Here are some thoughts which are essentially a complement to Branimir's answer. In many naturally occurring examples, in particular the theory of quantum groups, it is actually more natural to start with a noncommutative cotangent bundle and to then construct the spectral triple from it. More explicitly, start with a pre-$C^*$-algebra $A$ whose noncommutative geometry you are trying to understand.

1. Look for a dga (differential graded algebra) $\Omega^{\bullet}$ for which $A = \Omega^0$.

2. Endow $\Omega^{\bullet}$ with some type of ``Riemannian metric'', which is to say an inner product $\langle \cdot,\cdot \rangle$.

3. Confirm that the differential d of the dga is adjointable $\langle \cdot,\cdot \rangle$. Denote by d$^*$ the adjoint of d.

4. Complete $\Omega^{\bullet}$ to a Hilbert space.

5. Check that $D:=$d$+$d$^*$ is a closable operator, and that the closure satisfies the axioms of a spectral triple.

I should stress that this is a rough plan of attack, and will only work in situations which are, in some sense, close to the classical situation. For example, $q$-deformed $C^*$-algebras, in particular examples coming from Drinfeld-Jimbo quantum groups.

The prototypical example here is the Podles sphere $\mathcal{O}_q(S^2)$, the degree zero part of a $\mathbb{Z}$-grading on $\mathcal{O}_q(SU_2)$, which $q$-deforms the algebra of algebraic functions of the $2$-sphere. This pre-$C^*$-algebra admits an essentially unique $\mathcal{O}_q(SU_2)$-covariant dga, called the Podles calculus. The calculus carries a $q$-deformation of the Fubini-Study metric, with respect to which $d$ is adjointable, and which gives a spectral triple upon completion of $\Omega^{\bullet}$ and closure of $D = $d$+$d$^*$. The dga constructed from this spectral triple (in the sense Branimir explained above) will coincide with $\Omega^{\bullet}$. This spectral triple was first constructed by Dabrowski and Sitarz (or at least the Dolbeault-Dirac version) and later reconstructed by Majid from a more geometric point of view.

This construction can be extended to a much larger class of examples called quantum flag manifolds . . . but this is very much ongoing work.

Here are some thoughts which essentially complement Branimir's answer. In many naturally occurring examples, in particular the theory of quantum groups, it is actually more natural to start with a noncommutative cotangent bundle and to then construct the spectral triple from it. More explicitly, start with a pre-$C^*$-algebra $A$ whose noncommutative geometry you are trying to understand.

1. Look for a dga (differential graded algebra) $\Omega^{\bullet}$ for which $A = \Omega^0$.

2. Endow $\Omega^{\bullet}$ with some type of ``Riemannian metric'', which is to say an inner product $\langle \cdot,\cdot \rangle$.

3. Confirm that the differential d of the dga is adjointable $\langle \cdot,\cdot \rangle$. Denote by d$^*$ the adjoint of d.

4. Complete $\Omega^{\bullet}$ to a Hilbert space.

5. Check that the closure of the symmetric operator $D:=$d$+$d$^*$ is a satisfies the axioms of a spectral triple.

I should stress that this is a rough plan of attack, and will only work in situations which are, in some sense, close to the classical situation. For example, $q$-deformed $C^*$-algebras, in particular examples coming from Drinfeld-Jimbo quantum groups.

The prototypical example here is the Podles sphere $\mathcal{O}_q(S^2)$, the degree zero part of a $\mathbb{Z}$-grading on $\mathcal{O}_q(SU_2)$, which $q$-deforms the algebra of algebraic functions of the $2$-sphere. This pre-$C^*$-algebra admits an essentially unique $\mathcal{O}_q(SU_2)$-covariant dga, called the Podles calculus. The calculus carries a $q$-deformation of the Fubini-Study metric, with respect to which $d$ is adjointable, and which gives a spectral triple upon completion of $\Omega^{\bullet}$ and closure of $D = $d$+$d$^*$. The dga constructed from this spectral triple (in the sense Branimir explained above) will coincide with $\Omega^{\bullet}$. This spectral triple was first constructed by Dabrowski and Sitarz (or at least the Dolbeault-Dirac version) and later reconstructed by Majid from a more geometric point of view.

This construction can be extended to a much larger class of examples called quantum flag manifolds . . . but this is very much ongoing work.

Here are some thoughts which are essentially a complement to Branimir's answer. In many naturally occurring examples, in particular the theory of quantum groups, it is actually more natural to start with a noncommutative cotangent bundle and to then construct the spectral triple from it. More explicitly, start with a pre-$C^*$-algebra $A$ whose noncommutative geometry you are trying to understand.

1. Look for a dga (differential graded algebra) $\Omega^{\bullet}$ for which $A = \Omega^0$.

2. Endow $\Omega^{\bullet}$ with some type of ``Riemannian metric'', which is to say an inner product $\langle \cdot,\cdot \rangle$.

3. Confirm that the differential d of the dga is adjointable $\langle \cdot,\cdot \rangle$. Denote by d$^*$ the adjoint of d.

4. Complete $\Omega^{\bullet}$ to a Hilbert space.

5. Check that $D:=$d$+$d$^*$ is a closable operator, and that the closure satisfies the axioms of a spectral triple.

I should stress that this is a rough plan of attack, and will only work in situations which are, in some sense, close to the classical situation. For example, $q$-deformed $C^*$-algebras, in particular examples coming from Drinfeld-Jimbo quantum groups.

The prototypical example here is the Podles sphere $\mathcal{O}_q(S^2)$, the degree zero part of a $\mathbb{Z}$-grading on $\mathcal{O}_q(SU_2)$, which $q$-deforms the algebra of algebraic functions of the $2$-sphere. This pre-$C^*$-algebra admits an essentially unique dga admitting a left $\mathcal{O}_q(SU_2)$-coaction, called the Podles calculus. The calculus carries a $q$-deformation of the Fubini-Study metric, with respect to which $d$ is adjointable, and which gives a spectral triple unpon completion of $\Omega^{\bullet}$ and closure of $D = $d$+$d$^*$. The dga constructed from this spectral triple (in the sense Branimir explained above) will coincide with $\Omega^{\bullet}$. This spectral triple was first constructed by Dabrowski and Sitarz (or at least the Dolbeault-Dirac version) and later reconstructed by Majid from a more geometric point of view.

This construction can be extended to a much larger class of examples called quantum flag manifolds . . . but this is very much ongoing work.

Here are some thoughts which are essentially a complement to Branimir's answer. In many naturally occurring examples, in particular the theory of quantum groups, it is actually more natural to start with a noncommutative cotangent bundle and to then construct the spectral triple from it. More explicitly, start with a pre-$C^*$-algebra $A$ whose noncommutative geometry you are trying to understand.

1. Look for a dga (differential graded algebra) $\Omega^{\bullet}$ for which $A = \Omega^0$.

2. Endow $\Omega^{\bullet}$ with some type of ``Riemannian metric'', which is to say an inner product $\langle \cdot,\cdot \rangle$.

3. Confirm that the differential d of the dga is adjointable $\langle \cdot,\cdot \rangle$. Denote by d$^*$ the adjoint of d.

4. Complete $\Omega^{\bullet}$ to a Hilbert space.

5. Check that $D:=$d$+$d$^*$ is a closable operator, and that the closure satisfies the axioms of a spectral triple.

I should stress that this is a rough plan of attack, and will only work in situations which are, in some sense, close to the classical situation. For example, $q$-deformed $C^*$-algebras, in particular examples coming from Drinfeld-Jimbo quantum groups.

The prototypical example here is the Podles sphere $\mathcal{O}_q(S^2)$, the degree zero part of a $\mathbb{Z}$-grading on $\mathcal{O}_q(SU_2)$, which $q$-deforms the algebra of algebraic functions of the $2$-sphere. This pre-$C^*$-algebra admits an essentially unique $\mathcal{O}_q(SU_2)$-covariant dga, called the Podles calculus. The calculus carries a $q$-deformation of the Fubini-Study metric, with respect to which $d$ is adjointable, and which gives a spectral triple upon completion of $\Omega^{\bullet}$ and closure of $D = $d$+$d$^*$. The dga constructed from this spectral triple (in the sense Branimir explained above) will coincide with $\Omega^{\bullet}$. This spectral triple was first constructed by Dabrowski and Sitarz (or at least the Dolbeault-Dirac version) and later reconstructed by Majid from a more geometric point of view.

This construction can be extended to a much larger class of examples called quantum flag manifolds . . . but this is very much ongoing work.