Quantum plane ($k_q[X,Y]$)

For a field $k$ and $q\in k^{\times }$, the quantum plane is $k_q[X,Y]=k⟨X,Y⟩/(YX-qXY)$. Its ordered monomials $X^i{Y}^j$, $i,j\ge 0$, are a basis, and their multiplication is $X^i{Y}^j{X}^u{Y}^v=q^{ju}{X}^{i+u}{Y}^{j+v}$. Leading monomials show it is a noncommutative domain. Some sources interchange $q$ with $q^{-1}$ in the defining relation.

For $yx=qxy$, the quantum enveloping algebra of sl2 acts by $K(x^r{y}^s)=q^{r-s}{x}^r{y}^s$, $E(x^r{y}^s)=[s{]}_q{x}^{r+1}{y}^{s-1}$ and $F(x^r{y}^s)=[r{]}_q{x}^{r-1}{y}^{s+1}$. Negative exponents here have zero coefficients and are omitted. The quantum integer identity $[r][s+1]-[s][r+1]=[r-s]$ verifies the commutator. These operations preserve total degree and make the quantum plane a module algebra.

At a primitive cube root of unity, the cubic homogeneous module has basis $x^3,x^{2}y,x{y}^2,y^3$. Its only simple modules occurring as submodules are the one-dimensional subspaces of $⟨x^3,y^3⟩$, on which $K=1$ and $E=F=0$. Its quotient by this two-dimensional submodule is simple, but it has no simple complement: any lift of the highest $K$ weight space contains $x^{2}y$, whose $E$ image is $x^3\ne 0$. Thus the cubic module is not a semisimple module.

For $q\ne 0$, define a vector space with basis $u^i{v}^j$, $i\in \mathbb{Z}$, $j\ge 0$, and product $(u^i{v}^j)(u^k{v}^{\ell })=q^{-jk}{u}^{i+k}{v}^{j+\ell }$. Bilinearity of the exponent verifies associativity. This constructs the algebra and proves the basis independently of its presentation. It localizes the quantum plane at powers of $u$ and gives a quotient witness for noncommutativity and noncocommutativity of the coordinate Hopf algebra of quantum SL2.

The two-dimensional quantum torus is the algebra with invertible generators $X,Y$ and relation $YX=qXY$, for $q\in k^{\times }$. Its basis consists of $X^i{Y}^j$ with $i,j\in \mathbb{Z}$, with the same multiplication rule as the quantum plane. It is a right Noetherian ring: successive adjunction of $X^{-1}$, $Y$, and $Y^{-1}$ satisfies the hypotheses of the noncommutative Hilbert basis theorem.