$\alpha, \beta, \gamma \in k$ be scalars satisfying $$\label{eq: 4sklyanincoeffcondintro}
\alpha + \beta + \gamma + \alpha \beta \gamma = 0 \;\text{ and }\; \{\alpha,\beta, \gamma\} \cap \{0,\pm 1\}=\emptyset.$$ We define the *4-dimensional Sklyanin algebra* $A(\alpha,\beta,\gamma)$ associated to such parameters to be the quotient of the free $k$-algebra $k\{x_0,x_1,x_2,x_3\}$ by the ideal whose generators are the six quadratic relations in [@smith1992regularity Equation 0.2.2]: $$\label{eq: 4sklyaninrelnsintro}
\begin{array}{llll}
[x_0,x_1]-\alpha[x_2,x_3]_+, &&& [x_0,x_1]_+ -[x_2,x_3], \\ \relax
[x_0,x_2]-\beta[x_3,x_1]_+, &&& [x_0,x_2]_+ -[x_3,x_1], \\ \relax
[x_0,x_3]-\gamma[x_1,x_2]_+, &&& [x_0,x_3]_+ -[x_1,x_2]. 
\end{array}$$

The latter condition of allows us to avoid certain degenerate cases (see [@smith1992regularity §1]). We will omit the parameters of $A(\alpha,\beta,\gamma)$ for the remainder of §\[subsec: examples\].

The construction of $A$ can also be phrased in terms of a smooth elliptic curve $E$ and the translation automorphism $\sigma$ associated to a point on the curve (as discussed in detail in [@smith1992regularity §2.9-2.14]). By generalising this construction one can define $n$-dimensional Sklyanin algebras for all integers $n \geq 3$. Indeed, it is this family which was studied in [@tate1996homological]. The 4-dimensional Sklyanin algebra has many good properties, including being AS-regular of global dimension 4 and having Hilbert series $1/(1-t)^4$ by [@smith1992regularity Theorem 0.3].

There are many actions of 