Heisenberg Doubles for SnyderType Models
Abstract
:1. Introduction
2. Issues with the Heisenberg Double for the Snyder Model
Different Realization for Coproducts of Momenta
3. Unified Notation for the Snyder Algebra: Extended Snyder Model
 to the Snyder noncommutative spacetime relations (16):$$[{\widehat{x}}_{jN},{\widehat{x}}_{iN}]=[\frac{1}{\sqrt{\beta}}{\widehat{x}}_{j},\frac{1}{\sqrt{\beta}}{\widehat{x}}_{i}]=i\lambda ({\eta}_{ji}{\widehat{x}}_{NN}{\eta}_{Ni}{\widehat{x}}_{jN}+{\eta}_{NN}{\widehat{x}}_{ji}{\eta}_{jN}{\widehat{x}}_{Ni})=i\lambda {\widehat{x}}_{ji}$$(note that ${\widehat{x}}_{NN}=0$ due to antisymmetricity);
 to the commutation relations for Lorentz generators (16):$$[{\widehat{x}}_{ij},{\widehat{x}}_{kl}]=i\lambda ({\eta}_{ik}{\widehat{x}}_{jl}{\eta}_{il}{\widehat{x}}_{jk}{\eta}_{jk}{\widehat{x}}_{il}+{\eta}_{jl}{\widehat{x}}_{ik});$$
 and to crosscommutation relations of Lorentz generators acting on coordinates (17):$$[{\widehat{x}}_{jk},{\widehat{x}}_{iN}]=[{\widehat{x}}_{jk},\frac{1}{\sqrt{\beta}}{\widehat{x}}_{i}]=i\lambda ({\eta}_{ji}{\widehat{x}}_{kN}{\eta}_{ki}{\widehat{x}}_{jN}+{\eta}_{kN}{\widehat{x}}_{ji}{\eta}_{jN}{\widehat{x}}_{ki})$$$$=i\lambda \frac{1}{\sqrt{\beta}}({\eta}_{ji}{\widehat{x}}_{k}{\eta}_{ki}{\widehat{x}}_{j}).$$
3.1. Generalized Heisenberg Algebra
 the usual Heisenberg algebra sector, i.e., quantum mechanical phase space corresponding to the commutative (classical) spacetime:$$\begin{array}{ccc}\hfill \left[{x}_{i},{x}_{j}\right]& =& 0,\phantom{\rule{1.em}{0ex}}\left[{p}_{i},{p}_{j}\right]=0\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left[{p}_{j},{x}_{i}\right]& =& \left[{p}_{iN},{x}_{jN}\right]=i\left({\eta}_{ij}{\eta}_{NN}{\eta}_{iN}{\eta}_{Nj}\right)=i{\eta}_{ij};\hfill \end{array}$$
 and “the remaining part”, consisting of commutation relations between ${x}_{ij}$—tensorial coordinates—and ${p}_{ij}$—their corresponding canonical momenta:$$\begin{array}{ccc}\hfill \left[{x}_{ij},{x}_{kl}\right]& =& \left[{x}_{ij},{x}_{k}\right]=0,\phantom{\rule{1.em}{0ex}}\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left[{p}_{ij},{p}_{kl}\right]& =& \left[{p}_{ij},{p}_{k}\right]=0,\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left[{p}_{ij},{x}_{kl}\right]& =& i\left({\eta}_{ik}{\eta}_{jl}{\eta}_{il}{\eta}_{jk}\right),\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left[{p}_{i},{x}_{kl}\right]& =& \left[{p}_{ij},{x}_{k}\right]=0.\hfill \end{array}$$
4. Extended Snyder Space and Its Heisenberg Doubles
4.1. Extended Snyder Phase Space from the Heisenberg Double Construction
 commutation relations between Snyder coordinates and their coupled momenta:$$\begin{array}{ccc}\hfill \left[{p}_{k},{\widehat{x}}_{i}\right]& =& \left[{p}_{kN},{\widehat{x}}_{iN}\right]=i{\eta}_{ik}(1\frac{\beta {\lambda}^{2}}{12}{p}_{l}{p}_{l})\frac{i\beta {\lambda}^{2}}{12}{p}_{k}{p}_{i}+\frac{i\lambda}{2}{p}_{ki}+\frac{i{\lambda}^{2}}{12}{p}_{il}{p}_{kl}+O\left({\lambda}^{3}\right);\hfill \end{array}$$
 commutation relations between tensorial coordinates and their coupled momenta:$$\begin{array}{ccc}\hfill \left[{p}_{kl},{\widehat{x}}_{ij}\right]& =& i\left({\eta}_{ik}{\eta}_{jl}{\eta}_{jk}{\eta}_{il}\right)+i\frac{\lambda}{2}({\eta}_{ik}{p}_{lj}{\eta}_{jk}{p}_{li}{\eta}_{il}{p}_{kj}+{\eta}_{jl}{p}_{ki})+\frac{i{\lambda}^{2}}{12}[({\eta}_{ik}{p}_{jm}{p}_{lm}{\eta}_{jk}{p}_{im}{p}_{lm})\hfill \\ & & \left({\eta}_{il}{p}_{jm}{p}_{km}{\eta}_{jl}{p}_{im}{p}_{km}\right)2{p}_{ki}{p}_{lj}+2{p}_{kj}{p}_{li}]+O\left({\lambda}^{3}\right);\hfill \end{array}$$
 and mixed relations:$$\left[{p}_{kl},{\widehat{x}}_{i}\right]=\left[{p}_{kl},\sqrt{\beta}{\widehat{x}}_{iN}\right]=i\frac{\lambda}{2}\beta ({\eta}_{ik}{p}_{l}{\eta}_{il}{p}_{k})i\frac{{\lambda}^{2}}{12}\beta [{\eta}_{ik}{p}_{m}{p}_{lm}{\eta}_{il}{p}_{m}{p}_{km}+2{p}_{ki}{p}_{l}2{p}_{k}{p}_{li}]+O\left({\lambda}^{3}\right),$$$$\left[{p}_{k},{\widehat{x}}_{ij}\right]=\left[\frac{1}{\sqrt{\beta}}{p}_{kN},{\widehat{x}}_{ij}\right]=i\frac{\lambda}{2}({\eta}_{ik}{p}_{j}{\eta}_{jk}{p}_{i})i\frac{{\lambda}^{2}}{12}[\left({\eta}_{ik}{p}_{jl}{p}_{l}{\eta}_{jk}{p}_{il}{p}_{l}\right)2{p}_{ki}{p}_{j}+2{p}_{kj}{p}_{i}]+O\left({\lambda}^{3}\right).$$
4.2. Another Heisenberg Double for the Extended Snyder Algebra
 duality for the Snyder coordinates with Lorentz matrices:$$\begin{array}{ccc}& & <{\mathsf{\Lambda}}_{jk},{\widehat{x}}_{i}>=\sqrt{\beta}<{\mathsf{\Lambda}}_{jk},{\widehat{x}}_{iN}>=i\lambda \sqrt{\beta}({\eta}_{ji}{\eta}_{kN}{\eta}_{jN}{\eta}_{ki})=0,\hfill \end{array}$$$$\begin{array}{ccc}& & <{\mathsf{\Lambda}}_{jN},{\widehat{x}}_{i}>=\sqrt{\beta}<{\mathsf{\Lambda}}_{jN},{\widehat{x}}_{iN}>=i\lambda \sqrt{\beta}{\eta}_{ji},\hfill \end{array}$$$$\begin{array}{ccc}& & <{\mathsf{\Lambda}}_{Nk},{\widehat{x}}_{i}>=\sqrt{\beta}<{\mathsf{\Lambda}}_{Nk},{\widehat{x}}_{iN}>=i\lambda \sqrt{\beta}{\eta}_{ki},\hfill \end{array}$$$$\begin{array}{ccc}& & <{\mathsf{\Lambda}}_{NN},{\widehat{x}}_{i}>=\sqrt{\beta}<{\mathsf{\Lambda}}_{NN},{\widehat{x}}_{iN}>=0;\hfill \end{array}$$
 duality of the Lorentz generators with their dual Lorentz matrices:$$\begin{array}{ccc}& & <{\mathsf{\Lambda}}_{jk},{\widehat{x}}_{ip}>=i\lambda ({\eta}_{ji}{\eta}_{kp}{\eta}_{jp}{\eta}_{ki}),\hfill \end{array}$$$$\begin{array}{ccc}& & <{\mathsf{\Lambda}}_{jN},{\widehat{x}}_{ip}>=0=<{\mathsf{\Lambda}}_{Nk},{\widehat{x}}_{ip}>,\hfill \end{array}$$$$\begin{array}{ccc}& & <{\mathsf{\Lambda}}_{NN},{\widehat{x}}_{ip}>=0.\hfill \end{array}$$
 crosscommutation relations between the Snyder coordinates and Lorentz matrices:$$\begin{array}{ccc}\hfill \left[{\mathsf{\Lambda}}_{jk},{\widehat{x}}_{i}\right]& =& \sqrt{\beta}\left[{\mathsf{\Lambda}}_{jk},{\widehat{x}}_{iN}\right]=i\lambda \sqrt{\beta}{\eta}_{ji}{\mathsf{\Lambda}}_{Nk},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left[{\mathsf{\Lambda}}_{jN},{\widehat{x}}_{i}\right]& =& \sqrt{\beta}\left[{\mathsf{\Lambda}}_{jN},{\widehat{x}}_{iN}\right]=i\lambda \sqrt{\beta}{\eta}_{ji}{\mathsf{\Lambda}}_{NN},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left[{\mathsf{\Lambda}}_{Nk},{\widehat{x}}_{i}\right]& =& \sqrt{\beta}\left[{\mathsf{\Lambda}}_{Nk},{\widehat{x}}_{iN}\right]=i\lambda \sqrt{\beta}{\mathsf{\Lambda}}_{ik},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left[{\mathsf{\Lambda}}_{NN},{\widehat{x}}_{i}\right]& =& \sqrt{\beta}\left[{\mathsf{\Lambda}}_{NN},{\widehat{x}}_{iN}\right]=i\lambda \sqrt{\beta}{\mathsf{\Lambda}}_{iN};\hfill \end{array}$$
 and the crosscommutation relations between the Lorentz generators (of $so(1,N1)$) with the Lorentz matrices:$$\begin{array}{ccc}\hfill \left[{\mathsf{\Lambda}}_{jk},{\widehat{x}}_{ip}\right]& =& i\lambda ({\eta}_{ji}{\mathsf{\Lambda}}_{pk}{\eta}_{jp}{\mathsf{\Lambda}}_{ik}),\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left[{\mathsf{\Lambda}}_{jN},{\widehat{x}}_{ip}\right]& =& i\lambda ({\eta}_{ji}{\mathsf{\Lambda}}_{pN}{\eta}_{jp}{\mathsf{\Lambda}}_{iN}),\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left[{\mathsf{\Lambda}}_{Nk},{\widehat{x}}_{ip}\right]& =& i\lambda ({\eta}_{Ni}{\mathsf{\Lambda}}_{pk}{\eta}_{Np}{\mathsf{\Lambda}}_{ik})=0,\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left[{\mathsf{\Lambda}}_{NN},{\widehat{x}}_{ip}\right]& =& 0.\hfill \end{array}$$
4.3. Realizations for Lorentz Matrices
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Heisenberg Double Construction
 (1)
 If the generators of A have a primitive coproduct, then the above formula, for the generators, reduces to:$$\left[{a}^{*},a\right]=<{a}_{\left(1\right)}^{*},a>{a}_{\left(2\right)}^{*}.$$
 (2)
 If the coproduct on the algebra ${A}^{*}$ is opposite, i.e., $\mathsf{\Delta}{a}^{*}={a}_{\left(2\right)}^{*}\otimes {a}_{\left(1\right)}^{*}$, then the commutator becomes:$$\left[{a}^{*},a\right]=a{}_{\left(1\right)}<{a}_{\left(2\right)}^{*},{a}_{\left(2\right)}>{a}_{\left(1\right)}^{*}a\circ {a}^{*}.$$If, in addition, the generators of A have a primitive coproduct then, for the generators, we obtain:$$\left[{a}^{*},a\right]=<{a}_{\left(2\right)}^{*},a>{a}_{\left(1\right)}^{*}.$$
Notes
1  The pairing we propose here is satisfied on the generators (in the first power) only, and it may not be possible to extend it to the full algebra due to the noncoassociative nature of $\tilde{A}$. We also note that the proper definition of $\tilde{A}$ would require the quasiHopf algebra framework [26]. Defining the Heisenberg double within the quasiHopf algebra setting has been proposed, for example, in [27,28] and would be worth investigating further. 
2  The general realization for the coproduct of momenta corresponds to the general realization for the Snyder coordinates; see Equations (6) and (7) in [8], where ${\widehat{x}}_{i}\u25b91={x}_{i},\phantom{\rule{1.em}{0ex}}{M}_{ij}\u25b91=0$. 
3  The difference between the lefthand side and the righthand side of the coassociativity condition for the coproduct of momenta can be explicitly calculated and is as follows, in the first order in $\beta $:
$$\left((id\otimes \mathsf{\Delta})\circ \mathsf{\Delta}{p}_{i}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left((\mathsf{\Delta}\otimes id)\circ \mathsf{\Delta}{p}_{i}\right)=\frac{1}{2}\beta \left({p}_{i}\otimes {p}_{k}\otimes {p}^{k}{p}_{k}\otimes {p}_{i}\otimes {p}^{k}\right)+O\left({\beta}^{2}\right).$$

4  Many authors use the word “generalized” for the version of Snyder space in a different meaning (see, e.g., [18] or [24]); therefore, following [21], we shall call the version used here “extended” instead of generalized—since it is unified with the additional tensorial coordinates transforming as the Lorentz generators. 
5  A topological extension of the corresponding enveloping algebra ${U}_{so(1,N)}$ into an algebra of formal power series ${U}_{so(1,N)}\left[\left[\lambda \right]\right]$ in the formal parameter $\lambda $ is required here. This provides the $\lambda $adic topology (see, for example, Chapter 1.2.10 in [25]). 
6  
7  The Weyl realization for the extended Snyder space is defined as ${e}^{i{k}_{i}{\widehat{x}}_{i}+\frac{i}{2}{k}_{ij}{\widehat{x}}_{ij}}\u25b91={e}^{i{k}_{i}{x}_{i}+\frac{i}{2}{k}_{ij}{x}_{ij}}$ where ${\widehat{x}}_{i}\u25b91={x}_{i},\phantom{\rule{1.em}{0ex}}{\widehat{x}}_{ij}\u25b91={x}_{ij}$, see Equation (17) in [21]. Note that the action differs from the one described in Note 2. 
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Meljanac, S.; Pachoł, A. Heisenberg Doubles for SnyderType Models. Symmetry 2021, 13, 1055. https://doi.org/10.3390/sym13061055
Meljanac S, Pachoł A. Heisenberg Doubles for SnyderType Models. Symmetry. 2021; 13(6):1055. https://doi.org/10.3390/sym13061055
Chicago/Turabian StyleMeljanac, Stjepan, and Anna Pachoł. 2021. "Heisenberg Doubles for SnyderType Models" Symmetry 13, no. 6: 1055. https://doi.org/10.3390/sym13061055