American Institute of Mathematical Sciences

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

The $\boldsymbol{q}$-deformed Campbell-Baker-Hausdorff-Dynkin theorem

Rüdiger Achilles 1, , Andrea Bonfiglioli 1, and  Jacob Katriel 2,

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 $Q_{i,j}$.
Keywords: q-deformed CBHD series, $q$-commutator identities., $q$-calculus, Exponential theorem, Campbell-Baker-Hausdoff-Dynkin (CBHD) series.
Mathematics Subject Classification: 05A30, 81R50, 16T20 (Pri); 17B37, 05A15 (Sec.
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

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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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