On multivariable Zassenhaus formula

Wang, Linsong; Gao, Yun; Jing, Naihuan

doi:10.1007/s11464-019-0760-1

Cited by 4 publications

(3 citation statements)

References 13 publications

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“…In what follows we use the classical Baker-Campbell-Hausdorff (BCH) and Zassenhaus formulas. Let us recall these formulas following [Bonfiglioli et al, 2012, Li et al, 2019, Manetti, 2012, Wang et al, 2019. Let A be an associative algebra with unit over k. In the formal power series algebra A [[t]] the function exp is well defined for any series without constant term, and log is well defined for any series with the constant term equal 1.…”

Section: Classical Formulasmentioning

confidence: 99%

Tits-type Alternative for Groups Acting on Toric Affine Varieties

Arzhantsev

Zaidenberg

2021

International Mathematics Research Notices

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Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots ,U_s$ of $\mathop{\textrm{Aut}}(X)$, which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\ldots ,U_s$ verifies the following alternative of Tits type: either $G$ is a unipotent algebraic group or it contains a non-abelian free subgroup. We deduce that if $G$ is $2$-transitive on a $G$-orbit in $X$, then $G$ contains a non-abelian free subgroup and so is of exponential growth.

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Section: Classical Formulasmentioning

confidence: 99%

Tits-type Alternative for Groups Acting on Toric Affine Varieties

Arzhantsev

Zaidenberg

2021

International Mathematics Research Notices

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“…The Zassenhaus formula [1,2] plays an important role in various fields of physics, such as the Dirac monopole problem [3], quantum spin lattices [4], fluids dynamics [5] and the study of solitary waves [6], statistical mechanics [7], many-body theories or quantum optics. In particle accelerator physics, the Zassenhaus formula was successfully used to compute the relevant maps both in Taylor-series and factorized-product forms [8].…”

Section: Introductionmentioning

confidence: 99%

Closed forms of the Zassenhaus formula

Dupays

2023

J. Phys. A: Math. Theor.

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The Zassenhaus formula finds many applications in theoretical 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 expressed as the product of exponentials of operators. The exponential of the sum can then be decomposed as the product of the exponentials multiplied by a supplementary term which takes generally the form of an infinite product of exponentials. Such a procedure is often referred to as ``disentanglement''. However, for some special commutators, closed forms can be found. In this work, we propose a closed form for the Zassenhaus formula when the commutator of operators $\hX$ and $\hY$ satisfy the relation $[\hX,\hY]=u\hX+v\hY+c\mathbbm{1}$. Such an expression boils down to three equivalent versions, a left-sided, a centered and a right-sided formula:%\begin{eqnarray}e^{\hX+\hY}&=&e^{\hX}e^{\hY}e^{g_{r}(u,v)[\hX,\hY]}=e^{\hX}e^{g_{c}(u,v)[\hX,\hY]}e^{\hY}\nonumber\\&=&e^{g_{\ell}(u,v)[\hX,\hY]}e^{\hX}e^{\hY},\end{eqnarray}%with respective arguments,%\begin{eqnarray}g_{r}(u,v)&=&g_{c}(v,u)e^{u}=g_{\ell}(v,u)\nonumber\\&=&\frac{u\left(e^{u-v}-e^{u}\right)+v\left(e^{u}-1\right)}{vu(u-v)}\end{eqnarray}%for $u\ne v$ and %\begin{eqnarray}g_{r}(u,u)=\frac{u+1-e^u}{u^2}\;\;\;\;\mathrm{with}\;\;\;\; g_r(0,0)=-1/2.\end{eqnarray}With additional special case\begin{eqnarray} g_{r}(0,v)= -\frac{e^{-v}-1+v}{v^{2}}, \quad & g_{r}(u,0)=\frac{e^{u}(1-u)-1}{u^{2}}.\end{eqnarray}

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“…Zassenhaus formula found recently a regain of interest [1,2] due to its numerous applications in various fields such as Soliton physics, Dirac's monopole problem, quantum lattices of spins or even fluids dynamics [3,4,5,6]. It is also studied for mathematical fundamental purpose such as disentangling exponential operators [7,8] or in differential geometry [9].…”

Section: Introductionmentioning

confidence: 99%

Closed forms of Zassenhaus formula

Dupays¹

2021

Preprint

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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 + c½. Zassenhaus formula then reduces to the closed form e X+Y = e X e Y e gr (u,v)[X,Y ] = e X e gc(u,v)[X,Y ] e Y = e g l (u,v)[X,Y ] e X e Y ,(1) a left-sided, centered and right-sided formula are found, with respective arguments, gr(u, v) = gc(v, u)e u = g l (v, u) = u e u−v − e u + v (e u − 1) vu(u − v) .(2)

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On multivariable Zassenhaus formula

Abstract: In this paper, we give a recursive algorithm to compute the multivariable Zassenhaus formula e X1+X2+···+Xn = e X1 e X2 · · · e Xn ∞ k=2 e W k and derive an effective recursion formula of W

Cited by 4 publications

References 13 publications

Tits-type Alternative for Groups Acting on Toric Affine Varieties

Tits-type Alternative for Groups Acting on Toric Affine Varieties

Closed forms of the Zassenhaus formula

Closed forms of Zassenhaus formula

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