• Corpus ID: 235727388

Closed forms of Zassenhaus formula

@inproceedings{Dupays2021ClosedFO,
  title={Closed forms of Zassenhaus formula},
  author={L. Dupays},
  year={2021}
}

• L. Dupays
• Published 2 July 2021
• Mathematics

Zassenhaus formula is used in a wide range of fields in physics or mathematics, from fluid dynamics to differential geometry. The non commutativity of the elements of the algebra implies that the exponential of a sum of operators cannot be splitted in the product of exponential of operators. The exponential of the sum can then be decomposed as the product of the exponentials multiplied by a supplementary term which form is generally an infinite product of exponentials. However, for some special commutators, closed forms can be found. We propose a closed form for Zassenhaus formula when the commutator of the operators X and Y satisfy the relation [X, Y ] = uX + vY + c1. Zassenhaus formula then reduces to the closed form e X+Y = ee er ] = eec e = el ee , (1) a left-sided, centered and right-sided formula are found, with respective arguments, gr(u, v) = gc(v, u)e u = gl(v, u) = u ( e − e ) + v (e − 1) vu(u− v) . (2)