Noncommutative products of Euclidean spaces

Abstract

We present natural families of coordinate algebras on noncommutative products of Euclidean spaces ${\mathbb {R}}^{N_1} \times _{\mathcal {R}} {\mathbb {R}}^{N_2}$. These coordinate algebras are quadratic ones associated with an $\mathcal {R}$-matrix which is involutive and satisfies the Yang–Baxter equations. As a consequence, they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces ${\mathbb {R}}^{4} \times _{\mathcal {R}} {\mathbb {R}}^{4}$. Among these, particularly well behaved ones have deformation parameter $\mathbf{u} \in {\mathbb {S}}^2$. Quotients include seven spheres ${\mathbb {S}}^{7}_\mathbf{u}$ as well as noncommutative quaternionic tori ${\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u} = {\mathbb {S}}^3 \times _\mathbf{u} {\mathbb {S}}^3$. There is invariance for an action of ${{\mathrm{SU}}}(2) \times {{\mathrm{SU}}}(2)$ on the torus ${\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u}$ in parallel with the action of $\mathrm{U}(1) \times \mathrm{U}(1)$ on a ‘complex’ noncommutative torus ${\mathbb {T}}^2_\theta$ which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.

Appendix A: Quadratic algebras

To be definite, we take the ground field to be complex numbers ${\mathbb {C}}$. A homogeneous quadratic algebra [13, 14] is an associative algebra $\mathcal {A}$ of the form

$$\begin{aligned} \mathcal {A}=A(E,R)=T(E)/(R) \end{aligned}$$

with E a finite-dimensional vector space, R a subspace of $E\otimes E$ and (R) denoting the two-sided ideal of the tensor algebra T(E) over E generated by R. The space E is the space of generators of $\mathcal {A}$ and the subspace R of $E\otimes E$ is the space of relations of $\mathcal {A}$. The algebra $\mathcal {A}=A(E,R)$ is naturally a graded algebra $\mathcal {A}=\bigoplus _{n\in \mathbb N}\mathcal {A}_n$ which is connected, that is such that $\mathcal {A}_0={\mathbb {C}}{\mathbb {1}}$ and generated by the degree 1 part, $\mathcal {A}_1=E$.

To a quadratic algebra $\mathcal {A}=A(E,R)$ as above one associates another quadratic algebra, its Koszul dual $\mathcal {A}^!$, defined by

$$\begin{aligned} \mathcal {A}^!=A\left( E^*,R^\perp \right) \end{aligned}$$

where $E^*$ denotes the dual vector space of E and $R^\perp \subset E^*\otimes E^*$ is the orthogonal of the space of relations $R\subset E\otimes E$ defined by

$$\begin{aligned} R^\perp = \left\{ \omega \in E^*\otimes E^*\, ; \, \langle \omega ,r\rangle =0,\forall r\in R\right\} . \end{aligned}$$

As usual, by using the finite-dimensionality of E, one identifies $E^*\otimes E^*$ with the dual vector space $(E\otimes E)^*$ of $E\otimes E$. One has of course $(\mathcal {A}^!)^!=\mathcal {A}$ and the dual vector spaces $\mathcal {A}^{!*}_n$ of the homogeneous components $\mathcal {A}^!_n$ of $\mathcal {A}^!$ are

$$\begin{aligned} \mathcal {A}^{!*}_1=E \quad \hbox {and} \quad \mathcal {A}^{!*}_n=\bigcap _{r+s+2=n} E^{\otimes ^r}\otimes R \otimes E^{\otimes ^s} \end{aligned}$$

(A.1)

for $n\ge 2$, as easily verified. In particular $\mathcal {A}^{!*}_2=R$ and $\mathcal {A}^{!*}_n\subset E^{\otimes ^n}$ for any $n\in \mathbb N$.

Consider the sequence of free left $\mathcal {A}$-modules

$$\begin{aligned} K(\mathcal {A}): \quad \cdots {\mathop {\rightarrow }\limits ^{b}}\mathcal {A}\otimes \mathcal {A}^{!*}_{n+1}{\mathop {\rightarrow }\limits ^{b}}\mathcal {A}\otimes \mathcal {A}^{!*}_n\rightarrow \cdots \rightarrow \mathcal {A}\otimes \mathcal {A}^{!*}_2{\mathop {\rightarrow }\limits ^{b}}\mathcal {A}\otimes E{\mathop {\rightarrow }\limits ^{b}}\mathcal {A}\rightarrow 0 \end{aligned}$$

(A.2)

where $b:\mathcal {A}\otimes \mathcal {A}^{!*}_{n+1}\rightarrow \mathcal {A}\otimes \mathcal {A}^{!*}_n$ is induced by the left $\mathcal {A}$-module homomorphism of $\mathcal {A}\otimes E^{\otimes ^{n+1}}$ into $\mathcal {A}\otimes E^{\otimes ^n}$ defined by

$$\begin{aligned} b(a\otimes (x_0\otimes x_1\otimes \cdots \otimes x_n))=ax_0 \otimes (x_1\otimes \cdots \otimes x_n) \end{aligned}$$

for $a\in \mathcal {A}$, $x_k \in E$. It follows from (A.1) that $\mathcal {A}^{!*}_n\subset R\otimes E^{\otimes ^{n-2}}$ for $n\ge 2$, which implies that $b^2=0$. As a consequence, the sequence $K(\mathcal {A})$ in (A.2) is a chain complex of free left $\mathcal {A}$-modules called the Koszul complex of the quadratic algebra $\mathcal {A}$. The quadratic algebra $\mathcal {A}$ is said to be a Koszul algebra whenever its Koszul complex is acyclic in positive degrees, that is, whenever $H_n(K(\mathcal {A}))=0$ for $n\ge 1$. One shows easily that $\mathcal {A}$ is a Koszul algebra if and only if its Koszul dual $\mathcal {A}^!$ is a Koszul algebra.

It is important to realize that the presentation of $\mathcal {A}$ by generators and relations is equivalent to the exactness of the sequence

$$\begin{aligned} \mathcal {A}\otimes R {\mathop {\rightarrow }\limits ^{b}} \mathcal {A}\otimes E {\mathop {\rightarrow }\limits ^{b}}\mathcal {A}{\mathop {\rightarrow }\limits ^{\varepsilon }} {\mathbb {C}}\rightarrow 0 \end{aligned}$$

with $\varepsilon$ the map induced by the projection onto degree 0. Thus one always has

$$\begin{aligned} H_1(K(\mathcal {A}))=0 \quad \hbox {and} \quad H_0(K(\mathcal {A}))={\mathbb {C}}\end{aligned}$$

and, whenever $\mathcal {A}$ is Koszul, the sequence

$$\begin{aligned} K(\mathcal {A}){\mathop {\rightarrow }\limits ^{\varepsilon }} {\mathbb {C}}\rightarrow 0 , \end{aligned}$$

is a free resolution of the trivial module ${\mathbb {C}}$. This resolution is then a minimal projective resolution of ${\mathbb {C}}$ in the category of graded modules [5].

Let $\mathcal {A}=A(E,R)$ be a quadratic Koszul algebra such that $\mathcal {A}^!_D\not =0$ and $\mathcal {A}^!_n=0$ for $n>D$. Then the trivial (left) module ${\mathbb {C}}$ has projective dimension D which implies that $\mathcal {A}$ has global dimension D (see [5]). This also implies that the Hochschild dimension of $\mathcal {A}$ is D (see [2]). By applying the functor ${{\mathrm{Hom}}}_\mathcal {A}(\, \cdot \, , \mathcal {A})$ to the Koszul chain complex $K(\mathcal {A})$ of left $\mathcal {A}$-modules one obtains the cochain complex $L(\mathcal {A})$ of right $\mathcal {A}$-modules

$$\begin{aligned} L(\mathcal {A}) : \quad \quad 0\rightarrow \mathcal {A}{\mathop {\rightarrow }\limits ^{b'}} \cdots {\mathop {\rightarrow }\limits ^{b'}}\mathcal {A}^!_n \otimes \mathcal {A}{\mathop {\rightarrow }\limits ^{b'}} \mathcal {A}^!_{n+1} \otimes \mathcal {A}{\mathop {\rightarrow }\limits ^{b'}} \cdots . \end{aligned}$$

Here $b'$ is the left multiplication by $\sum _k\theta ^k\otimes e_k$ in $\mathcal {A}^!\otimes \mathcal {A}$ where $(e_k)$ is a basis of E with dual basis $(\theta ^k)$. The algebra $\mathcal {A}$ is said to be Koszul–Gorenstein if it is Koszul of finite global dimension D as above and if $H^n(L(\mathcal {A}))={\mathbb {C}}\, \delta ^n_D$. Notice that this implies that $\mathcal {A}^!_n\simeq \mathcal {A}^{!*}_{D-n}$ as vector spaces (this is a version of Poincaré duality).

Finally, a graded algebra $\mathcal {A}=\oplus _n \mathcal {A}_n$ is said to have polynomial growth whenever there are a positive C and $N \in {\mathbb {N}}$ such that, for any $n \in {\mathbb {N}}$,

$$\begin{aligned} \dim (\mathcal {A}_n)\le C n^{N-1}. \end{aligned}$$

As before, let E be a finite-dimensional vector space with the tensor algebra T(E) endowed with its natural filtration $F^n(T(E))=\bigoplus _{m\le n} E^{\otimes ^m}$. A nonhomogeneous quadratic algebra [4, 15], is an algebra ${\mathfrak {A}}$ of the form

$$\begin{aligned} {\mathfrak {A}}=A(E,P)=T(E)/(P) \end{aligned}$$

where P is a subspace of $F^2(T(E))$ and where (P) denotes as above the two-sided ideal of T(E) generated by P. The filtration of the tensor algebra T(E) induces a filtration $F^n({\mathfrak {A}})$ of ${\mathfrak {A}}$ and one associates to ${\mathfrak {A}}$ the graded algebra

$$\begin{aligned} \hbox {gr} ({\mathfrak {A}})=\oplus _n F^n({\mathfrak {A}})/F^{n-1}({\mathfrak {A}}). \end{aligned}$$

Let R be the image of P under the canonical projection of $F^2(T(E))$ onto $E\otimes E$ and let $\mathcal {A}=A(E,R)$ be the homogeneous quadratic algebra T(E) / (R); this $\mathcal {A}$ is referred to as the quadratic part of ${\mathfrak {A}}$. There is a canonical surjective-graded algebra homomorphism

$$\begin{aligned} \hbox {can} :\mathcal {A}\rightarrow \hbox {gr}({\mathfrak {A}}). \end{aligned}$$

One says that ${\mathfrak {A}}$ has the Poincaré–Birkhoff–Witt (PBW) property whenever this homomorphism is an isomorphism. The terminology comes from the example where ${\mathfrak {A}}=U({\mathfrak {g}})$ the universal enveloping algebra of a Lie algebra ${\mathfrak {g}}$. A central result (see [4, 14]) states that if ${\mathfrak {A}}$ has the PBW property then the following conditions are satisfied:

$$\begin{aligned}&\mathrm {(i)} \quad P\cap F^1 (T(E))=0 , \nonumber \\&\mathrm {(ii)} \quad (P \cdot E + E \cdot P)\cap F^2 (T(E)) \subset P. \end{aligned}$$

(A.3)

Conversely, if the quadratic part $\mathcal {A}$ is a Koszul algebra, the conditions $\mathrm {(i)}$ and $\mathrm {(ii)}$ imply that ${\mathfrak {A}}$ has the PBW property. Condition (i) means that P is obtained from R by adding to each nonzero element of R terms of degrees 1 and 0. That is there are linear mappings $\psi _1:R\rightarrow E$ and $\psi _0:R \rightarrow {\mathbb {C}}$ such that one has

$$\begin{aligned} P=\{r+\psi _1(r)+\psi _0(r) {\mathbb {1}}\,\, \vert \,\, r \in R\} \end{aligned}$$

(A.4)

giving P in terms of R. Condition (ii) is a generalization of the Jacobi identity (see [14]).