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The $\boldsymbol{q}$-deformed Campbell-Baker-Hausdorff-Dynkin theorem
1. | Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy, Italy |
2. | Department of Chemistry, Technion -- Isreal Institute of Technology, Haifa 32000, Israel |
References:
[1] |
R. Achilles, A. Bonfiglioli and J. Katriel, A sixth-order expansion of the $q$-Campbell-Baker-Hausdorff series,, preprint, (2014). Google Scholar |
[2] |
M. Arik and D. D. Coon, Hilbert spaces of analytic functions and generalized coherent states,, J. Mathematical Phys., 17 (1976), 524.
doi: 10.1063/1.522937. |
[3] |
D. Bonatsos and C. Daskaloyannis, Model of $n$ coupled generalized deformed oscillators for vibrations of polyatomic molecules,, Phys. Rev. A, 48 (1993), 3611.
doi: 10.1103/PhysRevA.48.3611. |
[4] |
F. Bonechi, E. Celeghini, R. Giachetti, C. M. Pereña, E. Sorace and M. Tarlini, Exponential mapping for nonsemisimple quantum groups,, J. Phys. A, 27 (1994), 1307.
doi: 10.1088/0305-4470/27/4/023. |
[5] |
A. Bonfiglioli and R. Fulci, Topics in Noncommutative Algebra. The Theorem of Campbell, Baker, Hausdorff and Dynkin,, Lecture Notes in Mathematics, (2034).
doi: 10.1007/978-3-642-22597-0. |
[6] |
A. Bonfiglioli and J. Katriel, The $q$-analogue of the Campbell-Baker-Hausdorff-Dynkin Theorem,, submitted, (2015). Google Scholar |
[7] |
J. Cigler, Operatormethoden für $q$-Identitäten,, Monatsh. Math., 88 (1979), 87.
doi: 10.1007/BF01319097. |
[8] |
V. G. Drinfel'd, Quantum groups,, J. Soviet Math., 41 (1988), 898.
doi: 10.1007/BF01247086. |
[9] |
V. G. Drinfel'd, On some unsolved problems in quantum group theory,, in Quantum Groups (Leningrad, (1990), 1.
doi: 10.1007/BFb0101175. |
[10] |
K. Ebrahimi-Fard and D. Manchon, A Magnus- and Fer-type formula in dendriform algebras,, Found. Comput. Math., 9 (2009), 295.
doi: 10.1007/s10208-008-9023-3. |
[11] |
K. Ebrahimi-Fard and D. Manchon, Dendriform equations,, J. Algebra, 322 (2009), 4053.
doi: 10.1016/j.jalgebra.2009.06.002. |
[12] |
K. Ebrahimi-Fard and D. Manchon, Twisted dendriform algebras and the pre-Lie Magnus expansion,, J. Pure Appl. Algebra, 215 (2011), 2615.
doi: 10.1016/j.jpaa.2011.03.004. |
[13] |
T. Ernst, A Comprehensive Treatment of $q$-Calculus,, Birkhäuser/Springer Basel AG, (2012).
doi: 10.1007/978-3-0348-0431-8. |
[14] |
A. M. Gavrilik and Yu. A. Mishchenko, Deformed Bose gas models aimed at taking into account both comositeness of particles and their interaction,, Ukr. J. Phys., 58 (2013), 1171. Google Scholar |
[15] |
A. C. Hearn, REDUCE, A portable general-purpose computer algebra system., Available from: , (). Google Scholar |
[16] |
A. Inomata and S. Kirchner, Bose-Einstein condensation of a quon gas,, Phys. Lett. A, 231 (1997), 311.
doi: 10.1016/S0375-9601(97)00345-9. |
[17] |
P. E. T. Jørgensen and R. F. Werner, Coherent states of the q-canonical commutation relations,, Comm. Math. Phys., 164 (1994), 455.
doi: 10.1007/BF02101486. |
[18] |
V. Kac and P. Cheung, Quantum Calculus,, Universitext; Springer-Verlag, (2002).
doi: 10.1007/978-1-4613-0071-7. |
[19] |
J. Katriel and G. Duchamp, Ordering relations for q-boson operators, continued fractions techniques and the q-CBH enigma,, J. Phys. A, 28 (1995), 7209.
doi: 10.1088/0305-4470/28/24/018. |
[20] |
J. Katriel, M. Rasetti and A. I. Solomon, The q-Zassenhaus formula,, Lett. Math. Phys., 37 (1996), 11.
doi: 10.1007/BF00400134. |
[21] |
J. Katriel and A. I. Solomon, A no-go theorem for a Lie-consistent $q$-Campbell-Baker-Hausdorff expansion,, J. Math. Phys., 35 (1994), 6172.
doi: 10.1063/1.530736. |
[22] |
J. Katriel and A. I. Solomon, A $q$-analogue of the Campbell-Baker-Hausdorff expansion,, J. Phys. A, 24 (1991).
doi: 10.1088/0305-4470/24/19/003. |
[23] |
C. Quesne, Disentangling $q$-exponentials: A general approach,, Internat. J. Theoret. Phys., 43 (2004), 545.
doi: 10.1023/B:IJTP.0000028885.42890.f5. |
[24] |
D. L. Reiner, A $q$-analog of the Campbell-Baker-Hausdorff formula,, Discrete Math., 43 (1983), 125.
doi: 10.1016/0012-365X(83)90030-4. |
[25] |
N. Reshetikhin, Quantization of Lie bialgebras,, Internat. Math. Res. Notices, (1992), 143.
doi: 10.1155/S1073792892000163. |
[26] |
M.-P. Schützenberger, Une interprétation de certains solutions de l'équation fonctionnelle: $F(x+y)=F(x)F(y)$,, C. R. Acad. Sci. Paris, 236 (1953), 352.
|
[27] |
R. Sridhar and R. Jagannathan, On the q-analogues of the Zassenhaus formula for disentangling exponential operators,, J. Comput. Appl. Math., 160 (2003), 297.
doi: 10.1016/S0377-0427(03)00633-2. |
[28] |
N. Ja. Vilenkin and A. U. Klimyk, Representations of Lie Groups and Special Functions. Vol. 3: Classical and Quantum Groups and Special Functions,, Mathematics and Its Applications (Soviet Series), (1992).
doi: 10.1007/978-94-017-2881-2. |
[29] |
H. Wachter, q-Exponentials on quantum spaces,, Eur. Phys. J. C Part. Fields, 37 (2004), 379.
doi: 10.1140/epjc/s2004-01999-5. |
show all references
References:
[1] |
R. Achilles, A. Bonfiglioli and J. Katriel, A sixth-order expansion of the $q$-Campbell-Baker-Hausdorff series,, preprint, (2014). Google Scholar |
[2] |
M. Arik and D. D. Coon, Hilbert spaces of analytic functions and generalized coherent states,, J. Mathematical Phys., 17 (1976), 524.
doi: 10.1063/1.522937. |
[3] |
D. Bonatsos and C. Daskaloyannis, Model of $n$ coupled generalized deformed oscillators for vibrations of polyatomic molecules,, Phys. Rev. A, 48 (1993), 3611.
doi: 10.1103/PhysRevA.48.3611. |
[4] |
F. Bonechi, E. Celeghini, R. Giachetti, C. M. Pereña, E. Sorace and M. Tarlini, Exponential mapping for nonsemisimple quantum groups,, J. Phys. A, 27 (1994), 1307.
doi: 10.1088/0305-4470/27/4/023. |
[5] |
A. Bonfiglioli and R. Fulci, Topics in Noncommutative Algebra. The Theorem of Campbell, Baker, Hausdorff and Dynkin,, Lecture Notes in Mathematics, (2034).
doi: 10.1007/978-3-642-22597-0. |
[6] |
A. Bonfiglioli and J. Katriel, The $q$-analogue of the Campbell-Baker-Hausdorff-Dynkin Theorem,, submitted, (2015). Google Scholar |
[7] |
J. Cigler, Operatormethoden für $q$-Identitäten,, Monatsh. Math., 88 (1979), 87.
doi: 10.1007/BF01319097. |
[8] |
V. G. Drinfel'd, Quantum groups,, J. Soviet Math., 41 (1988), 898.
doi: 10.1007/BF01247086. |
[9] |
V. G. Drinfel'd, On some unsolved problems in quantum group theory,, in Quantum Groups (Leningrad, (1990), 1.
doi: 10.1007/BFb0101175. |
[10] |
K. Ebrahimi-Fard and D. Manchon, A Magnus- and Fer-type formula in dendriform algebras,, Found. Comput. Math., 9 (2009), 295.
doi: 10.1007/s10208-008-9023-3. |
[11] |
K. Ebrahimi-Fard and D. Manchon, Dendriform equations,, J. Algebra, 322 (2009), 4053.
doi: 10.1016/j.jalgebra.2009.06.002. |
[12] |
K. Ebrahimi-Fard and D. Manchon, Twisted dendriform algebras and the pre-Lie Magnus expansion,, J. Pure Appl. Algebra, 215 (2011), 2615.
doi: 10.1016/j.jpaa.2011.03.004. |
[13] |
T. Ernst, A Comprehensive Treatment of $q$-Calculus,, Birkhäuser/Springer Basel AG, (2012).
doi: 10.1007/978-3-0348-0431-8. |
[14] |
A. M. Gavrilik and Yu. A. Mishchenko, Deformed Bose gas models aimed at taking into account both comositeness of particles and their interaction,, Ukr. J. Phys., 58 (2013), 1171. Google Scholar |
[15] |
A. C. Hearn, REDUCE, A portable general-purpose computer algebra system., Available from: , (). Google Scholar |
[16] |
A. Inomata and S. Kirchner, Bose-Einstein condensation of a quon gas,, Phys. Lett. A, 231 (1997), 311.
doi: 10.1016/S0375-9601(97)00345-9. |
[17] |
P. E. T. Jørgensen and R. F. Werner, Coherent states of the q-canonical commutation relations,, Comm. Math. Phys., 164 (1994), 455.
doi: 10.1007/BF02101486. |
[18] |
V. Kac and P. Cheung, Quantum Calculus,, Universitext; Springer-Verlag, (2002).
doi: 10.1007/978-1-4613-0071-7. |
[19] |
J. Katriel and G. Duchamp, Ordering relations for q-boson operators, continued fractions techniques and the q-CBH enigma,, J. Phys. A, 28 (1995), 7209.
doi: 10.1088/0305-4470/28/24/018. |
[20] |
J. Katriel, M. Rasetti and A. I. Solomon, The q-Zassenhaus formula,, Lett. Math. Phys., 37 (1996), 11.
doi: 10.1007/BF00400134. |
[21] |
J. Katriel and A. I. Solomon, A no-go theorem for a Lie-consistent $q$-Campbell-Baker-Hausdorff expansion,, J. Math. Phys., 35 (1994), 6172.
doi: 10.1063/1.530736. |
[22] |
J. Katriel and A. I. Solomon, A $q$-analogue of the Campbell-Baker-Hausdorff expansion,, J. Phys. A, 24 (1991).
doi: 10.1088/0305-4470/24/19/003. |
[23] |
C. Quesne, Disentangling $q$-exponentials: A general approach,, Internat. J. Theoret. Phys., 43 (2004), 545.
doi: 10.1023/B:IJTP.0000028885.42890.f5. |
[24] |
D. L. Reiner, A $q$-analog of the Campbell-Baker-Hausdorff formula,, Discrete Math., 43 (1983), 125.
doi: 10.1016/0012-365X(83)90030-4. |
[25] |
N. Reshetikhin, Quantization of Lie bialgebras,, Internat. Math. Res. Notices, (1992), 143.
doi: 10.1155/S1073792892000163. |
[26] |
M.-P. Schützenberger, Une interprétation de certains solutions de l'équation fonctionnelle: $F(x+y)=F(x)F(y)$,, C. R. Acad. Sci. Paris, 236 (1953), 352.
|
[27] |
R. Sridhar and R. Jagannathan, On the q-analogues of the Zassenhaus formula for disentangling exponential operators,, J. Comput. Appl. Math., 160 (2003), 297.
doi: 10.1016/S0377-0427(03)00633-2. |
[28] |
N. Ja. Vilenkin and A. U. Klimyk, Representations of Lie Groups and Special Functions. Vol. 3: Classical and Quantum Groups and Special Functions,, Mathematics and Its Applications (Soviet Series), (1992).
doi: 10.1007/978-94-017-2881-2. |
[29] |
H. Wachter, q-Exponentials on quantum spaces,, Eur. Phys. J. C Part. Fields, 37 (2004), 379.
doi: 10.1140/epjc/s2004-01999-5. |
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