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Classification of commutator algebras leading to the new type of closed Baker–Campbell–Hausdorff formulas
- Source :
- Journal of Geometry and Physics. 97:34-43
- Publication Year :
- 2015
- Publisher :
- Elsevier BV, 2015.
-
Abstract
- We show that there are {\it 13 types} of commutator algebras leading to the new closed forms of the Baker-Campbell-Hausdorff (BCH) formula $$\exp(X)\exp(Y)\exp(Z)=\exp({AX+BZ+CY+DI}) \ , $$ derived in arXiv:1502.06589, JHEP {\bf 1505} (2015) 113. This includes, as a particular case, $\exp(X) \exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$. The algorithm exploits the associativity of the BCH formula and is based on the decomposition $\exp(X)\exp(Y)\exp(Z)=\exp(X)\exp({\alpha Y}) \exp({(1-\alpha) Y}) \exp(Z)$, with $\alpha$ fixed in such a way that it reduces to $\exp({\tilde X})\exp({\tilde Y})$, with $\tilde X$ and $\tilde Y$ satisfying the Van-Brunt and Visser condition $[\tilde X,\tilde Y]=\tilde u\tilde X+\tilde v\tilde Y+\tilde cI$. It turns out that $e^\alpha$ satisfies, in the generic case, an algebraic equation whose exponents depend on the parameters defining the commutator algebra. In nine {\it types} of commutator algebras, such an equation leads to rational solutions for $\alpha$. We find all the equations that characterize the solution of the above decomposition problem by combining it with the Jacobi identity.<br />Comment: 15 pages. Typos corrected. JGP version
- Subjects :
- High Energy Physics - Theory
Jacobi identity
FOS: Physical sciences
General Physics and Astronomy
Type (model theory)
Combinatorics
Baker-Campbell-Hausdorff formula
Commutators
Lie algebras
Mathematical Physics
Physics and Astronomy (all)
Geometry and Topology
symbols.namesake
Lie algebra
FOS: Mathematics
Representation Theory (math.RT)
Algebra over a field
Computer Science::Data Structures and Algorithms
Mathematics
Quantum Physics
Commutator
Mathematical analysis
Mathematical Physics (math-ph)
Algebraic equation
High Energy Physics - Theory (hep-th)
Baker–Campbell–Hausdorff formula
symbols
Quantum Physics (quant-ph)
Mathematics - Representation Theory
Decomposition problem
Subjects
Details
- ISSN :
- 03930440
- Volume :
- 97
- Database :
- OpenAIRE
- Journal :
- Journal of Geometry and Physics
- Accession number :
- edsair.doi.dedup.....2e12fe944b5c9d9f8ade59a76622b5ae