Closed Form of the Baker-Campbell-Hausdorff Formula for the Generators of Semisimple Complex Lie Algebras

Recently it has been introduced an algorithm Baker-Campbell-Hausdorff (BCH) formula, which extends the Van-Brunt and Visser recent results, leading to new closed forms of BCH formula. More recently, it has been shown that there are {\it 13 types} of such commutator algebras. We show, by providing the explicit solutions, that these include the generators of the semisimple complex Lie algebras. More precisely, for any pair, $X$, $Y$ of the Cartan-Weyl basis, we find $W$, linear combination of $X$, $Y$, such that $$ \exp(X) \exp(Y)=\exp(W) $$ The derivation of such closed forms follows, in part, by using the above mentioned recent results. The complete derivation is provided by considering the structure of of the root system. Furthermore, if $X$, $Y$ and $Z$ are three generators of the Cartan-Weyl basis, we find, for a wide class of cases, $W$, linear combination of $X$, $Y$ and $Z$, such that $$ \exp(X) \exp(Y) \exp(Z)=\exp(W) $$ It turns out that the relevant commutator algebras are {\it type 1c-i}, {\it type 4} and {\it type 5}. A key result concerns an iterative application of the algorithm leading to relevant extensions of the cases admitting closed forms of the BCH formula. Here we provide the main steps of such an iteration that will be developed in a forthcoming paper.


Introduction
The Baker-Campbell-Hausdorff (BCH) formula is of considerable interest in Quantum Mechanics, Quantum Field Theory, Conformal Field Theory, Statistical Mechanics, Quantum Computing, Optics, etc. (see, for example [1]- [7]). It is then clear that it is of interest to find cases when the BCH formula admits a closed form. Such a formula expresses Z = ln(e X e Y ) , (1.1) in terms of an infinite series of commutators of X and Y .
In [8] it has been introduced a simple algorithm leading, for a wide class of cases, including some Virasoro subalgebras, to closed forms of the BCH formula. Such an algorithm exploits the associativity of the BCH formula and implement the Jacobi identity. In [9] it has been shown that there are 13 types of commutator algebras admitting such simplified versions of the BCH formula. In [10] the closed form of the BCH formula has been used to covariantize the conformal group. The main points of the algorithm are the following. First, following the Van-Brunt and Visser remarkable result, consider two elements, X and Y , of an associative algebra, with commutator [X, Y ] = uX + vY + cI , (1.2) with u, v and c, being c-numbers and I a central element. Van-Brunt and Visser proved that [11] (see also [12,13]) where f (u, v) is the symmetric function (1.4) In [8] it has been considered the BCH problem of finding the closed form of W in exp(X) exp(Y ) exp(Z) = exp(W ) . (1.5) The first step of the algorithm is to consider the decomposition (1.9) that we rewrite in the form Next, imposing that exp(X) exp(Ỹ ) satisfies the condition to apply (1.3), that is we get the solution of (1.5) [8] (1.14) The condition (1.11) fixes α,c,ũ andṽ, namelỹ with α solution of the equation (1.16) Inserting (1.7) and (1.13) in the Jacobi identity leads to the following linear system for e, m, n and p (1.17) Such a system has thirteen different solutions, reported in the table, depending on the values and algebraic relations of c, d, u, v, w and z. Note that the case cw = dv = 0 corresponds to five different conditions (see [9]).
The thirteen cases of the Jacobi identity Consider the symbol [9] case Jacobi constraints parameters of [X, Z] unf ixed D The first slot specifies under which conditions the linear system (1.17) is solved. This classifies the types of commutator algebras. The constraints are the ones on the commutator parameters that follow by the Jacobi identity. The third slot reports which ones, between the parameters m, n, p and e in [X, Z], remains unfixed. D is the number of the commutator parameters unfixed by the Jacobi identity. In this paper we explicitly show that the above algorithm leads to closed forms for the BCH formula in the case of semisimple complex Lie algebras. In particular, according to the above classification the commutator algebras, we will see that, in several cases, the commutator algebras associated to the BCH problem for semisimple complex Lie algebras corresponds to the type 1c-i, type 4 and type 5. This implies that if X, Y and Z are three generators of the Cartan-Weyl basis, then, for a wide class of cases, W , defined by (1.5) is explicitly expressed as a linear combination of X, Y and Z. In the last section we derive, by iteration, a basic generalization of the algorithm introduced in [8]. This provides important extensions of the cases for which the BCH formula admits a closed form. Before discussing the case of arbitrary semisimple complex Lie algebras, we consider the BCH problem for sl 3 (C).

BCH formulas for the generators of sl 3 (C)
Consider the commutators for sl 3 (C) (see, for example, [14]) In the following we will consider the BCH problem of finding the closed form of W in with X and Z all the possible generators of sl 3 (C). Let us start with E 1 Next, consider E 1 + and E θ − . Also in this case, since

8)
j, k = 1, 2. All these are of the same kind, so that we consider only j = k = 1. By (1.3) Let us now consider the nontrivial cases We focus on (2.10) with k = 1, the case k = 2 is equivalent. This is a particular case of the type 4 commutator algebras [9], and has been worked out, as an example, in [8] for

12)
m, n = −1, 0, 1. A straightforward application of the algorithm leads to [8] exp (2.14) Let us explicitly write down the case of exp(E 1 + ) exp(E 1 − ). Setting λ 0 = 0, corresponding to λ − = −λ + , in Eq.(2.13) and using the identification An identical expression holds in the case of E 2 − , E 2 + , one has just to replace H 1 by H 2 . The same happens for exp(E θ − ) exp(E θ + ). Actually, setting we see that the relevant algebra to treat exp( is again given by (2.16) by obvious substitutions.

BCH formulas for the generators of semisimple complex Lie algebras
Let Φ(g) be the root system of a semisimple complex Lie algebra g. Denote by (α, β) the standard non-degenerate inner product on the dual of the Cartan subalgebra g * 0 , and .
For any g, consider its Cartan-Weyl basis where e αβ , defined for any α = −β, is non-zero if α + β ∈ Φ(g), and zero otherwise. In the following, for any X, Y and Z, identified with any triple of the Cartan-Weyl generators H α and E β , we consider the problem of finding the closed form of W defined by (1.5).
f We will explicitly show that, for a wide class of cases, such closed forms are the ones classified in [9]. In particular, the relevant commutator algebras correspond to the type 1c-i, type 4 and type 5 commutator algebras, leading to the closed forms of the BCH formula introduced in [8].
The type 4 corresponds to the case u = z = 0, namely x u , where x := e α , satisfies the equation Let us first consider the case exp(λ α H α ) exp(µ β E β ). This corresponds to (1.3), so that Therefore, This leads to Let us now consider the BCH problem exp(X) exp(Y ) = exp(W ), with X = µ α E α , Y = µ β E β , α = −β, α + β ∈ Φ(g). In this case it is convenient to use the expansion would be proportional to E 2(α+β) . On the other hand, since α + β is a root, it follows that 2(α + β) / ∈ Φ(g). So that (3.11) vanishes. In order to consider the higher order terms in (3.10), we first recall that the length of root strings is at most five. Here we do not consider the special cases in which either 2α + 3β or 3α + 2β are roots, so that we specialize to the cases in which the length of root strings is at most four. Denote by n X (n Y ) the number of X (Y ) appearing in each term of the expansion (3.10). Then observe that all the terms in the dots of (3.10) correspond to either n X ≥ 4 and n Y ≥ 1, or n Y ≥ 4 and n X ≥ 1, or n X ≥ 2 and n Y ≥ 2. Therefore the terms in (3.10) which are next to (3.11) are proportional to either E 4α+nβ , n ≥ 1, or E nα+4β , n ≥ 1, or E mα+nβ , m, n ≥ 2. On the other hand, as explained above, if α + β ∈ Φ(g), then 4α + nβ, nα + 4β, n ≥ 1, and mα + nβ, m, n ≥ 2, cannot be roots, and the corresponding commutators vanish. Summarizing, in the case α + β ∈ Φ(g), there are three possibilities. If neither 2α + β nor α + 2β belong to Φ(g), then (3.14) The above results can be used as building blocks to solve more elaborated cases, such as Let us now consider exp(µ α E α ) exp(λ α H α ) exp(µ −α E −α ). Again, this is a particular case of the type 4 commutator algebras [9], worked out in [8]. We have where µ −α , λ α , µ α ∈ C, λ − + λ + = λ α , and Note that λ α = 0 reproduces the closed form for exp( As we will discuss later, this corresponds to the type 5 commutator algebras. However, it is instructive to directly derive the solution. Consider the identity where again λ − β + λ + β = λ β . Then note that (3.20) and (3.21) Imposing that the commutator between the exponents on the right hand side of (3.20) and (3.21) vanishes yields It follows that (3.23)

Generalization of the algorithm by iteration
In this section we iterate the algorithm introduced in [8] to extend the Van-Brut and Visser original formula. The problem then is to find, in more general cases, the closed form W , defined by exp(X) exp(Z) = exp(W ) . (4.1) A first result in this direction was already obtained in [8]. The point is to start with the BCH problem for the product of the three exponentials in (1.5), choosing Y in such a way that (1.7) and (1.13) are satisfied. Next, one reproduces the steps of the algorithm leading to (1.12) where Y is replaced by λY and then considers exp(X) exp(Z) = lim λ→0 exp(X) exp(λY ) exp(Z) . It follows that the expression on the right hand side of (4.2), before taking the λ → 0 limit, is given by (1.12) with u, c, z, d and Y multiplied by λ X = g α (λu, v)X + h α (λu, v)(λY ) + l α (λu, v)(λc)I , Y = h β (λz, w)(λY ) + g β (λz, w)Z + l β (λz, w)(λd)I .  It is clear that the above construction can be iterated to generalize the algorithm leading to (1.12). Namely, one can repeat the steps of the algorithm, where now the Van-Brut and Visser condition is replaced by the weaker one given by Lemma 1. In particular, the first step of the algorithm is again to consider the decomposition exp(X) exp(Y ) exp(Z) = exp(X) exp(αY ) exp(βY ) exp(Z) , (4.10) with α + β = 1. However, now each one of the two conditions (1.7) should be replaced by the weaker conditions in Lemma 1. Namely, if there exist Y ′ and Y ′′ satisfying the conditions and that we rewrite as By (4.11), (4.12), (4.14) and (4.15) it follows that now X and Z satisfy the weaker condition [X, Z] = mX + nY + pY ′ + qY ′′ + rỸ ′ + sZ + tI . (4.16) One then can apply the rest of the algorithm and classify all the possible algebras leading to the closed BCH formula, in a similar way to the classification developed in [9]. The next step is to investigate again (4.2), but now using the weaker conditions (4.11), (4.12) and (4.15). This is the content of Lemma 2. analogous to Lemma 1. but with the previous weaker conditions on the commutators.
Lemma 2. If [X, Z] = mX + nY + pY ′ + qY ′′ + rỸ ′ + sZ + tI , Such an algorithm can be iterated indefinitely by applying again the original algorithm to (4.10) but now using Lemma 2., applying the result to (4.2) will lead to Lemma 3., with weaker conditions with respect to Lemma 2., etc.. The full consequences of such a construction are under investigation.