving $$L=x(1-x)\partial_x^2+\bigl(a+1-(a+b+2)x\bigr)\partial_x,$$5, $$L=x(1-x)\partial_x^2+\bigl(a+1-(a+b+2)x\bigr)\partial_x,$$6, and related pairs. The four edge-centralizers are centrally extended rank-one Jacobi algebras, while the bottom centralizer $$L=x(1-x)\partial_x^2+\bigl(a+1-(a+b+2)x\bigr)\partial_x,$$7 is a centrally extended rank-one Racah algebra (Crampé et al., 9 Sep 2025).

This pentagonal organization governs basis changes. The “top” basis $$L=x(1-x)\partial_x^2+\bigl(a+1-(a+b+2)x\bigr)\partial_x,$$8 diagonalizes $$L=x(1-x)\partial_x^2+\bigl(a+1-(a+b+2)x\bigr)\partial_x,$$9 and $X$0, while the “bottom-left” basis $X$1 diagonalizes $X$2 and $X$3, and the “bottom-right” basis diagonalizes $X$4 and $X$5. Moving along an edge corresponds to a rank-one Jacobi overlap, so the standard product formula for $X$6 is recovered by factoring the top-to-bottom overlap through an intermediate basis. The mutual overlap of the two bottom bases is governed by Racah polynomials. This explains why permutations of the variables $X$7 and the parameters $X$8 produce Racah expansion coefficients. The same framework encodes the order-three symmetry associated with the dihedral group $X$9, acting on the three polynomial families

$Xf(x)=x f(x)$0

(Crampé et al., 9 Sep 2025).

5. Dual realizations, bispectrality, and superintegrable dynamics

The algebra has two complementary realizations. In the variable representation, the generators act as differential or multiplication operators on functions of $Xf(x)=x f(x)$1. In the degree representation, they act as difference operators on the polynomial basis $Xf(x)=x f(x)$2, where

$Xf(x)=x f(x)$3

The diagonal operators are

$Xf(x)=x f(x)$4

The operator $Xf(x)=x f(x)$5 is tridiagonal in $Xf(x)=x f(x)$6, $Xf(x)=x f(x)$7 is tridiagonal in $Xf(x)=x f(x)$8, and $Xf(x)=x f(x)$9 acts by a nine-term relation mixing both $K_1=L,\qquad K_2=X,\qquad K_3=[K_1,K_2]$0 and $K_1=L,\qquad K_2=X,\qquad K_3=[K_1,K_2]$1. The same commutation relations hold in the degree model: $K_1=L,\qquad K_2=X,\qquad K_3=[K_1,K_2]$2 for each generator $K_1=L,\qquad K_2=X,\qquad K_3=[K_1,K_2]$3. This is the algebraic form of bispectrality: differential equations in the variables and recurrence relations in the degrees are encoded by one and the same quadratic algebra (Crampe et al., 10 Jul 2025).

The commutator algebra also yields explicit structure relations. Among the three main examples are

$K_1=L,\qquad K_2=X,\qquad K_3=[K_1,K_2]$4

$K_1=L,\qquad K_2=X,\qquad K_3=[K_1,K_2]$5

$K_1=L,\qquad K_2=X,\qquad K_3=[K_1,K_2]$6

These are multivariate analogues of classical rank-one structure relations.

A further realization identifies the same rank-two Jacobi algebra as the dynamical algebra of the generic second-order superintegrable model on the unit two-sphere. In that setting, the Hamiltonian is realized inside $K_1=L,\qquad K_2=X,\qquad K_3=[K_1,K_2]$7, the symmetry algebra is the rank-one Racah algebra generated by intermediate Casimirs, and adjoining the position-like operators

$K_1=L,\qquad K_2=X,\qquad K_3=[K_1,K_2]$8

produces the full dynamical algebra $K_1=L,\qquad K_2=X,\qquad K_3=[K_1,K_2]$9. In the differential model,

$$J_{n,k}^{(a,b,c)}(x,y) = J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right), \qquad n\ge k\ge 0,$$0

and on the sphere one imposes $$J_{n,k}^{(a,b,c)}(x,y) = J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right), \qquad n\ge k\ge 0,$$1. The exact spectrum is

$$J_{n,k}^{(a,b,c)}(x,y) = J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right), \qquad n\ge k\ge 0,$$2

with degeneracy $$J_{n,k}^{(a,b,c)}(x,y) = J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right), \qquad n\ge k\ge 0,$$3. The physical wavefunctions are

$$J_{n,k}^{(a,b,c)}(x,y) = J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right), \qquad n\ge k\ge 0,$$4

so the two-variable Jacobi polynomials arise directly as matrix elements of the rank-two Jacobi algebra representation (Crampé et al., 28 Apr 2026).

6. Terminological scope and related but distinct uses

The phrase “rank-two Jacobi algebra” is not unique to the bivariate orthogonal-polynomial setting. One distinct usage appears in the theory of invariant differential operator algebras for scalar slash actions of real Jacobi groups. There the relevant algebra is

$$J_{n,k}^{(a,b,c)}(x,y) = J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right), \qquad n\ge k\ge 0,$$5

and the rank-two case is the first genuinely higher-rank example beyond the classical rank-one situation. For invertible index $$J_{n,k}^{(a,b,c)}(x,y) = J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right), \qquad n\ge k\ge 0,$$6, this invariant differential operator algebra is noncommutative, generated by operators of orders $$J_{n,k}^{(a,b,c)}(x,y) = J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right), \qquad n\ge k\ge 0,$$7 and $$J_{n,k}^{(a,b,c)}(x,y) = J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right), \qquad n\ge k\ge 0,$$8, and its center is a polynomial algebra in one variable generated by the image of the Jacobi Casimir operator. In rank exceeding $$J_{n,k}^{(a,b,c)}(x,y) = J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right), \qquad n\ge k\ge 0,$$9 there are exactly two characters, in contrast with the richer rank-one case (Conley et al., 2014).

Another distinct usage appears in the theory of weak Jacobi forms indexed by integral lattices of rank two. There the object called the rank-two Jacobi algebra is a bigraded ring

$0\le x,\qquad 0\le y,\qquad x+y\le 1.$0

which is finitely generated, and for each fixed lattice index $0\le x,\qquad 0\le y,\qquad x+y\le 1.$1, the module $0\le x,\qquad 0\le y,\qquad x+y\le 1.$2 is free over $0\le x,\qquad 0\le y,\qquad x+y\le 1.$3 of rank $0\le x,\qquad 0\le y,\qquad x+y\le 1.$4. Its generators come from rank-one Jacobi forms in the three directions $0\le x,\qquad 0\le y,\qquad x+y\le 1.$5, $0\le x,\qquad 0\le y,\qquad x+y\le 1.$6, and $0\le x,\qquad 0\le y,\qquad x+y\le 1.$7, together with a small number of genuinely rank-two generators (Wang et al., 2021).

A further possible confusion is with work on quadratic polynomial maps with Jacobian rank two, where “quadratic” and “rank two” refer to polynomial maps and low-rank Jacobian matrices rather than to any Jacobi algebra. That literature classifies Jacobian matrices of degree at most $0\le x,\qquad 0\le y,\qquad x+y\le 1.$8 and rank at most $0\le x,\qquad 0\le y,\qquad x+y\le 1.$9, proves normal forms after linear changes of coordinates, and derives consequences for transcendence degree and nilpotent triangularization (Bondt, 2016).

Within this terminological landscape, the quadratic rank two Jacobi algebra in the narrow sense is the five-generator quadratic algebra attached to the two-variable Jacobi polynomials on the triangle. Its distinctive features are the simultaneous differential and difference realizations, the embedded Jacobi and Racah subalgebras, the pentagonal organization of commuting pairs, and the superintegrable $L,L_1,L_3$00 realization. These features identify it as the bivariate extension of the classical quadratic Jacobi algebra, rather than as a Jacobi-group invariant differential operator algebra, a ring of lattice-index Jacobi forms, or a low-rank Jacobian classification problem.