Graded Down-Up Algebra: Structure & Applications

• Graded down-up algebras are noncommutative AS-regular algebras defined by two generators with cubic relations and an explicit grading structure.
• They provide a framework for studying deformation theory, noncommutative projective geometry, and representation theory through graded and filtered structures.
• Their rigidity under finite group coactions and links to Beilinson algebras offer critical insights into homological invariants and noncommutative algebraic geometry.

A graded down-up algebra is a family of noncommutative, typically Artin–Schelter (AS) regular algebras generated by two elements subject to cubic relations and equipped with an explicit grading structure. These algebras, parametrized by scalars and, in more general forms, degrees, provide a rich exploration ground for noncommutative projective geometry, deformation theory, actions of (quantum) groups, and representation theory. Their rigidity phenomena under Hopf coactions, graded structures, associated Beilinson algebras, and connections to quiver and poset combinatorics have motivated extensive developments in algebraic research.

1. Definition and Structural Properties

Let $k$ be an algebraically closed field of characteristic zero. The classical graded down-up algebra $D(\alpha, \beta)$ is generated by elements $u, d$ of degree 1, subject to the cubic relations

$$u^2 d = \alpha\, u d u + \beta\, d u^2, \qquad u d^2 = \alpha\, d u d + \beta\, d^2 u,$$

with $\alpha, \beta \in k$ and $\beta \neq 0$ for regularity. The algebra is naturally $\mathbb{N}$-graded. It is noetherian and AS-regular of global dimension 3, with Hilbert series

$H_{D}(t) = \frac{1}{(1-t)^2 (1-t^2)},$

and a PBW basis $\{ u^i (du)^j d^k : i, j, k \ge 0 \}$ (Chen et al., 2016).

Generalizations allow graded down-up algebras with arbitrary coprime weights $\deg x = n$, $\deg y = m$ ($n, m \geq 1$, $\gcd(n, m)=1$), using the presentation

$$A(\alpha, \beta) = k \langle x, y \rangle\, /\, (x^2 y - \beta y x^2 - \alpha x y x,\; x y^2 - \beta y^2 x - \alpha y x y),$$

with the Gorenstein parameter $\ell = 2(n+m)$ (Itaba et al., 25 Nov 2025).

2. Graded and Filtered Structures, Associated Algebras

The graded structure governs both homological invariants and deformation and quantization approaches. For generalized or multi-parameter down-up algebras $A(\alpha, \beta, \gamma; \lambda, \omega)$ generated by $x, y, z$, relations are

$$z x = \alpha x z + \gamma z, \quad x y = \beta y x + \gamma y, \quad z y = \omega y z + \lambda x.$$

Assigning degree 1 to all generators yields a filtered algebra whose associated graded algebra is

$$\operatorname{gr} A \cong K\langle x, y, z \rangle / (z x - \alpha x z,\; x y - \beta y x,\; z y - \omega y z),$$

which is a solvable polynomial algebra (PBW type) for generic parameters, thus admitting an effective Gröbner basis theory and global dimension 3. The Rees algebra $R(A)$ is constructed via a central variable $t$ to encode the filtration, leading to a Koszul graded algebra of dimension 4 (Tuniyaz et al., 2022).

3. Rigidity under Group Coactions and Fixed Subring Results

Graded down-up algebras display remarkable rigidity properties under finite group coactions. Given a connected $\mathbb{N}$-graded AS-regular down-up algebra $A = D(\alpha, \beta)$, if a nontrivial finite group $G$ coacts inner faithfully and homogeneously, then the fixed subring $A^{co\,G}$ is not isomorphic to $A$ and fails to be AS-regular. The proof proceeds by distinguishing abelian and non-abelian $G$:

• For abelian $G$, coactions correspond to automorphism group actions; in all cases, the fixed subring loses AS-regularity.
• For non-abelian $G$, classification via the parameters $(\alpha, \beta)$ and a detailed combinatorial enumeration (using Bergman’s Diamond Lemma and analysis of monomial degrees and G-gradings) shows that the fixed subring always fails the generation and homological conditions for AS-regularity in dimension three.

An explicit example: for $A = D(0,1)$, $G = D_8$ (dihedral group order 8), the fixed subring is $$k[(du)^2, (ud)^2, d^4, u^2] \cong k[x, y, z, t]/(xy - z t^2)$$, a commutative complete intersection that is AS-Gorenstein but not AS-regular. This demonstrates that the homological determinant criterion for Gorenstein property fails under coactions (Chen et al., 2016).

4. Beilinson Algebra and Derived Equivalences

Given a graded AS-regular algebra $A$ of dimension $d$ and Gorenstein parameter $\ell$, the Beilinson algebra $\nabla A$ is the upper-triangular matrix algebra built out of the graded components of $A$: $$\nabla A = \left( \begin{array}{cccc} A_0 & A_1 & \cdots & A_{\ell-1} \ 0 & A_0 & \cdots & A_{\ell-2} \ \vdots & \ddots & \ddots & \vdots \ 0 & \cdots & 0 & A_0 \end{array} \right).$$ For graded down-up algebras, $d=3$, $\ell=2(n+m)$; thus, $\operatorname{gldim}(\nabla A) = 2$. The noncommutative projective geometry of $A$ is realized as the derived category of finite modules over $\nabla A$: $D^b(\text{tails}~A) \cong D^b(\bmod\,\nabla A)$ (Itaba et al., 25 Nov 2025).

5. Hochschild Cohomology and Derived Category Constraints

Explicit computations of $\dim HH^i(\nabla A)$ are available for weights $(n, m)$, revealing:

• $\dim HH^0 = 1$.
• $\dim HH^1$ and $\dim HH^2$ depend on combinatorial data (for $(n, m)$, parity, parameters).
• All higher Hochschild cohomology vanishes: $HH^{r\geq3} = 0$.

The ring structure is given by an exterior algebra truncated in degree 3, with the number and relations among generators depending on $(n, m)$. For $n, m > 1$, it is shown via Euler characteristic and Grothendieck group arguments that for $m > n > 1$, the derived category $D^b(\text{tails}~A)$ cannot be equivalent to that of any smooth projective surface. The independence of the Hochschild–cohomology Euler characteristic from the expected value for surfaces (e.g., $\chi_{HH} = n + m \neq 2(n+m)$) obstructs such an equivalence (Itaba et al., 25 Nov 2025, Itaba et al., 2019).

6. Augmented Down-Up Algebras and Uniform Posets

Augmented down-up (ADU) algebras, introduced by Terwilliger and Worawannotai, generalize down-up algebras by adjoining a toral generator $k^{\pm1}$ and Laurent polynomial parameters. In this framework, two equivalent presentations arise: one via generators $e, f, k^{\pm1}$ with cubic relations involving Laurent polynomials, another using central elements and specialized products. The ADU algebras introduce a $\mathbb{Z}$-grading with a basis where each degree is spanned by monomials of the form $K^h C_s^i C_t^j\,$ or involving powers of $E, F$.

A key result is that the center of an ADU algebra is always a polynomial algebra in two variables. ADU algebras naturally act as endomorphism algebras on incidence spaces for seven families of uniform posets—the action of $e, f, k$ correspond to lowering, raising, and rank operators, respectively. The parameters for each poset family are explicitly given by integer data and Laurent polynomials (Terwilliger et al., 2012).

7. Open Directions and Significance

Important open problems include a complete characterization of when the fixed subring of a down-up algebra under a finite group coaction is AS-Gorenstein, as rigidity does not preclude this property. Another direction concerns possible generalizations to rigidity under semisimple Hopf algebra coactions, for which only partial results exist. The close relationship with weighted projective planes, matrix algebra techniques (Beilinson algebra), PBW and Gröbner basis techniques for noncommutative solvable polynomial algebras, and the use of poset combinatorics suggest broad applicability to noncommutative algebraic geometry, quantum algebra, and representation theory (Chen et al., 2016, Itaba et al., 25 Nov 2025, Terwilliger et al., 2012, Tuniyaz et al., 2022).

A plausible implication is that the interplay between rigidity phenomena, homological constraints, and graded structure situates graded down-up algebras as archetypal examples for testing noncommutative analogues of classical algebraic geometry, deformation theory, and representation-equivalence criteria.