2015, 22: 32-45. doi: 10.3934/era.2015.22.32

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

Received  August 2014 Published  July 2015

We announce an analogue of the celebrated theorem by Campbell, Baker, Hausdorff, and Dynkin for the $q$-exponential $\exp_q(x)=\sum_{n=0}^{\infty} \frac{x^n}{[n]_q!}$, with the usual notation for $q$-factorials: $[n]_q!:=[n-1]_q!\cdot(q^n-1)/(q-1)$ and $[0]_q!:=1$. Our result states that if $x$ and $y$ are non-commuting indeterminates and $[y,x]_q$ is the $q$-commutator $yx-q\,xy$, then there exist linear combinations $Q_{i,j}(x,y)$ of iterated $q$-commutators with exactly $i$ $x$'s, $j$ $y$'s and $[y,x]_q$ in their central position, such that $\exp_q(x)\exp_q(y)=\exp_q\!\big(x+y+\sum_{i,j\geq 1}Q_{i,j}(x,y)\big)$. Our expansion is consistent with the well-known result by Schützenberger ensuring that one has $\exp_q(x)\exp_q(y)=\exp_q(x+y)$ if and only if $[y,x]_q=0$, and it improves former partial results on $q$-deformed exponentiation. Furthermore, we give an algorithm which produces conjecturally a minimal generating set for the relations between $[y,x]_q$-centered $q$-commutators of any bidegree $(i,j)$, and it allows us to compute all possible $Q_{i,j}$.
Citation: Rüdiger Achilles, Andrea Bonfiglioli, Jacob Katriel. The $\boldsymbol{q}$-deformed Campbell-Baker-Hausdorff-Dynkin theorem. Electronic Research Announcements, 2015, 22: 32-45. doi: 10.3934/era.2015.22.32
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), 524-527. doi: 10.1063/1.522937.  Google Scholar

[3]

D. Bonatsos and C. Daskaloyannis, Model of $n$ coupled generalized deformed oscillators for vibrations of polyatomic molecules, Phys. Rev. A, 48 (1993), 3611-3616. doi: 10.1103/PhysRevA.48.3611.  Google Scholar

[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), 1307-1315. doi: 10.1088/0305-4470/27/4/023.  Google Scholar

[5]

A. Bonfiglioli and R. Fulci, Topics in Noncommutative Algebra. The Theorem of Campbell, Baker, Hausdorff and Dynkin, Lecture Notes in Mathematics, 2034, Springer-Verlag, Heidelberg, 2012. doi: 10.1007/978-3-642-22597-0.  Google Scholar

[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), 87-105. doi: 10.1007/BF01319097.  Google Scholar

[8]

V. G. Drinfel'd, Quantum groups, J. Soviet Math., 41 (1988), 898-915. doi: 10.1007/BF01247086.  Google Scholar

[9]

V. G. Drinfel'd, On some unsolved problems in quantum group theory, in Quantum Groups (Leningrad, 1990), Lecture Notes in Math., 1510, Springer, Berlin, 1992, 1-8. doi: 10.1007/BFb0101175.  Google Scholar

[10]

K. Ebrahimi-Fard and D. Manchon, A Magnus- and Fer-type formula in dendriform algebras, Found. Comput. Math., 9 (2009), 295-316. doi: 10.1007/s10208-008-9023-3.  Google Scholar

[11]

K. Ebrahimi-Fard and D. Manchon, Dendriform equations, J. Algebra, 322 (2009), 4053-4079. doi: 10.1016/j.jalgebra.2009.06.002.  Google Scholar

[12]

K. Ebrahimi-Fard and D. Manchon, Twisted dendriform algebras and the pre-Lie Magnus expansion, J. Pure Appl. Algebra, 215 (2011), 2615-2627. doi: 10.1016/j.jpaa.2011.03.004.  Google Scholar

[13]

T. Ernst, A Comprehensive Treatment of $q$-Calculus, Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0431-8.  Google Scholar

[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), 1171-1177. 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), 311-314. doi: 10.1016/S0375-9601(97)00345-9.  Google Scholar

[17]

P. E. T. Jørgensen and R. F. Werner, Coherent states of the q-canonical commutation relations, Comm. Math. Phys., 164 (1994), 455-471. doi: 10.1007/BF02101486.  Google Scholar

[18]

V. Kac and P. Cheung, Quantum Calculus, Universitext; Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.  Google Scholar

[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), 7209-7225. doi: 10.1088/0305-4470/28/24/018.  Google Scholar

[20]

J. Katriel, M. Rasetti and A. I. Solomon, The q-Zassenhaus formula, Lett. Math. Phys., 37 (1996), 11-13. doi: 10.1007/BF00400134.  Google Scholar

[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), 6172-6178. doi: 10.1063/1.530736.  Google Scholar

[22]

J. Katriel and A. I. Solomon, A $q$-analogue of the Campbell-Baker-Hausdorff expansion, J. Phys. A, 24 (1991), L1139-L1142. doi: 10.1088/0305-4470/24/19/003.  Google Scholar

[23]

C. Quesne, Disentangling $q$-exponentials: A general approach, Internat. J. Theoret. Phys., 43 (2004), 545-559. doi: 10.1023/B:IJTP.0000028885.42890.f5.  Google Scholar

[24]

D. L. Reiner, A $q$-analog of the Campbell-Baker-Hausdorff formula, Discrete Math., 43 (1983), 125-129. doi: 10.1016/0012-365X(83)90030-4.  Google Scholar

[25]

N. Reshetikhin, Quantization of Lie bialgebras, Internat. Math. Res. Notices, (1992), 143-151. doi: 10.1155/S1073792892000163.  Google Scholar

[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), 352-353.  Google Scholar

[27]

R. Sridhar and R. Jagannathan, On the q-analogues of the Zassenhaus formula for disentangling exponential operators, J. Comput. Appl. Math., 160 (2003), 297-305. doi: 10.1016/S0377-0427(03)00633-2.  Google Scholar

[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), 75, Springer, Netherlands, 1992. doi: 10.1007/978-94-017-2881-2.  Google Scholar

[29]

H. Wachter, q-Exponentials on quantum spaces, Eur. Phys. J. C Part. Fields, 37 (2004), 379-389. doi: 10.1140/epjc/s2004-01999-5.  Google Scholar

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), 524-527. doi: 10.1063/1.522937.  Google Scholar

[3]

D. Bonatsos and C. Daskaloyannis, Model of $n$ coupled generalized deformed oscillators for vibrations of polyatomic molecules, Phys. Rev. A, 48 (1993), 3611-3616. doi: 10.1103/PhysRevA.48.3611.  Google Scholar

[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), 1307-1315. doi: 10.1088/0305-4470/27/4/023.  Google Scholar

[5]

A. Bonfiglioli and R. Fulci, Topics in Noncommutative Algebra. The Theorem of Campbell, Baker, Hausdorff and Dynkin, Lecture Notes in Mathematics, 2034, Springer-Verlag, Heidelberg, 2012. doi: 10.1007/978-3-642-22597-0.  Google Scholar

[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), 87-105. doi: 10.1007/BF01319097.  Google Scholar

[8]

V. G. Drinfel'd, Quantum groups, J. Soviet Math., 41 (1988), 898-915. doi: 10.1007/BF01247086.  Google Scholar

[9]

V. G. Drinfel'd, On some unsolved problems in quantum group theory, in Quantum Groups (Leningrad, 1990), Lecture Notes in Math., 1510, Springer, Berlin, 1992, 1-8. doi: 10.1007/BFb0101175.  Google Scholar

[10]

K. Ebrahimi-Fard and D. Manchon, A Magnus- and Fer-type formula in dendriform algebras, Found. Comput. Math., 9 (2009), 295-316. doi: 10.1007/s10208-008-9023-3.  Google Scholar

[11]

K. Ebrahimi-Fard and D. Manchon, Dendriform equations, J. Algebra, 322 (2009), 4053-4079. doi: 10.1016/j.jalgebra.2009.06.002.  Google Scholar

[12]

K. Ebrahimi-Fard and D. Manchon, Twisted dendriform algebras and the pre-Lie Magnus expansion, J. Pure Appl. Algebra, 215 (2011), 2615-2627. doi: 10.1016/j.jpaa.2011.03.004.  Google Scholar

[13]

T. Ernst, A Comprehensive Treatment of $q$-Calculus, Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0431-8.  Google Scholar

[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), 1171-1177. 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), 311-314. doi: 10.1016/S0375-9601(97)00345-9.  Google Scholar

[17]

P. E. T. Jørgensen and R. F. Werner, Coherent states of the q-canonical commutation relations, Comm. Math. Phys., 164 (1994), 455-471. doi: 10.1007/BF02101486.  Google Scholar

[18]

V. Kac and P. Cheung, Quantum Calculus, Universitext; Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.  Google Scholar

[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), 7209-7225. doi: 10.1088/0305-4470/28/24/018.  Google Scholar

[20]

J. Katriel, M. Rasetti and A. I. Solomon, The q-Zassenhaus formula, Lett. Math. Phys., 37 (1996), 11-13. doi: 10.1007/BF00400134.  Google Scholar

[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), 6172-6178. doi: 10.1063/1.530736.  Google Scholar

[22]

J. Katriel and A. I. Solomon, A $q$-analogue of the Campbell-Baker-Hausdorff expansion, J. Phys. A, 24 (1991), L1139-L1142. doi: 10.1088/0305-4470/24/19/003.  Google Scholar

[23]

C. Quesne, Disentangling $q$-exponentials: A general approach, Internat. J. Theoret. Phys., 43 (2004), 545-559. doi: 10.1023/B:IJTP.0000028885.42890.f5.  Google Scholar

[24]

D. L. Reiner, A $q$-analog of the Campbell-Baker-Hausdorff formula, Discrete Math., 43 (1983), 125-129. doi: 10.1016/0012-365X(83)90030-4.  Google Scholar

[25]

N. Reshetikhin, Quantization of Lie bialgebras, Internat. Math. Res. Notices, (1992), 143-151. doi: 10.1155/S1073792892000163.  Google Scholar

[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), 352-353.  Google Scholar

[27]

R. Sridhar and R. Jagannathan, On the q-analogues of the Zassenhaus formula for disentangling exponential operators, J. Comput. Appl. Math., 160 (2003), 297-305. doi: 10.1016/S0377-0427(03)00633-2.  Google Scholar

[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), 75, Springer, Netherlands, 1992. doi: 10.1007/978-94-017-2881-2.  Google Scholar

[29]

H. Wachter, q-Exponentials on quantum spaces, Eur. Phys. J. C Part. Fields, 37 (2004), 379-389. doi: 10.1140/epjc/s2004-01999-5.  Google Scholar

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