Constructions of Algebras and Their Field Theory Realizations
We construct algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity. We prove that any such bracket can be extended to a 2-term algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator. While these algebras always exist, they generally do not realize a nontrivial symmetry in a field theory. In order to define algebras with genuine field theory realizations, we prove the significantly more general theorem that if the Jacobiator takes values in the image of any linear map that defines an ideal there is a 3-term algebra with a generally nontrivial 4-bracket. We discuss special cases such as the commutator algebra of octonions, its contraction to the “R-flux algebra,” and the Courant algebroid.
Lie groups are ubiquitous in mathematics and theoretical physics as the structures formalizing the notion of continuous symmetries. Their infinitesimal objects are Lie algebras: vector spaces equipped with an antisymmetric bracket satisfying the Jacobi identity. In various contexts it is advantageous (if not strictly required) to generalize the notion of a Lie algebra so that the brackets do not satisfy the Jacobi identity. Rather, in addition to the “2-bracket,” general “-brackets” are introduced on a graded vector space for , satisfying generalized Jacobi identities involving all brackets. Such structures, referred to as or strongly homotopy Lie algebras, first appeared in the physics literature in closed string field theory  and in the mathematics literature in topology [2–4]. A closely related cousin of algebras is algebras, which generalize associative algebras to structures without associativity [5, 6].
Our goal in this paper is to prove general theorems about the existence of structures for given “initial data” such as an antisymmetric bracket and to discuss their possible field theory realizations. First, as a warm-up, we answer the following natural question: Given a vector space with an antisymmetric bracket , under which conditions can this algebra be extended to an algebra with ? We will show that this is always possible. More specifically, we will prove the following theorem: The graded vector space , where is the space of degree zero and is isomorphic to and of degree one, carries a 2-term structure, meaning that the highest nontrivial product is , which encodes the “Jacobiator” (i.e., the anomaly due to the failure of the original bracket to satisfy the Jacobi identity). We have been informed that this theorem is known to some experts, and one instance of it has been stated in , but we have not been able to find a proof in the literature. (See also [8, 9] for examples of finite-dimensional algebras.)
At first sight the above theorem may shed doubt on the usefulness of algebras, since it states that any generally non-Lie algebra can be extended to an algebra. It should be emphasized, however, that for a generic bracket the resulting structure is quite degenerate in that the 2-term algebra may not be extendable further in a nontrivial way, say by including a vector space . Such extensions are particularly important for applications in theoretical physics as here encodes the “space of physical fields”, the space of “gauge parameters,” and the space of “trivial parameters” whose action on fields vanishes . Thus, if is isomorphic to there is no nontrivial action of on the physical fields and hence no genuine field theory realization of the algebra. In order to obtain nontrivial field theory realizations we will next prove a much more general theorem that covers the case of the Jacobiator being of a special form. Specifically, we will prove that if the Jacobiator takes values in the image of a linear operator that defines an ideal of the original algebra then there exists a 3-term algebra whose highest bracket in general is a nontrivial . A special case is the Courant bracket investigated by Roytenberg and Weinstein , for which the 4-bracket trivializes, but which is extendable and realized in string theory, in the form of double field theory [10, 12, 13].
We will illustrate these results with examples. Our investigation arose in fact out of the question whether the nonassociative octonions (more precisely, the 7-dimensional commutator algebra of imaginary octonions) can be viewed as part of an algebra. Our first theorem implies that the answer is affirmative, with the total graded space being 14-dimensional, which we will see is minimal. However, given the theorem, the existence of this structure does not express a nontrivial fact about the octonions. Moreover, this structure is not extendable, which implies with the results of  that the octonions, at least when realized as a 2-term algebra, cannot realize a nontrivial gauge symmetry in field theory.
As recently discovered in  and further investigated in [15, 16], the octonions are related to the phase space of non-geometric backgrounds in M-theory (nongeometric R-flux or non-geometric Kaluza-Klein monopoles in M-theory). Furthermore, a contraction of the octonions leads to the string theory “R-flux algebra” of [7, 17–20] and also to the “magnetic monopole algebra” of [20–26]. The Jacobiator of the R-flux algebra only takes values in a one-dimensional subspace, and therefore these contracted nonassociative algebras may in fact be extendable. Here it is sufficient to take to be one-dimensional, leading to an 8-dimensional algebra. (A 14-dimensional and hence nonminimal realization of the R-flux algebra has already been given in .)
The remainder of this paper is organized as follows. In Section 2 we briefly review the axioms of algebras. In Section 3 we prove the theorem that for arbitrary 2-bracket as initial data there is an structure on the “doubled” vector space. This theorem will then be significantly generalized in Section 4. In Section 5 we discuss examples, such as the octonions, the “R-flux algebra,” and the Courant algebroid. In the appendix we prove an analogous result for algebras.
2. Axioms of Algebras
We begin by stating the axioms of an algebra. It is defined on a graded vector spaceand we refer to elements in as having degree . We also refer to algebras with for all with as a -term algebra. There are a potentially infinite number of generalized multilinear products or brackets having inputs and intrinsic degree , meaning that they take values in a vector space whose degree is given byFor instance, has intrinsic degree , implying that it acts on the graded vector space according toMoreover, the brackets are graded (anti-)commutative in that, e.g., satisfiesand similarly for all other brackets.
The brackets have to satisfy a (potentially infinite) number of generalized Jacobi identities. In order to state these identities we have to define the Koszul sign for any in the permutation group of objects and a choice of such objects. It can be defined implicitly by considering a graded commutative algebra withwhere in exponents denotes the degree of the corresponding element. The Koszul sign is then inferred fromThe relations are given byfor each , which indicates the total number of inputs. Here gives a plus sign if the permutation is even and a minus sign if the permutation is odd. Moreover, the inner sum runs, for a given , over all permutations of objects whose arguments are partially ordered (“unshuffles”), satisfying
We will now state these relations explicitly for the values of relevant for our subsequent analysis. For the identity reduces tostating that is nilpotent, so that (3) is a chain complex. For the identity readsmeaning that acts like a derivation on the product . For one obtainsWe recognize the last three lines as the usual Jacobiator. Thus, this relation encodes the failure of the 2-bracket to satisfy the Jacobi identity in terms of a 1- and 3-bracket and the failure of to act as a derivation on . Finally, the relations readwhere we named the l.h.s. for later convenience. For a 2-term algebra there are no 4-brackets and hence the above right-hand side is zero. The relation then poses a nontrivial constraint on and , while all higher relations will be automatically satisfied.
3. A Warm-Up Theorem
We now prove the first theorem stated in the introduction. Consider an algebra with bilinear antisymmetric 2-bracket, i.e.,but we do not assume that the bracket satisfies any further constraints. In particular, the Jacobi identity is generally not satisfied, so that the Jacobiatorin general is nonzero. We then have the following.
Theorem 1. The graded vector spacewhere and with a vector space isomorphic to , carries a 2-term structure whose nontrivial brackets are given by
Comment. We denote the elements of by , etc., and the isomorphism byand similarly for its inverse. For instance, if carries a nondegenerate metric we can take to be the dual vector space of and the isomorphism to be the canonical isomorphism. (More simply, we can think of as a second copy of and of the isomorphism as the identity, but at least for notational reasons it is important to view and as different objects.)
Proof. The proof proceeds straightforwardly by fixing the products so that the relations are partially satisfied and then verifying that in fact all relations are satisfied. First, maps to , and we take it to be given by the (inverse) isomorphism (20),The second relation in here is necessary because there is no space in (15). The relations then hold trivially.
Next, we fix the product by requiring on and imposing the relation (10). For arguments of total degree 0 this relation is trivial because of the second relation in (21). For arguments of total degree 1 we havewhere we used (21). Using (21) on the l.h.s. we inferSince there is no space we have . This is consistent with the relation (10) for arguments of total degree 2:where we used (23). Thus, all relations are satisfied.
Let us now consider the relations (11). For arguments of total degree 0 (i.e., all taking values in ), it reads from which we inferDue to the antisymmetry of the bracket , the Jacobiator is completely antisymmetric in all arguments, and (26) is consistent with the required graded commutativity of . Since there is no space , is trivial for any arguments in . We have thus determined all nontrivial -brackets.
So far we have verified the relations and the relation for arguments of total degree 0. We now verify the remaining relations. The relation for arguments of total degree 1 readsand is thus satisfied. The relations for arguments of total degree larger than 1 are trivially satisfied, completing the proof of all relations.
Finally, we have to verify the relations. Since there is no nontrivial these require that the left-hand side of (12) vanishes identically for and defined above. This follows by a direct computation that we display in detail. First, for arguments of total degree 0 one may verify that (12) is completely antisymmetric in the four arguments. Writing for the totally antisymmetrized sum (carrying terms and pre-factor ) we then compute for the left-hand side of (12)Here we used repeatedly the total antisymmetry in the four arguments, in particular in the last step that under the sum then vanishes. The relations for arguments of total degree 1 or higher are trivially satisfied because they would have to take values in spaces of degree 2 or higher, which do not exist. The relations for are trivially satisfied for the same reason. This completes the proof.
4. Main Theorem
The above theorem states that an arbitrary bracket can be extended to an algebra. For generic brackets, this structure is, however, quite degenerate in that it may not be extendable further, say by adding a further space . Indeed, if the violation of the Jacobi identity is “maximal” and the Jacobiator takes values in all of , the space has to be as large as , and the image of the map equals . Consequently, one cannot introduce a further space together with a nontrivial satisfying . Since in physical applications serves as the space of fields, such brackets do not lead to algebras encoding a nontrivial gauge symmetry.
More interesting situations arise when the Jacobiator takes values in a proper subspace , for then it is sufficient to set and to take to be the “inclusion” defined for any by , viewing as an element of . Indeed, it is easy to verify, provided the subspace forms an ideal (i.e., ), that the above proof goes through as before. In this case, further extensions of the algebra may exist. In the following we will prove a yet more general theorem that is applicable to situations where the Jacobiator takes values in the image of a linear map that itself may have a nontrivial kernel. Then there is an extension to a 3-term algebra that generally requires a nontrivial 4-bracket.
Theorem 2. Let be an algebra with bilinear antisymmetric 2-bracket as in Section 3, and let be a linear map satisfying the closure conditionstogether with the Jacobiator relationwhere and denote image and kernel of , respectively. Then there exists a 3-term structure with on the graded vector space withwhere , , and denotes the inclusion of into . The highest nontrivial bracket in general is given by the 4-bracket (and the complete list of nontrivial brackets is given in eq. (54) below).
Notation and Comments. We denote the elements of by , the elements of by and the elements of by The condition (30) implies that there is a multilinear and totally antisymmetric map so thatThe condition (29) states that the bracket of an arbitrary with , , lies in the image of , i.e., we can writeWe can think of the operation on the r.h.s. as defining for each a map on , . This map is defined by (33) only up contributions in the kernel, as is the function in (32), but the following construction goes through for any choice of functions satisfying (33), (32). (The algebras resulting for different choices of these functions are almost certainly equivalent under suitably defined isomorphisms; see, e.g., , but we leave a detailed analysis for future work.)
Proof. As for Theorem 1, the proof proceeds by determining the -brackets from the relations as far as possible and then proving that in fact all relations are satisfied. The relations for defined in (31) are satisfied by definition since for all . In the following we systematically go through all relations for .
relations: The relations are satisfied for arguments of total degree zero, since acts trivially on . For arguments , of total degree 1 we needwhere we used (33). As the l.h.s. equals , this relation is satisfied if we setFor arguments of total weight 2 we computeusing (35) in the last step. As on the l.h.s. acts by inclusion, we can satisfy this relation by settingbut it remains to prove that the r.h.s. indeed takes values in the kernel. This follows by setting in (33):using the fact that the bracket is antisymmetric. Note that (37) is properly symmetric in its two arguments, in agreement with the graded commutativity (4). Another choice of arguments of total degree 2 is , for which we requirewhere we used (35) in the last step, recalling . Thus, using on the l.h.s. together with the graded symmetry we haveWe can also write this as (Here we employ the map on induced by via , which lies in as a consequence of and (33))We next consider arguments of total degree 3, for which must vanish as there is no vector space . This leads to a constraint from the relation:where we used (37) and . This relation is satisfied for (41). Finally, the relations are trivially satisfied for arguments of total degree 4 or higher, completing the proof of all relations.
relations: We now consider the relations for arguments of total degree zero:Recalling (32) and that when acting on , we infer that this relation is satisfied forNext, for arguments of total weight 1 the relation readswhere we used repeatedly (35). Moreover, we used (44) and that on which acts as the inclusion. We will next prove that the functiontakes values in the subspace . We have to prove that the r.h.s. is annihilated by . To this end we compute for the first term with (32)where we repeatedly used (33). This show that the r.h.s. of (46) is annihilated by , proving that takes values in . We can thus satisfy (45) by settingWe next recall that there can be no nontrivial for arguments of total degree 2. Thus, the relation for these arguments has to be satisfied for the products already defined. We then compute from (11), noting that it is symmetric in and writing for the symmetrized sum,where we used (35), (37), and, in the third equality, (48). It is now easy to see that under the symmetrized sum all terms cancel, using in particular that is totally antisymmetric. Thus, this relation is satisfied. Since the relations for total degree 3 or higher are trivially satisfied, we have completed the proof of all relations.
relations: We consider the relations (12) for arguments of total degree 0. Precisely as in (28) we computeIn contrast to (28) this is not zero in general, but we can now have a nontrivial taking values in . We next prove that the function defined bytakes values in . To this end we have to show that it is annihilated by : Thus, the relation can be satisfied by settingWe have now determined all nontrivial brackets, which we summarize here:with the functions defined in (46) and (51), respectively. All further relations have to be satisfied identically. Let us next consider the relations (12) for arguments of total degree 1. It is easy to see that (12) is then totally antisymmetric in , and writing for the antisymmetric sum over these three arguments we computewhere we used the products already defined, in particular (48), and the relation (32) for the Jacobiator. We observe that various terms cancelled under the totally antisymmetric sum. In order to satisfy the relation (12), the remaining terms need to be equal to . To see this note that writing (53) with an antisymmetrized sum over only the first three arguments one obtainsSpecializing this to we infer that it equals (55), completing the proof of this relation. It is easy to see that for arguments of total degree 2 or higher the relations are trivially satisfied. Thus, we have verified all relations.
relations: We have not displayed the relations in Section 2 for explicitly as these get increasingly laborious. However, it is easy to see that here the only nontrivial relation has arguments , which are of even degree so that the Koszul sign becomes . Moreover, is trivial, and it is then easy to verify that (7) reduces towhere the sum antisymmetrizes over all five arguments. Upon inserting the products in (54), it is a straightforward direct calculation, largely analogous to (55), to verify that this relation is identically satisfied. As these are the only nontrivial relations for or higher, this completes the proof.
Specializations. As a special case of Theorem 2 let us assume that the Jacobiator takes values in a subspace , which forms an ideal of the bracket. In this case we can take to be the inclusion map . Since its kernel is trivial, we have , and the algebra can be reduced to a 2-term algebra. Indeed, the action of on that is implicit in (33) then reduces toUsing this and , it is straightforward to verify that all products in (54) that take values in trivialize. In particular, trivializes. Theorem 1 is contained as a special case, for which .
We will now discuss a few examples, which get increasingly less trivial, with the goal to illustrate the scope of the above theorems.
The octonions: The seven imaginary octonions , satisfy the algebraand thus the commutation relationswhere the structure constants are defined as follows. Splitting the index as , where , is the totally antisymmetric tensor defined bywith the three-dimensional Levi-Civita symbol satisfying . (This coincides with the conventions of .) satisfy the following relationsUsing these it is straightforward to compute the Jacobiator:It is easy to verify with this expression that each generator appears on the right-hand side, see . Thus, the Jacobiator does not take values in a proper subspace, and therefore the extension requires a doubling to a 14-dimensional space (with basis ) as in Theorem 1, with the nontrivial brackets being given in addition to (60) byThere is no further nontrivial extension; in particular, this algebra cannot describe a nontrivial gauge symmetry in a field theory.
The R-flux algebra: This algebra, introduced in [17–19], is a contraction of the algebra of imaginary octonions in the following sense : (As shown in , the algebra of octonions can be also contracted in an analogous way to the magnetic monopole algebra, which is isomorphic to the R-flux algebra upon exchange of position and momentum variables.) We decompose , with , and introduce a scaling parameter to defineExpressing the algebra (60) now in terms of and sending one obtains the -flux algebrawhere is a central element that commutes with everything. It is easy to see that the only nonvanishing Jacobiator isThus, the Jacobiator takes values in the one-dimensional subspace spanned by . According to the specialization discussed after the proof of Theorem 2, we can then define an structure on , where and . In addition to the 2-brackets defined by (66) we have the nontrivial products
The Courant algebroid: The Courant bracket of generalized geometry or the “C-bracket” of double field theory has a nonvanishing Jacobiator. Denoting the arguments of this bracket, i.e., the elements of , by , etc., it is given bywhere denotes the invariant metric and is the exterior derivative in generalized geometry or the doubled partial derivative in double field theory. The bracket satisfies for a function so that for our current notation we read off with (33)It was established by Roytenberg and Weinstein that the Courant algebroid defines a 2-term algebra with the highest bracket being , which is defined by , and being the space of functions . The space of constants (the kernel of the differential operator ) is not needed as all brackets in (54) taking values in vanish. For instance, for two functions becomesIn double field theory language this is zero because of the “strong constraint,” and it is also one of the axioms of a Courant algebroid (see definition 3.2, axiom 4 in ). The vanishing of all other products taking values in can be verified similarly using the relations given, for instance, in . Thus, the existence of an structure on the Courant algebroid is a corollary of the more general Theorem 2.
We established general theorems about the existence of algebras for a given bracket and discussed possible field theory realizations. This includes well-known examples such as the Courant algebroid as special cases. Most importantly, it then remains to construct explicit examples of algebras that obey the conditions of Theorem 2 and that really do use the full structure possible, particularly a nontrivial 4-bracket. This may require identifying a structure that relaxes some of the axioms of a Courant algebroid.
Moreover, it is clear that there will be further generalizations of this theorem. For instance, the construction of Theorem 2 could be extended by taking the map not to be the inclusion map but rather a nontrivial operator that again could have a nontrivial kernel, which in turn would necessitate a new space and higher brackets beyond a 4-bracket. These may be useful for generalizations of double and exceptional field theory [28, 29]. Indeed, it is to be expected that the gauge structure of exceptional field theory requires algebras with arbitrarily high brackets , as is also the case in closed string field theory . Moreover, in order to obtain interesting algebras with nontrivial field theory realizations, for special cases it is instrumental to take an appropriate bracket as starting point. For instance, for the theory in  the naive bracket does not yield a Jacobiator living in the image of an appropriate operator (or, equivalently, the naive bracket does not transform covariantly under its own “adjoint” action ), but rather the vector space has to be suitably enlarged from the beginning, leading to a so-called Leibniz-Loday structure .
and Nonassociative Algebras
In analogy to the doubling of vector spaces introduced for the realization of Theorem 1 we will show that every nonassociative algebra has a realization as an algebra. An algebra is a graded vector space together with a collection of multilinear maps of internal degree satisfying the following fundamental identity for every . The first four equations read explicitly (i), :(ii), :(iii), :(iv), :
Let be a nonassociative algebra and a vector space isomorphic to with the isomorphism denoted by . The graded vector space of the algebra is then defined asIn addition we define the following productsUsing this construction, the equation is trivially satisfied. For the second equation we computefrom which we concludeFor two arguments of degree 1 we computefrom which we concludeNote that the -products have no a priori symmetry properties, so the -product has to be specified for each order of entries individually.
The equations readfrom which we infer that the 3-product is defined by the associator:Moreover, for total degree 1 we compute