Return to Answer

• Applications of a more conceptual or fundamental nature:
  -->Such as for example the introduction and study of non-commutative space-times. Here, the non-commutativity is frequently expressed through deformed commutation relations between the coordinates or the functions on the space-time manifolds, leading to deformed algebras and non-commutative geometries. Fundamental quantization problems are frequently discussed in this setting: See for example this paper, this paper and this one (among lots of works in a similar spirit). The relation of $q$-mathematics and quantum groups to deformation quantization is another hot topic with a significant number -imo- of open questions. In Minimal areas from q-deformed oscillator algebras the authors argue that non-commutative space-times with dynamical commutation relations between the coordinates imply $q$-deformed algebras of observables and a kind of converse argument is also supplied.
  -->The quantum inverse scattering method and the definition(s) and study of quantum integrability (see p. 269) have given rise to quantum groups and quantum algebras. The mathematical developments associated with them have been greatly inspired and have -in return- significantly contributed to such directions of study. Jule Lamers' answer provides further info and references at that point. S. Majid's book is also worth to be mentioned as it contains a cataclysm of such ideas and applications.
  -->The case of deformed particles (deformed bosons or fermions) which interpolate between different statistics has been another line of interesting applications of this kind.

• Applications of a more conceptual or fundamental nature:
  -->Such as for example the introduction and study of non-commutative space-times. Here, the non-commutativity is frequently expressed through deformed commutation relations between the coordinates or the functions on the space-time manifolds, leading to deformed algebras and non-commutative geometries. Fundamental quantization problems are frequently discussed in this setting: See for example this paper, this paper and this one (among lots of works in a similar spirit). The relation of $q$-mathematics and quantum groups to deformation quantization is another hot topic with a significant number -imo- of open questions. In Minimal areas from q-deformed oscillator algebras the authors argue that non-commutative space-times with dynamical commutation relations between the coordinates imply $q$-deformed algebras of observables and a kind of converse argument is also supplied.
  -->The quantum inverse scattering method and the definition(s) and study of quantum integrability (see p. 269) have given rise to quantum groups and quantum algebras. The mathematical developments associated with them have been greatly inspired and have -in return- significantly contributed to such directions of study. Jule Lamers' answer provides further details and references on that point. S. Majid's book is also worth to be mentioned as it contains a cataclysm of such ideas and applications.
  -->The case of deformed particles (deformed bosons or fermions) which interpolate between different statistics has been another line of interesting applications of this kind.

• Applications of phenomenological nature: Various $q$-deformed oscillators or $q$-deformed rotators have been used. Their spectrums, transition rates, matrix elements etc are proved to depend on the deformation parameter(s). "Playing" with the parameters allows curve fitting to experimental data. In lots of cases deformations based on more than one parameters (i.e. $q,p,..$-deformations) have been used. An interesting characteristic of such applications is that in many cases -for example in the fitting of the vibrational and rotational spectra of nuclei and molecules- the number of phenomenological $q$-parameters needed is significantly fewer than the number of traditional phenomenological parameters required to fit the same spectral data. (In many cases, the physical interpretation of such deformation parameters still lingers and imposes ideas of a more fundamental nature). In modern times, such ideas are also finding applications in semi-phenomenological cosmological models. See for example 9 where the dark energy is essentialy considered to be a $q$-deformed scalar field.
• Applications of a more conceptual or fundamental nature:
  -->Such as for example the introduction and study of non-commutative space-times. Here, the non-commutativity is frequently expressed through deformed commutation relations between the coordinates or the functions on the space-time manifolds, leading to deformed algebras and non-commutative geometries. Fundamental quantization problems are frequently discussed in this setting: See for example this paper, this paper and this one (among lots of works in a similar spirit). The relation of $q$-mathematics and quantum groups to deformation quantization is another hot topic with a significant number -imo- of open questions. In Minimal areas from q-deformed oscillator algebras the authors argue that non-commutative space-times with dynamical commutation relations between the coordinates imply $q$-deformed algebras of observables and a kind of converse argument is also supplied.
  -->The rise of quantum groups and quantum algebras and the mathematical developments associated with them has been greatly inspired and has -in return- significantly contributed to such directions of study. Jule Lamers' answer provides further info and references at that point. S. Majid's book is also worth to be mentioned as it contains a cataclysm of such ideas and applications.
  -->The case of deformed particles (deformed bosons or fermions) which interpolate between different statistics has been another line of interesting applications of this kind.

Just to mention a few papers (my former phd advisor has quite some work on the field):

A significant amount of related literature can be found at mathematical physics journals such as the Journal of Mathematical Physics, Journal of Physics A: Mathematical and General, Communications of Mathematical Physics, etc.

• Applications of phenomenological nature:
  Various $q$-deformed oscillators or $q$-deformed rotators have been used. Their spectrums, transition rates, matrix elements etc are proved to depend on the deformation parameter(s). "Playing" with the parameters allows curve fitting to experimental data. In lots of cases deformations based on more than one parameters (i.e. $q,p,..$-deformations) have been used. An interesting characteristic of such applications is that in many cases -for example in the fitting of the vibrational and rotational spectra of nuclei and molecules- the number of phenomenological $q$-parameters needed is significantly fewer than the number of traditional phenomenological parameters required to fit the same spectral data. (In many cases, the physical interpretation of such deformation parameters still lingers -see p.179- and imposes ideas of a more fundamental nature). In modern times, such ideas are also finding applications in semi-phenomenological cosmological models. See for example 9 where the dark energy is essentialy considered to be a $q$-deformed scalar field.
  Just to mention a few papers in this category (my former phd advisor has quite some work on the field):

• Applications of a more conceptual or fundamental nature:
  -->Such as for example the introduction and study of non-commutative space-times. Here, the non-commutativity is frequently expressed through deformed commutation relations between the coordinates or the functions on the space-time manifolds, leading to deformed algebras and non-commutative geometries. Fundamental quantization problems are frequently discussed in this setting: See for example this paper, this paper and this one (among lots of works in a similar spirit). The relation of $q$-mathematics and quantum groups to deformation quantization is another hot topic with a significant number -imo- of open questions. In Minimal areas from q-deformed oscillator algebras the authors argue that non-commutative space-times with dynamical commutation relations between the coordinates imply $q$-deformed algebras of observables and a kind of converse argument is also supplied.
  -->The quantum inverse scattering method and the definition(s) and study of quantum integrability (see p. 269) have given rise to quantum groups and quantum algebras. The mathematical developments associated with them have been greatly inspired and have -in return- significantly contributed to such directions of study. Jule Lamers' answer provides further info and references at that point. S. Majid's book is also worth to be mentioned as it contains a cataclysm of such ideas and applications.
  -->The case of deformed particles (deformed bosons or fermions) which interpolate between different statistics has been another line of interesting applications of this kind.

A significant amount of related literature can be found at mathematical physics journals such as the Journal of Mathematical Physics, Journal of Physics A: Mathematical and General, Communications of Mathematical Physics, SIGMA etc.

• Applications of phenomenological nature: Various $q$-deformed oscillators or $q$-deformed rotators have been used. Their spectrums, transition rates, matrix elements etc are proved to depend on the deformation parameter(s). "Playing" with the parameters allows curve fitting to experimental data. In lots of cases deformations based on more than one parameters (i.e. $q,p,..$-deformations) have been used. An interesting characteristic of such applications is that in many cases -for example in the fitting of the vibrational and rotational spectra of nuclei and molecules- the number of phenomenological $q$-parameters needed is significantly fewer than the number of traditional phenomenological parameters required to fit the same spectral data. (In many cases, the physical interpretation of such deformation parameters still lingers and imposes ideas of a more fundamental nature). In modern times, such ideas are also finding applications in semi-phenomenological cosmological models. See for example 9 where the dark energy is essentialy considered to be a $q$-deformed scalar field.
• Applications of a more conceptual or fundamental nature:
  -->Such as for example the introduction and study of non-commutative space-times. Here, the non-commutativity is frequently expressed through deformed commutation relations between the coordinates or the functions on the space-time manifolds, leading to deformed algebras and non-commutative geometries. Fundamental quantization problems are frequently discussed in this setting: See for example this paper, this paper and this one (among lots of works in a similar spirit). The relation of $q$-mathematics and quantum groups to deformation quantization is another -quite unexplored imo- hot topic. In Minimal areas from q-deformed oscillator algebras the authors argue that non-commutative space-times with dynamical commutation relations between the coordinates imply $q$-deformed algebras of observables and a kind of converse argument is also supplied.
  -->The rise of quantum groups and quantum algebras and the mathematical developments associated with them has been greatly inspired and has -in return- significantly contributed to such directions of study. Jule Lamers' answer provides further info and references at that point. S. Majid's book is also worth to be mentioned as it contains a cataclysm of such ideas and applications.
  -->The case of deformed particles (deformed bosons or fermions) which interpolate between different statistics has been another line of interesting applications of this kind.

• Applications of phenomenological nature: Various $q$-deformed oscillators or $q$-deformed rotators have been used. Their spectrums, transition rates, matrix elements etc are proved to depend on the deformation parameter(s). "Playing" with the parameters allows curve fitting to experimental data. In lots of cases deformations based on more than one parameters (i.e. $q,p,..$-deformations) have been used. An interesting characteristic of such applications is that in many cases -for example in the fitting of the vibrational and rotational spectra of nuclei and molecules- the number of phenomenological $q$-parameters needed is significantly fewer than the number of traditional phenomenological parameters required to fit the same spectral data. (In many cases, the physical interpretation of such deformation parameters still lingers and imposes ideas of a more fundamental nature). In modern times, such ideas are also finding applications in semi-phenomenological cosmological models. See for example 9 where the dark energy is essentialy considered to be a $q$-deformed scalar field.
• Applications of a more conceptual or fundamental nature:
  -->Such as for example the introduction and study of non-commutative space-times. Here, the non-commutativity is frequently expressed through deformed commutation relations between the coordinates or the functions on the space-time manifolds, leading to deformed algebras and non-commutative geometries. Fundamental quantization problems are frequently discussed in this setting: See for example this paper, this paper and this one (among lots of works in a similar spirit). The relation of $q$-mathematics and quantum groups to deformation quantization is another hot topic with a significant number -imo- of open questions. In Minimal areas from q-deformed oscillator algebras the authors argue that non-commutative space-times with dynamical commutation relations between the coordinates imply $q$-deformed algebras of observables and a kind of converse argument is also supplied.
  -->The rise of quantum groups and quantum algebras and the mathematical developments associated with them has been greatly inspired and has -in return- significantly contributed to such directions of study. Jule Lamers' answer provides further info and references at that point. S. Majid's book is also worth to be mentioned as it contains a cataclysm of such ideas and applications.
  -->The case of deformed particles (deformed bosons or fermions) which interpolate between different statistics has been another line of interesting applications of this kind.