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. 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. 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. 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). 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. doi: 10.1007/BF01319097. Google Scholar

[8]

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

[9]

V. G. Drinfel'd, On some unsolved problems in quantum group theory,, in Quantum Groups (Leningrad, (1990), 1. 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. doi: 10.1007/s10208-008-9023-3. Google Scholar

[11]

K. Ebrahimi-Fard and D. Manchon, Dendriform equations,, J. Algebra, 322 (2009), 4053. 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. doi: 10.1016/j.jpaa.2011.03.004. Google Scholar

[13]

T. Ernst, A Comprehensive Treatment of $q$-Calculus,, Birkhäuser/Springer Basel AG, (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. 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. 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. doi: 10.1007/BF02101486. Google Scholar

[18]

V. Kac and P. Cheung, Quantum Calculus,, Universitext; Springer-Verlag, (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. 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. 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. 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). 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. 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. doi: 10.1016/0012-365X(83)90030-4. Google Scholar

[25]

N. Reshetikhin, Quantization of Lie bialgebras,, Internat. Math. Res. Notices, (1992), 143. 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. 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. 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), (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. 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. 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. 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. 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). 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. doi: 10.1007/BF01319097. Google Scholar

[8]

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

[9]

V. G. Drinfel'd, On some unsolved problems in quantum group theory,, in Quantum Groups (Leningrad, (1990), 1. 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. doi: 10.1007/s10208-008-9023-3. Google Scholar

[11]

K. Ebrahimi-Fard and D. Manchon, Dendriform equations,, J. Algebra, 322 (2009), 4053. 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. doi: 10.1016/j.jpaa.2011.03.004. Google Scholar

[13]

T. Ernst, A Comprehensive Treatment of $q$-Calculus,, Birkhäuser/Springer Basel AG, (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. 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. 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. doi: 10.1007/BF02101486. Google Scholar

[18]

V. Kac and P. Cheung, Quantum Calculus,, Universitext; Springer-Verlag, (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. 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. 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. 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). 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. 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. doi: 10.1016/0012-365X(83)90030-4. Google Scholar

[25]

N. Reshetikhin, Quantization of Lie bialgebras,, Internat. Math. Res. Notices, (1992), 143. 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. 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. 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), (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. doi: 10.1140/epjc/s2004-01999-5. Google Scholar

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