Explicit Baker-Campbell-Hausdorff formulae for some specific Lie algebras

In a previous article, [arXiv:1501.02506, JPhysA {\bf48} (2015) 225207], we demonstrated that whenever $[X,Y] = u X + vY + cI$ the Baker-Campbell-Hausdorff formula reduces to the tractable closed-form expression \[ Z(X,Y)=\ln( e^X e^Y ) = X+Y+ f(u,v) \; [X,Y], \] where $f(u,v)=f(v,u)$ is explicitly given by \[ f(u,v) = {(u-v)e^{u+v}-(ue^u-ve^v)\over u v (e^u - e^v)} = {(u-v)-(ue^{-v}-ve^{-u})\over u v (e^{-v} - e^{-u})}. \] This is much more general than the results usually presented for either the Heisenberg commutator $[P,Q]=-i\hbar I$, or the creation-destruction commutator $[a,a^\dagger]=I$. In the current article we shall further generalize and extend this result, primarily by relaxing the input assumptions. We shall work with the structure constants $f_{ab}{}^c$ of the Lie algebra, (defined by $[T_a,T_b] = f_{ab}{}^c \; T_c$), and identify suitable constraints one can place on the structure constants to make the Baker--Campbell--Hausdorff formula tractable. We shall also develop related results using the commutator sub-algebra $[\mathfrak{g},\mathfrak{g}]$ of the relevant Lie algebra $\mathfrak{g}$. Under suitable conditions, and taking $L_A B = [A,B]$ as usual, we shall demonstrate that \[ \ln( e^X e^Y ) = X + Y + {I \over e^{-L_X} - e^{+L_Y} } \left( {I-e^{-L_X}\over L_X} + {I-e^{+L_Y}\over L_Y} \right) [X,Y]. \]

. This is much more general than the results usually presented for either the Heisenberg commutator [P, Q] = −i I, or the creation-destruction commutator [a, a † ] = I. In the current article we shall further generalize and extend this result, primarily by relaxing the input assumptions. We shall work with the structure constants f ab c of the Lie algebra, (defined by [T a , T b ] = f ab c T c ), and identify suitable constraints one can place on the structure constants to make the Baker-Campbell-Hausdorff formula tractable. We shall also develop related results using the commutator sub-algebra [g, g] of the relevant Lie algebra g. Under suitable conditions, and taking L A B = [A, B] as usual, we shall demonstrate that ln(e X e Y ) = X + Y + I e −L X − e +L Y

Introduction
The Baker-Campbell-Hausdorff formula for Z(X, Y ) = ln(e X e Y ) when X and Y are non-commutative quantities is a general multi-purpose result of considerable interest in not only both pure and applied mathematics [1][2][3][4][5][6][7][8][9][10][11][12], but also within the fields of theoretical physics, physical chemistry, the theory of numerical integration, and other disciplines [11][12][13][14][15][16][17][18]. Applications include topics as apparently remote and unconnected as the embedding problem for stochastic matrices. For our current purposes, the general Baker-Campbell-Hausdorff formula can best be written as [11] ln(e X e Y ) = X + Y + Here as usual L A B = [A, B]. If one makes no further simplifying assumptions, then this expression expands to an infinite series of nested commutators, with the first few well-known terms being [1,2] ln Higher-order terms in the expansion quickly become very unwieldy. See for instance references [1][2][3][4][5][6][7][8][9][10][11][12]. In contrast, by making specific simplifying assumptions about the commutator [X, Y ] one can sometimes obtain a terminating series, or develop other ways of simplifying the expansion. The most common terminating series results are: • If [X, Y ] = 0, then: ln(e X e Y ) = X + Y .
• If [X, Y ] = vY , then: Observe that [X, Y ] = vY implies that X acts as a "shift operator", a "ladder operator", for Y , thus allowing one to invoke the techniques of Sack [6]. This particular result can also be extracted from equation (7.9) of Wilcox [7]; but only after some nontrivial manipulations.
Considerably more subtle is our recent result [11]: • If [X, Y ] = uX + vY + cI, then: It is often more useful to write this as Sometimes the structure is more clearly brought out by writing this in the form In a series of very recent articles, Matone [19][20][21], has generalized this result in various ways. 1 In the current article we shall also develop several generalizations -but shall work towards a rather different goal by instead seeking to weaken the conditions under which this simplified form of the Baker-Campbell-Hausdorff [BCH] formula applies.

Strategy
In reference [11] our strategy was to use (1 − e v e −tu ) n−1 n(n + 1)

Structure constants
Let us work in a Lie algebra with basis T a , and define the structure constants f ab c by taking [T a , T b ] = f ab c T c . Then setting We shall now systematically build up to our most general result in several incremental stages. The art lies in choosing structure constants appropriately.

Case 1: Reproducing the special commutator
Let us first choose This is our special commutator of equation (2.1), under the identifications Thus this particular set of structure constants has not actually generalized our previous result -instead it has provided a natural way in which the specific commutator (2.1) will automatically arise.

Case 2: Commutator algebras of dimension unity
Let us now choose f ab c = ω ab n c .
Note that the special commutator of equation (2.1) can certainly be put into this form. Specifically, by taking T a = (X, Y, I) we have But we shall now work with completely arbitrary n c and ω ab , thereby generalizing our previous result. Let us define This establishes equation (2.2) without equation (2.1). 2 Consequently for the same function f (u, v) as previously encountered. More explicitly we now have We can also write this as That is, (at least in this particular class of Lie algebras), we see that we can view the Baker-Campbell-Hausdorff formula as a generalized notion of "addition". By defining the generalized "addition" operator ⊕ via (x ⊕ y) c T c = ln(e X e Y ), we explicitly have Now the statement that f ab c = ω ab n c can be rephrased as the statement that the commutator sub-algebra [g, g], (the sub-algebra formed from the commutators of the ambient Lie algebra g), is of dimension unity. 3 Note that the object [g, g] is also called the first derived sub-algebra, or the first lower central sub-algebra, (aka first descending central sub-algebra), though these two series of sub-algebras will differ once one goes to higher levels. Observe that: • If the commutator sub-algebra [g, g] is of dimension zero, then the Lie algebra is Abelian, and the Baker-Campbell-Hausdorff result is trivial: ln(e X e Y ) = X + Y .
• If the commutator sub-algebra [g, g] is of dimension one then [T a , • We can naturally split this into 2 sub-cases: That is, the second lower central sub-algebra, (and so all the higher-order lower central sub-algebras), all equal the first lower central sub-algebra. This can also be phrased as the demand that the commutator sub-algebra be an ideal of the underlying Lie algebra.
In short, the explicit Baker-Campbell-Hausdorff formula (3.21) holds whenever the commutator sub-algebra [g, g] is of dimension unity.

Case 3: Nilpotent Lie algebras
Consider now the higher terms in the lower central series, defined iteratively by g 0 = g; g 1 = [g, g]; g n = [g, g n−1 ]. (3.28) If, for some n, we have g n = 0 then the Lie algebra g is said to be "nilpotent". In this case all nth-order and higher commutators vanish and the Baker-Campbell-Hausdorff series truncates -but this result has previously been (implicitly) used when developing the Reinsch algorithm [9,10], and our own variant thereof [12]. That algorithm works by utilizing a faithful representation for the first n nested commutators of a free Lie algebra in terms of strictly upper triangular (n + 1) × (n + 1) matrices with entires only on the first super-diagonal. That is: working with a level-n nilpotent Lie algebra "merely" reproduces the first n terms in the Baker-Campbell-Hausdorff formula, and gives zeros thereafter. So while certainly useful, this is not really new [9,10,12]. In terms of the structure constants, nilpotency is achieved if at some stage

Case 4: Abelian commutator algebras
Can the discussion above be generalized even further? Note that in all generality That is, the commutator sub-algebra is Abelian. This is a special case of a "solvable" Lie algebra. Can anything be done for more general solvable Lie algebras? Let us now consider the situation where the the commutator sub-algebra is Abelian, but we do not demand that the commutator algebra is one dimensional. The Jacobi identity leads to  (3.39) and treat the L X and L Y as though they commute. But then, summing and integrating as previously, we have  Furthermore, we note the series expansion which verifies that, (as it should), the operator f (L X , −L Y ) contains only non-negative powers of L X and L Y .

Case 5: [X, Y ] is in the centre of the commutator algebra
Let us now relax the conditions for the validity of this result even further: The key step is to realize that In terms of the structure constants this is equivalent to the weakened constraint Under this milder condition we still have (effective) commutativity of L X with L Y , thereby allowing us to treat the L X and L Y appearing in the Baker-Campbell-Hausdorff formula (1.1) as though they commute. Integrating and summing the series we again see 3.6 Case 6: [X, Y ] is in the centralizer of {L m X L n Y [X, Y ]} As our final weakening of the input assumptions, (while still keeping the same strength conclusions), take an arbitrary but fixed ambient Lie algebra g and consider the set The construction of this set is inspired by considering the form of the terms which appear in the Baker-Campbell-Hausdorff expansion (1.1). 5 If we now demand merely that [X, Y ] commute with all the elements of S, (that is, [[X, Y ], S] = 0 or equivalently L [X,Y ] S = 0, so that [X, Y ] is in the so-called centralizer of the set S), then the Jacobi identity, (in the form (3.37)), implies That is, under these conditions L X and L Y can still be treated as though they commute in the Baker-Campbell-Hausdorff expansion. Under these conditions, all of the terms appearing in the Baker-Campbell-Hausdorff expansion of equation (1.1) can now be reduced to elements of the set S. Integrating and summing the series we again see Careful inspection of the above quickly verifies that the only terms present when one expands the above are of the form L m X L n Y [X, Y ], (the elements of the set S), and that our simplifying assumption has eliminated all terms such as L [X,Y ] L m X L n Y [X, Y ], and variants thereof. By summing over the integers m and n the centralizer condition can also be restated as This is as far as we have currently been able to weaken the input assumptions we originally started with, while still keeping a reasonably close analogue of our initial result involving the function f (u, v).

Discussion
The Baker-Campbell-Hausdorff formula is a general purpose tool that has found many applications both in pure and applied mathematics [1][2][3][4][5][6][7][8][9][10][11][12], and generally in the physical sciences [11][12][13][14][15][16][17][18]. Via the study of the embeddability problem for stochastic matrices (Markov processes) there are even potential applications in the social sciences and financial sector. Explicit closed-form results are relatively rare, see the Introduction for examples. In this present article we have significantly extended our previous results reported in reference [11] by systematically weakening the input assumptions. In a number of increasingly general situations we have shown that the Baker-Campbell-Hausdorff expansion can be written in closed form as where f (u, v) is the symmetric function This was first demonstrated in reference [11] for the very explicit commutator [X, Y ] = uX + vY + cI. Herein, (with suitable expressions for u and v), a structurally identical result is established for Lie algebras with a one-dimensional commutator sub-algebra. More generally, whenever the commutator sub-algebra is Abelian, one has ln(e X e Y ) = X + Y + f (L X , −L Y )[X, Y ]. This result furthermore extends to the weaker input condition [X, Y ] ∈ Z [g,g] , that is, [X, Y ] being an element of the centre of the commutator algebra. Even more generally, this result extends to [X, Y ] being an element of the centralizer of those Lie brackets that appear in the Baker-Campbell-Hausdorff expansion. Overall, we find it quite remarkable just how far we have been able to push this result. There are of course many other directions that one might also wish to explore -we have concentrated our efforts on directions in which it seems that relatively concrete and explicit results might be readily extractable.