Branching rules of semisimple Lie algebras
using affine extensions
Abstract
We present a closed formula for the branching coefficients of an embedding of two finitedimensional semisimple Lie algebras. The formula is based on the untwisted affine extension of . It leads to an alternative proof of a simple algorithm for the computation of branching rules which is an analog of the RacahSpeiser algorithm for tensor products. We present some simple applications and describe how integral representations for branching coefficients can be obtained. In the last part we comment on the relation of our approach to the theory of NIMreps of the fusion ring in WZW models with chiral algebra . In fact, it turns out that for these models each embedding induces a NIMrep at level . In cases where these NIMreps can be extended to finite level, we obtain a Verlindelike formula for branching coefficients. Reviewing this question we propose a solution to a puzzle which remained open in related work by Alekseev, Fredenhagen, Quella and Schomerus.
AEI2001133, mathph/0111020
PACS: 02.20.Sv, 11.25.Hf MCS: 17B10, 81R10
1 Introduction
Given a module of a Lie algebra , it is an important and natural question to ask how this module decomposes under restriction of the action to a subalgebra . This decomposition is described by nonnegative integer numbers, the socalled branching coefficients. The aim of this paper is to provide new tools for determining branching coefficients in the case where both and are finitedimensional semisimple Lie algebras. Several techniques have been developed to deal with this question. Among them are the use of generating functions, Schur functions and a generalization of Kostant’s multiplicity formula as well as different kinds of algorithms. For details we refer the reader to [1, 2, 3, 4, 5] and references therein.
In this paper we develop a new approach which uses the fact that a semisimple Lie algebra is naturally embedded in its affine extension . This makes available the powerful techniques of affine KacMoody algebras (see e.g. [6]) and conformal field theories related to such algebras (see [4] for instance). To give an example, we remind the reader that Verlinde’s formula [7] for fusion coefficients in WessZuminoWitten (WZW) theories gives a generalization of the concept of tensor product coefficients of . We will show that analogous relations hold for branching coefficients if we extend either or its subalgebra to the corresponding affine KacMoody algebra. In particular, in the first case there exists a relation to the theory of conformal boundary conditions and to the theory of fusion rings in WZW models [8].
The paper is organized as follows. In Section 2 we first provide some background on semisimple Lie algebras and their affine extensions. Subsequently, we present a closed formula for branching coefficients based on the extension of the subalgebra to . This formula is used in turn to give a simple derivation of a RacahSpeiser like algorithm in Section 3. Our results are applied to derive properties of branching coefficients and specialized to tensor product coefficients in Section 4. In addition we present a general procedure to obtain integral representations for branching coefficients. As an illustration of this method, we derive an integral representation for branching coefficients of the diagonal embedding . In Section 5, we consider a different approach based on representations of the fusion ring in WZW models. This leads to a Verlindelike formula for branching coefficients and induces a second type of integral representations. We exploit the latter to obtain an explicit nontrivial integral representation for the branching coefficients of with embedding index . In addition we indicate that for the series the fusion ring representation contains informations about two different embeddings at the same time. This solves some puzzle which remained open in [8].
2 A closed formula for branching coefficients
We want to describe an embedding of one finitedimensional semisimple Lie algebra into another. For notational simplicity let us assume that actually is a simple Lie algebra but this does not restrict the validity of our results. Denote the weight lattices of and by and , respectively. Here and in what follows we will always use the convention that and . The finitedimensional irreducible representations of the Lie algebras and are in onetoone correspondence to the weights with nonnegative integral Dynkin labels. These sets of socalled integrable highest weights of are denoted by with Weyl group and similarly for . Let and be the weight systems of the representations and including the multiplicities. The embedding can be characterized by a projection where means the span of the lattice over . Under this projection, the weight system of the representation of decomposes into weight systems of representations of according to
(1) 
The numbers are called branching coefficients. Our aim is to find an explicit and general formula for the coefficients with and . To achieve this, we consider the untwisted affine extension of . The level has to be chosen large enough and depends on the value of . This statement will be made precise below. The integrable highest weights of are given by the set where we used the decomposition of the affine Weyl group into a semidirect product of finite Weyl group and translations by times the coroot lattice . If we introduce the notation where is the highest root of we may write . The bracket denotes the scalar product on the weight space which is induced by the Killing form. It is given in terms of the quadratic form matrix if the weights are written using Dynkin labels, i.e. . In the following we will always identify in a natural way an integrable highest weight representation of with a highest weight of .
Before we continue, let us briefly introduce further objects that will be needed as we proceed. The character of an highest weight representation of is defined as
(2) 
and analogously for . The second ingredient of our formula is the modular S matrix of which, for , is given by the KacPeterson formula [6]
(3) 
This formula involves the rank of the Lie algebra , the number of positive roots , the Weyl vector , the dual Coxeter number and a sum over the Weyl group including its sign function . We omit the index because we will not encounter the corresponding objects for the Lie algebra . Due to Weyl’s character formula we may write
(4) 
We are now prepared to state the first result of this paper.
Theorem 1.
Consider an embedding of two finitedimensional semisimple Lie algebras. Let be the projection matrix characterizing the embedding and be two arbitrary but integrable highest weights. Define a map and let be a number such that . Then we have
(5) 
Proof.
For notational simplicity we assume to be simple. Let us first note that exists as all weight systems involved are finite. We then start by writing down the identity
(6) 
If we multiply both sides of (6) with and sum over all we obtain the desired result due to the unitarity of the S matrix. Thus we only have to motivate (6). The left equality simply results from (4) and the condition on the level , but the right equality is more interesting. Let be the weight system of the representation including all multiplicities. We insert the definition (2) of the characters into (6). After this substitution, the sum on the right hand side of (6) is over and involves scalar products . In contrast to this, the sum in the middle is over the projected weights and therefore involves scalar products of the form . The sum in both cases runs essentially over the same set . Therefore the equality in (6) holds if we can identify the scalar products according to . Writing this relation in terms of quadratic form matrices, we see that was constructed exactly in a way that this identity holds. ∎
Notice the following remarkable observation. If we could rewrite as for some integrable highest weight of at a certain level , we could apply eq. (4) and eq. (5) would reduce to a Verlindelike formula [7] for branching coefficients. In general, this does not seem to be possible because might cause negative entries in . We will see however in Section 5 that in some specific cases we are able to recover a Verlindelike formula using a different approach.
Let us briefly comment on the changes if is finitedimensional and semisimple but not simple. Under these circumstances we have a decomposition of into simple Lie algebras . In the affine extension, each simple factor obtains its own level: with . All relevant structures like the weight lattice, the Weyl group, the quadratic form matrix and the modular S matrix ’factorize’ in some sense, i.e. they are given by a direct sum, a product, a block diagonal matrix or factorize in the original sense of the word. Obviously, the proof of theorem 1 still remains valid if one takes these notational difficulties into account. In particular, the condition actually means in this case.
3 An alternative derivation of a RacahSpeiser like algorithm for branching rules
We will now use formula (5) to give an easy derivation of a wellknown algorithm [5] for the calculation of branching coefficients which is the basis of many computer algebra programs^{2}^{2}2I am grateful to M. van Leeuwen for providing this information.. The algorithm exhibits some similarity with the RacahSpeiser algorithm for the calculation of tensor product multiplicities (see also [6, 9, 10, 11, 12, 13] for its extension to fusion rules).
Theorem 2.
Consider an embedding of finitedimensional semisimple Lie algebras. Let be a highest weight of and be the projection matrix characterizing the embedding. The decomposition can be obtained by the following algorithm^{3}^{3}3The algorithm and the proof are based on [14] in which a slightly different algorithm for calculating NIMreps for twisted boundary conditions in WZW models is proved..

Calculate the weight system of the representation including the multiplicities. This gives some set .

Project this set to and add the Weyl vector of the subalgebra . Now we are dealing with the set including the multiplicities.

For each weight of use a Weyl reflection to map it into the fundamental Weyl chamber where all Dynkin labels are nonnegative. An algorithm in terms of elementary Weyl reflections can be found in [3] for example.

Drop all weights lying on the boundary of the fundamental Weyl chamber and subtract the Weyl vector of the subalgebra from the remaining ones.

Add up all these contributions including the signs of the relevant Weyl reflections and the multiplicities. The coefficient obtained for each weight is just the number .
Proof.
Again we assume to be simple without loss of generality. Essentially, the idea is to evaluate equation (5) for . We insert the definitions (2),(3) for the characters and the S matrix. Denoting the prefactor by we obtain
(7) 
where we already made use of the defining relation for . The next step consists in evaluating the sum over . We define a function by . The function as read of from eq. (7) has two important properties. First, it satisfies for all . Indeed, the Weyl reflection may be absorbed into a redefinition^{4}^{4}4Note that the weight system which belongs to an arbitrary representation is invariant under Weyl transformations. In particular this holds for the set . of and . To derive the second property let us define the set . It turns out that exactly contains the elements of which do not lie at the boundary of the corresponding affine Weyl chamber. This boundary is given by the set of all weights which are invariant under at least one elementary Weyl reflection including the shifted reflection at the dependent hyperplane described by . One may show that if is invariant under an affine fundamental Weyl reflection. To see this, note that the function which enters satisfies with respect to any affine Weyl transformation . These considerations lead to the simple relation
(8) 
We are now in a situation where we are able to perform the sum over . The sum over the exponentials in eq. (7) exactly gives a nonvanishing result if . In this case it obviously compensates the normalization factor . In the limit this condition reduces to a Kronecker symbol and we are left with the independent expression
(9) 
Next shift to the other side of the Kronecker symbol () and resum as well as . The expression under the sum then obviously does not depend on anymore. By summing over , we compensate the factor . The final result is
(10) 
For each weight lying at the boundary of a Weyl chamber there always exists an elementary Weyl reflection which leaves it fixed. These weights may be omitted because they would contribute twice with different sign. Inserting our result into equation (1) proves the theorem. ∎
4 Applications and an integral formula for branching coefficients
Using theorem 1 and formula (5) one may explicitly check some well known properties of branching coefficients. Thus one obtains
Corollary 1.
Let be an embedding of finitedimensional semisimple Lie algebras and denote the integrable highest weights by and and respectively. The branching coefficients have the following properties.

The trivial representation decomposes according to .

Denoting the conjugate representation by , the relation holds.

The branching coefficients of the embedding are related by .

In the decomposition of a tensor product both reductions are equivalent, i.e. the branching coefficients satisfy .
Proof.
The first relation holds because . For the second relation one needs that the charge conjugation matrix satisfies as well as and . The third relation is due to the fact that . The last property can be checked using the Verlinde formula for (this is valid if we choose large enough, see corollary 2), the unitarity of the S matrix and the property of characters. ∎
The diagonal embedding is special in the sense that its branching coefficients correspond to the tensor products in . In this case theorem 1 implies
Corollary 2.
Let be a finite dimensional semisimple Lie algebra and two fixed integrable highest weight modules. There exists some such that the coefficients in the decomposition may be expressed by the Verlinde formula
for all integers .
Proof.
This is a simple consequence of theorem 1 and the fact that the branching coefficients for the diagonal embedding with projection are given by the tensor product multiplicities of . Using the definition one obtains . The character of in (5) decomposes into a product of two characters of with argument . Applying equation (4) gives the desired result. ∎
The last remarks concern integral formulae for branching coefficients which may be deduced from theorem 1. We will not give a proof that this is always possible but only give the idea and a simple example for illustration. First we observe that the S matrices and the character in (5) both have a dependence on the summation index . In addition, the two S matrices give a total prefactor of the form where is the rank of the subalgebra, i.e. the number of independent components of . Therefore it is likely that in many (if not all) cases we may rewrite the sum as an integral in the limit and in this way recover an integral representation of branching coefficients.
We show how this works in a very simple example and rederive some integral formula for the (of course wellknown) tensor product multiplicities of representations of , i.e. the branching rules of the diagonal embedding . The characters of read
For the last equality we consider the sum to be a Riemann sum with an equidistant partition of the interval into intervals of length . Due to continuity we may extend the interval to . As the integral exists, it is given by the previous series in the limit . While such integral representations for general branching coefficients seem to be new, similar statements for tensor products can for example be found in [4, p. 534].
5 Relation to conformal field theory and a Verlindelike formula for branching coefficients
Let us mention that there exists an interesting relation of our work to the classification of boundary conditions in a special class of conformal field theories [4], the socalled WZW models with affine symmetry . It can be shown that to every consistent set of conformal boundary conditions there exists a socalled NIMrep of the corresponding fusion ring [15]. A NIMrep is given by nonnegative integral matrices satisfying and where the numbers are the fusion rules of the model. One can show that every NIMrep (at least the finite ones) can be diagonalized by an unitary matrix and one obtains a Verlindelike formula of the form
(11) 
with some map . For recent work on NIMreps and the connection to the classification of conformal boundary conditions see [15, 16]. Explicit formulae for may be found in [17, 18]. An approach based on graphs is given in [15]. Note that not all NIMreps have physical significance [16].
We now want to show how our construction is related to the theory of NIMreps. Let be a subalgebra of . Denote the tensor product multiplicities of by and the branching coefficients by . One can easily show that the matrices constitute a NIMrep of the fusion ring of the WZW model associated with at level . In this limit the fusion rules reduce to the tensor product multiplicities of . The proof of the NIMrep properties relies on the fact that the two possibilities of decomposing a module of into modules of are equivalent (compare corollary 1) and on the associativity of tensor products. It is easy to generalize the considerations of the Sections 2 and 3 to obtain
(12) 
for sufficiently large values of the level . Note that we did not rely on methods of conformal field theory to obtain this result. We just provided a completely algebraic treatment along the lines of the first four Sections.
Our next task is to relate the purely algebraic NIMreps of the last paragraph to results from conformal field theory. Indeed, one may prove [8] that NIMreps which come along with certain kinds of boundary conditions^{5}^{5}5These socalled twisted boundary conditions are connected to nontrivial symmetries of the Dynkin diagram of . in WZW theories coincide with the expressions given in (12) in the limit . This means that NIMreps which may be described as in equation (11) for finite values of , reduce to the expression (12) in the limit for certain distinguished subalgebras . In particular, this holds true for the special matrix elements . Starting from (11), we thus obtain another representation of branching coefficients for these distinguished embeddings. On one hand this yields another version of a RacahSpeiser like algorithm [14] invented originally for the calculation of NIMreps. On the other hand it may be used to derive alternative integral representations for branching coefficients along the lines of Section 4 if one takes the explicit expressions for the matrices (see for example [18]) and the results of [8] into account.
Table 1 contains a list of embeddings to which these considerations are known or conjectured to be applicable. A large part of these identifications are taken from [8]. Note, that the corresponding subalgebra in almost all examples is given by the subalgebra invariant under the Lie algebra automorphism induced by the Dynkin diagram symmetry to which the NIMrep belongs. It remained obscure, however, why in the case of the relevant subalgebra is given by (the socalled orbit Lie algebra [19] of ) and not by the subalgebra , invariant with respect to the nontrivial diagram automorphism of . Below we will partly fill this gap and show that one and the same NIMrep may lead to two different subalgebras under two distinct identifications of NIMrep labels. We will prove this remarkable feature of NIMreps in the case of and comment on the case of with afterwards. It is an open problem whether all NIMreps of the type may be extended to finite values of . This is certainly true for NIMreps related to the embeddings given in Table 1 (with some caveat regarding embeddings of the type for ) or to diagonal embeddings , but to our knowledge nothing is known for arbitrary embeddings .
Let us illustrate our considerations with an example. The Lie algebra has exactly one automorphism related to a nontrivial Dynkin diagram symmetry, where it acts as a permutation of nodes. On the level of weights it thus acts as a permutation of Dynkin labels . As is well known, induces a conformal boundary condition in the WZW model. Following [18] the boundary labels are given by halfinteger symmetric weights . Here, the symbol denotes the largest integer number smaller or equal to . The relevant NIMreps
(13) 
may be calculated [18] using the explicit formula
(14) 
where we identified the tupel with one of its (identical) entries. The obvious similarity of this expression with the S matrix of in mind we may ask whether the NIMrep (13) in the limit reduces to a NIMrep of the type (12) coming from an embedding . To check this assertion we have to identify the halfinteger symmetric NIMrep label with weights of via some map . Unfortunately there are two of these embeddings at our disposal and we have to worry which is the correct one. In [8] a map has been proposed which leads to the embedding with projection and embedding index . We will show below, however, that there is another map yielding the embedding with projection and embedding index .
We will discuss the first case first and derive an integral representation for branching coefficients of the embedding with embedding index . In this case one has to use the identification map [8]. In order to be able to apply equation (13) we further need the special quotient
of S matrices of which may be computed using the KacPeterson formula (3). Following [8] one may write
Performing the continuum limit we arrive at
We thus obtained a nontrivial integral formula for the branching coefficients of the embedding with embedding index .
As stated above there is another identification of NIMrep labels with weights of , leading to the embedding with . We will assume to be even in what follows. For any even weight of define . Before we continue, let us mention two obvious differences compared to the previous identification map . First, the identification map involves the level explicitly. Second, the map is only welldefined for a subset of weights of , i.e. the even ones. One may easily check however, that this restriction corresponds exactly to a general selection rule of the branching coefficients of with . We use our new identification map to rewrite (14) according to
Apart from a factor this is just the S matrix of at level . Using (4) we are now able to write equation (13) as
Remembering the definitions of in Theorem 1 and of in equation (4), the argument of the character can be identified to be . By setting the index to zero, Theorem 1 implies
This equality holds because we are allowed to use the prefactor to extend the range of from to . Taking the considerations of the previous paragraph into account, we just proved that the NIMrep for the twisted boundary conditions in the WZW model contains informations on both embeddings , with embedding index or respectively, at the same time. We leave it to the reader to write down the integral representation for branching coefficients of with .
After the detailed discussion of the case, we now want to comment on the series for . Numerical analysis indicates that a treatment similar to the one just presented leads to embeddings , in addition to the embeddings which are proposed in [8]. Following [18], the NIMrep labels are given by fractional symmetric weights of . To be more concrete, the Dynkin labels have to satisfy the relations , and . Like before we assume the level to be even. The map from weights of to the NIMrep labels is then given by
Again this map involves explicitly and is only welldefined for weights satisfying the relevant branching selection rule. We may use the projection to calculate the branching rules of according to Theorem 2 and compare them to NIMrep calculations at which have been performed using the algorithm proved in [14]. Taking our new identification of subalgebra weights with NIMrep labels into account, full agreement has been observed. Up to now, however, we have no rigorous proof to support this observation. As a last remark, note that even in the case of our new identification requires a maximally embedded in contrast to the result in [8].
6 Conclusions
In our paper we derived an explicit formula for the branching rules of embeddings of two semisimple Lie algebras. Starting from this result, we gave an alternative proof for an algorithm which can be used to calculate branching rules. We have also been able to check some simple properties of branching coefficients explicitly and argued that our formula induces integral representations for them. In two examples, these integral representations have been derived explicitly. Finally, we discussed the relation of embeddings to NIMreps of WZW models at infinite level. In particular we solved some puzzle which remained open in [8] and found that one NIMrep may contain informations about several embeddings at the same time by reinterpretation of NIMrep labels. A possible continuation of general NIMreps of the type (12) to finite values of using a Verlindelike formula (11) might be of importance for a representation theoretic understanding of embeddings of quantum groups at roots of unity as it provides a natural analogue to the transition from tensor product to fusion coefficients (cmp. [12, 13]). This last point has to be clarified in future work. Note that there has been some progress recently in understanding subgroups of quantum groups [20, 21, 22, 23, 24].
Another approach to express the branching coefficients of semisimple Lie algebras by using affine extensions of both Lie algebras at the same time would be to consider the grade zero part of the corresponding branching functions. A general expression for branching functions was found in [25]. However, it does not seem to provide a considerable simplification in our context.
Acknowledgements
The author likes to thank S. Fredenhagen, J. Fuchs, I. Runkel, V. Schomerus and Ch. Schweigert for useful discussions and careful reading of the manuscript. In particular he is grateful to I. Runkel and Ch. Schweigert for the collaboration on [14]. This work was financially supported by the Studienstiftung des deutschen Volkes.
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