Let F be a field of characteristic different from 2. It is shown that the problems of classifying
(i) local commutative associative algebras over F with zero cube radical,
(ii) Lie algebras over F with central commutator subalgebra of dimension 3, and
(iii) finite p-groups of exponent p with central commutator subgroup of order are hopeless since each of them contains
• the problem of classifying symmetric bilinear mappings UxU → V , or
• the problem of classifying skew-symmetric bilinear mappings UxU → V ,
in which U and V are vector spaces over F (consisting of p elements for p-groups (iii)) and V is 3-dimensional. The latter two problems are hopeless since they are wild; i.e., each of them contains the problem of classifying pairs of matrices over F up to similarity.
We compare the dynamics of maximal three-dimensional gauged supergravity in appropriate truncations with the equations of motion that follow from a one-dimensional E10/K(E10) coset model at the first few levels. The constant embedding tensor, which describes gauge deformations and also constitutes an M-theoretic degree of freedom beyond eleven-dimensional supergravity, arises naturally as an integration constant of the geodesic model. In a detailed analysis, we find complete agreement at the lowest levels. At higher levels there appear mismatches, as in previous studies. We discuss the origin of these mismatches.
We construct generalized diffeomorphisms for E-9 exceptional field theory. The transformations, which like in the E-8 case contain constrained local transformations, close when acting on fields. This is the first example of a generalized diffeomorphism algebra based on an infinite-dimensional Lie algebra and an infinite-dimensional coordinate module. As a byproduct, we give a simple generic expression for the invariant tensors used in any extended geometry. We perform a generalized Scherk-Schwarz reduction and verify that our transformations reproduce the structure of gauged supergravity in two dimensions. The results are valid also for other affine algebras.
We study the non-linear realisation of E-11 originally proposed by West with particular emphasis on the issue of linearised gauge invariance. Our analysis shows even at low levels that the conjectured equations can only be invariant under local gauge transformations if a certain section condition that has appeared in a different context in the E-11 literature is satisfied. This section condition also generalises the one known from exceptional field theory. Even with the section condition, the E-11 duality equation for gravity is known to miss the trace component of the spin connection. We propose an extended scheme based on an infinite-dimensional Lie superalgebra, called the tensor hierarchy algebra, that incorporates the section condition and resolves the above issue. The tensor hierarchy algebra defines a generalised differential complex, which provides a systematic description of gauge invariance and Bianchi identities. It furthermore provides an E-11 representation for the field strengths, for which we define a twisted first order self-duality equation underlying the dynamics.
A class of generalized Verma modules over sl(n+2) are constructed from sl(n+1)-modules which are u(h(n))-free modules of rank 1. The necessary and sufficient conditions for these sl(n+2)-modules to be simple are determined. This leads to a class of new simple sl(n+2)-modules.
In Kac's classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For example, A(n - 1,0) = sl(1 vertical bar n) can be constructed by adding a "gray" node to the Dynkin diagram of A(n-1) = sl(n), corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is W(n), the derivation algebra of the Grassmann algebra on n generators. Here we present a novel construction of W(n), from the same Dynkin diagram as A(n - 1,0), but with additional generators and relations.
We give an analog of a Chevalley-Serre presentation for the Lie superalgebras W(n) and S(n) of Cartan type. These are part of a wider class of Lie superalgebras, the so-called tensor hierarchy algebras, denoted W(g) and S(g), where g denotes the Kac-Moody algebra A(r), D-r or E-r. Then W(A(n-1)) and S(A(n-1)) are the Lie superalgebras W(n) and S(n). The algebras W(g) and S(g) are constructed from the Dynkin diagram of the Borcherds-Kac-Moody superalgebras B(g) obtained by adding a single grey node (representing an odd null root) to the Dynkin diagram of g. We redefine the algebras W(A(r)) and S(A(r)) in terms of Chevalley generators and defining relations. We prove that all relations follow from the defining ones at level >= -2. The analogous definitions of the algebras in the D- and E-series are given. In the latter case the full set of defining relations is conjectured.
The pure spinor superfield formalism reveals that, in any dimension and with any amount of supersymmetry, one particular supermultiplet is distinguished from all others. This “canonical supermultiplet” is equipped with an additional structure that is not apparent in any component-field formalism: a (homotopy) commutative algebra structure on the space of fields. The structure is physically relevant in several ways; it is responsible for the interactions in ten-dimensional super Yang–Mills theory, as well as crucial to any first-quantised interpretation. We study the L∞ algebra structure that is Koszul dual to this commutative algebra, both in general and in numerous examples, and prove that it is equivalent to the subalgebra of the Koszul dual to functions on the space of generalised pure spinors in internal degree greater than or equal to three. In many examples, the latter is the positive part of a Borcherds–Kac–Moody superalgebra. Using this result, we can interpret the canonical multiplet as the homotopy fiber of the map from generalised pure spinor space to its derived replacement. This generalises and extends work of Movshev–Schwarz and Gálvez–Gorbounov–Shaikh–Tonks in the same spirit. We also comment on some issues with physical interpretations of the canonical multiplet, which are illustrated by an example related to the complex Cayley plane, and on possible extensions of our construction, which appear relevant in an example with symmetry type G2 × A1 .
We present a unified and completely general formulation of extended geometry, characterised by a Kac-Moody algebra and a highest weight coordinate module. Generalised diffeomorphisms are constructed, as well as solutions to the section constraint. Generically, additional ("ancillary") gauge transformations are present, and we give a concrete criterion determining when they appear. A universal form of the (pseudo-) action determines the dynamics in all cases without ancillary transformations, and also for a restricted set of cases based on the adjoint representation of a finite-dimensional simple Lie group. Our construction reproduces(the internal sector of) all previously considered cases of double and exceptional field theories.
Extended geometry provides a unified framework for double geometry, exceptional geometry, etc., i.e., for the geometrisations of the string theory and M-theory dualities. In this talk, we will explain the structure of gauge transformations (generalised diffeomorphisms) in these models. They are generically infinitely reducible, and arise as derived brackets from an underlying Borcherds superalgebra or tensor hierarchy algebra. The infinite reducibility gives rise to an L-infinity structure, the brackets of which have universal expressions in terms of the underlying superalgebra.
We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin-Vilkovisky framework, or equivalently, an L∞ algebra. The L∞ brackets are given as derived brackets constructed using an underlying Borcherds superalgebra B(gr+1), which is a double extension of the structure algebra gr. The construction includes a set of ancillary ghosts. All brackets involving the infinite sequence of ghosts are given explicitly. All even brackets above the 2-brackets vanish, and the coefficients appearing in the brackets are given by Bernoulli numbers. The results are valid in the absence of ancillary transformations at ghost number 1. We present evidence that in order to go further, the underlying algebra should be the corresponding tensor hierarchy algebra.
We consider Borcherds superalgebras obtained from semisimple finite-dimensional Lie algebras by adding an odd null root to the simple roots. The additional Serre relations can be expressed in a covariant way. The spectrum of generators at positive levels are associated to partition functions for a certain set of constrained bosonic variables, the constraints on which are complementary to the Serre relations in the symmetric product. We give some examples, focusing on superalgebras related to pure spinors, exceptional geometry and tensor hierarchies, of how construction of the content of the algebra at arbitrary levels is simplified.
Tensor hierarchy algebras are infinite-dimensional generalisations of Cartantype Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of attention due to the fundamental role they play for extended geometry. In the present paper, we examine tensor hierarchy algebras which are super-extensions of over-extended (often, hyperbolic) Kac-Moody algebras. They contain novel algebraic structures. Of particular interest is the extension of a over-extended algebra by its fundamental module, an extension that contains and generalises the extension of an affine Kac-Moody algebra by a Virasoro derivation L-1. A conjecture about the complete superalgebra is formulated, relating it to the corresponding Borcherds superalgebra.
Tensor hierarchy algebras constitute a class of non-contragredient Lie superalgebras, whose finite-dimensional members are the "Cartan-type" Lie superalgebras in Kac's classification. They have applications in mathematical physics, especially in extended geometry and gauged supergravity. We further develop the recently proposed definition of tensor hierarchy algebras in terms of generators and relations encoded in a Dynkin diagram (which coincides with the diagram for a related Borcherds superalgebra). We apply it to cases where a grey node is added to the Dynkin diagram of a rank r + 1 Kac-Moody algebra g(+), which in turn is an extension of a rank r finite-dimensional semisimple simply laced Lie algebra g. The algebras are specified by g together with a dominant integral weight λ. As a by-product, a remarkable identity involving representation matrices for arbitrary integral highest weight representations of g is proven. An accompanying paper describes the application of tensor hierarchy algebras to the gauge structure and dynamics in models of extended geometry.
The recent investigation of the gauge structure of extended geometry is generalised to situations when ancillary transformations appear in the commutator of two generalised diffeomorphisms. The relevant underlying algebraic structure turns out to be a tensor hierarchy algebra rather than a Borcherds superalgebra. This tensor hierarchy algebra is a non-contragredient superalgebra, generically infinite-dimensional, which is a double extension of the structure algebra of the extended geometry. We use it to perform a (partial) analysis of the gauge structure in terms of an L-infinity algebra for extended geometries based on finite-dimensional structure groups. An invariant pseudo-action is also given in these cases. We comment on the continuation to infinite-dimensional structure groups. An accompanying paper [1] deals with the mathematical construction of the tensor hierarchy algebras.
We give an explicit expression for the primitive E-8-invariant tensor with eight symmetric indices. The result is presented in a manifestly spin(16)/Z(2)-covariant notation.
We study how elementary divisors and minimal indices of a skew-symmetric matrix polynomial of odd degree may change under small perturbations of the matrix coefficients. We investigate these changes qualitatively by constructing the stratifications (closure hierarchy graphs) of orbits and bundles for skew-symmetric linearizations. We also derive the necessary and sufficient conditions for the existence of a skew-symmetric matrix polynomial with prescribed degree, elementary divisors, and minimal indices.
The set POLd,rm×n of m×n complex matrix polynomials of grade d and (normal) rank at most r in a complex (d+1)mn dimensional space is studied. For r=1,...,min{m,n}−1, we show that POLd,rm×n is the union of the closures of the rd+1 sets of matrix polynomials with rank r, degree exactly d, and explicitly described complete eigenstructures. In addition, for the full-rank rectangular polynomials, i.e. r=min{m,n} and m≠n, we show that POLd,rm×n coincides with the closure of a single set of the polynomials with rank r, degree exactly d, and the described complete eigenstructure. These complete eigenstructures correspond to generic m×n matrix polynomials of grade d and rank at most r.
We show that the set of m×m complex skew-symmetric matrix polynomials of odd grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m×m complex skew-symmetric matrix polynomials of odd grade d and rank at most 2r. In particular, this result includes the case of skew-symmetric matrix pencils (d=1).
The homogeneous system of matrix equations (X(T)A + AX, (XB)-B-T + BX) = (0, 0), where (A, B) is a pair of skew-symmetric matrices of the same size is considered: we establish the general solution and calculate the codimension of the orbit of (A, B) under congruence. These results will be useful in the development of the stratification theory for orbits of skew-symmetric matrix pencils.
The set of all solutions to the homogeneous system of matrix equations (X-T A + AX, X-T B + BX) = (0, 0), where (A, B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A, B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. Kagstrom, and V. V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375-3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.
We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group G, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular we show that they have finite images, and that for surfaces of genus at least 3 their restriction to the Torelli group is non-trivial iff G is non-abelian.
We construct free Lie algebras which, together with the algebra of spatial rotations, form infinite-dimensional extensions of finite-dimensional Galilei Maxwell algebras appearing as global spacetime symmetries of extended non-relativistic objects and non-relativistic gravity theories. We show how various extensions of the ordinary Galilei algebra can be obtained by truncations and contractions, in some cases via an affine Kac-Moody algebra. The infinite-dimensional Lie algebras could be useful in the construction of generalized Newton-Cartan theories gravity theories and the objects that couple to them.
We study systematically various extensions of the Poincare superalgebra. The most general structure starting from a set of spinorial supercharges Q is a free Lie superalgebra that we discuss in detail. We explain how this universal extension of the Poincare superalgebra gives rise to many other algebras as quotients, some of which have appeared previously in various places in the literature. In particular, we show how some quotients can be very neatly related to Borcherds superalgebras. The ideas put forward also offer some new angles on exotic branes and extended symmetry structures in M-theory.
We construct finite- and infinite-dimensional non-relativistic extensions of the Newton-Hooke and Carroll (A)dS algebras using the algebra expansion method, starting from the (anti-)de Sitter relativistic algebra in D dimensions. These algebras are also shown to be embedded in different affine Kac-Moody algebras. In the three-dimensional case, we construct Chern-Simons actions invariant under these symmetries. This leads to a sequence of non-relativistic gravity theories, where the simplest examples correspond to extended Newton-Hooke and extended (post-)Newtonian gravity together with their Carrollian counterparts.
In this Letter we study an infinite extension of the Galilei symmetry group in any dimension that can be thought of as a nonrelativistic or post-Galilean expansion of the Poincare symmetry. We find an infinite-dimensional vector space on which this generalized Galilei group acts and usual Minkowski space can be modeled by our construction. We also construct particle and string actions that are invariant under these transformations.
A compact formulation of the field-strengths, Bianchi identities and gauge transformations for tensor hierarchies in gauged maximal supergravity theories is given. A key role in the construction is played by the recently-introduced tensor hierarchy algebra.
The forms in D-dimensional (half-) maximal supergravity theories are discussed for 3 <= D <= 11. Superspace methods are used to derive consistent sets of Bianchi identities for all the forms for all degrees, and to show that they are soluble and fully compatible with supersymmetry. The Bianchi identities determine Lie superalgebras that can be extended to Borcherds superalgebras of a special type. It is shown that any Borcherds superalgebra of this type gives the same form spectrum, up to an arbitrary degree, as an associated Kac-Moody algebra. For maximal supergravity up to D-form potentials, this is the very extended Kac-Moody algebra E-11. It is also shown how gauging can be carried out in a simple fashion by deforming the Bianchi identities by means of a new algebraic element related to the embedding tensor. In this case the appropriate extension of the form algebra is a truncated version of the so-called tensor hierarchy algebra.
Heltalen och polynom tycks ha flera gemensamma egenskaper. En av heltalens egenskaper är aritmetikens fundamentalsats som säger att alla heltal kan skrivas som en produkt av primtal. Polynomen har en motsvarande egenskap, faktorsatsen, som innebär att varje polynom kan skrivas som en produkt av rotfaktorer. Denna och flera andra egenskaper som heltal och polynom har som motsvarar varandra beror inte på en slump utan på att de är besläktade. Egenskaper hos många välanvända mängder, de reella talen, de rationella talen samt heltalen kan beskrivas med gruppteori. Dessa egenskaper gäller endast över en binär operation men många intressanta och användbara egenskaper kräver två operationer. Inom denna uppsats undersöks den algebraiska strukturen ringar där många egenskaper som tas för givet beror på speciella egenskaper och därmed inte alltid finns närvarande. Efteråt studeras en speciell typ av ring kallad Euklidiska domän. Där många egenskaper som tillhör heltalen existerar i generaliserade former inom denna ring. Detta kapitel innehåller bevis som har generaliserats. Även polynomens struktur studeras och visar sig vara en Euklidisk domän. I studien används ett annat tillvägagångsätt än den traditionella där det bevisas genom idealer och PID. Uppsatsen avslutas med en kort studie av flervariabelpolynom där de egna bevisen finns varvid det ses att flervariabelpolynom med samma mängdvariabler är isomorfa.
We study extremal black hole solutions of the S-3 model (obtained by setting S-T-U in the STU model) using group theoretical methods. Upon dimensional reduction over time, the S 3 model exhibits the pseudo-Riemannian coset structure G/(K) over tilde with G = G(2(2)) and (K) over tilde = SO0(2; 2). We study nilpotent (K) over tilde -orbits of G(2(2)) corresponding to non-rotating single-center extremal solutions. We find six such distinct (K) over tilde -orbits. Three of these orbits are supersymmetric, one is non-supersymmetric, and two are unphysical. We write general solutions and discuss examples in all four physical orbits. We show that all solutions in supersymmetric orbits when uplifted to five-dimensional minimal supergravity have single-center Gibbons-Hawking space as their four-dimensional Euclidean hyper-Kahler base space. We construct hitherto unknown extremal (supersymmetric as well as non-supersymmetric) pressureless black strings of minimal five-dimensional supergravity and briefly discuss their relation to black rings.
We discuss a generalization of N = 6 three-algebras to N = 5 three-algebras in connection to anti-Lie triple systems and basic Lie superalgebras of type II. We then show that the structure constants defined in anti-Lie triple systems agree with those of N = 5 superconformal theories in three dimensions.
We analyse the M-theoretic generalisation of the tangent space structure group after reduction of the D = 11 supergravity theory to two space-time dimensions in the context of hidden Kac-Moody symmetries. The action of the resulting infinite-dimensional 'R symmetry' group K(E-9) on certain unfaithful, finite-dimensional spinor representations inherited from K(E-10) is studied. We explain in detail how these representations are related to certain finite codimension ideals within K(E-9), which we exhibit explicitly, and how the known, as well as new finite-dimensional 'generalised holonomy groups' arise as quotients of K(E-9) by these ideals. In terms of the loop algebra realisations of E-9 and K(E-9) on the fields of maximal supergravity in two space-time dimensions, these quotients are shown to correspond to (generalised) evaluation maps, in agreement with previous results of [1]. The outstanding question is now whether the related unfaithful representations of K(E-10) can be understood in a similar way.
We study the recently discovered isomorphisms between hyperbolic Weyl groups and modular groups over integer domains in normed division algebras. We show how to realize the group action via fractional linear transformations on generalized upper half-planes over the division algebras, focusing on the cases involving quaternions and octonions. For these we construct automorphic forms, whose explicit expressions depend crucially on the underlying arithmetic properties of the integer domains. Another main new result is the explicit octavian realization of W+(E-10), which contains as a special case a new realization of W+(E-8) in terms of unit octavians and their automorphism group.
The tensor hierarchy of maximal supergravity in D dimensions is known to be closely related to a Borcherds (super) algebra that is constructed from the global symmetry group E11-D. We here explain how the Borcherds algebras in different dimensions are embedded into each other and can be constructed from a unifying Borcherds algebra. The construction also has a natural physical explanation in terms of oxidation. We then go on to show that the Hodge duality that is present in the tensor hierarchy has an algebraic counterpart. For D > 8 the Borcherds algebras we find differ from the ones existing in the literature although they generate the same tensor hierarchy.
Rank data, in which each row is a complete or partial ranking of available items (columns), is ubiquitous. Among others, itcan be used to represent preferences of users, levels of gene expression, and outcomes of sports events. It can have many types ofpatterns, among which consistent rankings of a subset of the items in multiple rows, and multiple rows that rank the same subset of theitems highly. In this article, we show that the problems of finding such patterns can be formulated within a single generic framework thatis based on the concept of semiring matrix factorisation. In this framework, we employ the max-product semiring rather than theplus-product semiring common in traditional linear algebra. We apply this semiring matrix factorisation framework on two tasks: sparserank matrix factorisation and rank matrix tiling. Experiments on both synthetic and real world datasets show that the framework iscapable of discovering different types of structure as well as obtaining high quality solutions.
Three-dimensional conformal theories with six supersymmetries and SU(4) R-symmetry describing stacks of M2-branes are here proposed to be related to generalized Jordan triple systems. Writing the four-index structure constants in an appropriate form, the Chern-Simons part of the action immediately suggests a connection to such triple systems. In contrast to the previously considered 3-algebras, the additional structure of a generalized Jordan triple system is associated with a graded Lie algebra, which corresponds to an extension of the gauge group. In this paper we show that the whole theory with six manifest supersymmetries can be naturally expressed in terms of such a graded Lie algebra. Also the Bagger, Lambert and Gustavsson theory with eight supersymmetries is included as a special case.
We introduce kProbLog as a declarative logical language for machine learning. kProbLog is a simple algebraic extension of Prolog with facts and rules annotated by semi-ring labels. It allows to elegantly combine algebraic expressions with logic programs. We introduce the semantics of kProbLog, its inference algorithm, its implementation and provide convergence guarantees. We provide several code examples to illustrate its potential for a wide range of machine learning techniques. In particular, we show the encodings of state-of-the-art graph kernels such as Weisfeiler-Lehman graph kernels, propagation kernels and an instance of graph invariant kernels, a recent framework for graph kernels with continuous attributes. However, kProbLog is not limited to kernel methods and it can concisely express declarative formulations of tensor-based algorithms such as matrix factorization and energy-based models, and it can exploit semirings of dual numbers to perform algorithmic differentiation. Furthermore, experiments show that kProbLog is not only of theoretical interest, but can also be applied to real-world datasets. At the technical level, kProbLog extends aProbLog (an algebraic Prolog) by allowing multiple semirings to coexist in a single program and by introducing meta-functions for manipulating algebraic values.
The Kantor-Koecher-Tits construction associates a Lie algebra to any Jordan algebra. We generalize this construction to include also extensions of the associated Lie algebra. In particular, the conformal realization of so(p + 1, q + 1) generalizes to so(p + n, q + n), for arbitrary n, with a linearly realized subalgebra so(p, q). We also show that the construction applied to 3 × 3 matrices over the division algebras R, C, H, O gives rise to the exceptional Lie algebras f4, e6, e7, e8, as well as to their affine, hyperbolic and further extensions.
We present a nonlinear realization of the 5-graded Lie algebra associated to a Kantor triple system. Any simple Lie algebra can be realized in this way, starting from an arbitrary 5-grading. In particular, we get a unified realization of the exceptional Lie algebras f4, e6, e7, e8, in which they are respectively related to the division algebras R,C,H,O.
We study the Borcherds superalgebra obtained by adding an odd (fermionic) null root to the set of simple roots of a simple finite-dimensional Lie algebra. We compare it to the Kac-Moody algebra obtained by replacing the odd null root by an ordinary simple root, and then adding more simple roots, such that each node that we add to the Dynkin diagram is connected to the previous one with a single line. This generalizes the situation in maximal supergravity, where the E-n symmetry algebra can be extended either to a Borcherds superalgebra or to the kac-Moody algebra E-11, and both extensions can be used to derive the spectrum of p-form potentials in the theory. We show that also in the general case, the Borcherds and Kac-Moody extensions lead to the same 'p-form spectrum' of representations of the simple finite-dimensional Lie algebra.
We compute the one-loop correction to the dispersion relations of the excitations of the sigma model dual to ABJM theory, expanded around the cusp background. The results parallel those of SYM. As in that case, the dispersion relations are compatible with the predictions from the Bethe ansatz for the GKP string, though showing some known discrepancies on which we comment.
In this thesis we study algebraic structures in M-theory, in particular the exceptional Lie algebras arising in dimensional reduction of its low energy limit, eleven-dimensional supergravity. We focus on e8 and its infinite-dimensional extensions e9 and e10. We review the dynamical equivalence, up to truncations on both sides, between eleven-dimensional supergravity and a geodesic sigma model based on the coset E10/K(E10), where K(E10) is the maximal compact subgroup. The description of e10 as a graded Lie algebra is crucial for this equivalence. We study generalized Jordan triple systems, which are closely related to graded Lie algebras, and which may also play a role in the description of M2-branes using three-dimensional superconformal theories. The introductory part is followed by five research papers. In Paper I we show that the spinor and vector-spinor representations of k(e10) in the fermionic extension of the original E10 coset model lead, upon restriction to k(e9), to the R-symmetry transformations in eleven-dimensional supergravity reduced to two dimensions. Paper II provides an explicit expression for the primitive E8 invariant tensor with eight symmetric indices, which is expected to appear in M-theory corrections in the reduction to three dimensions. In Paper III we show that e8, e9 and e10 can be constructed in a unified way from a Jordan algebra, via generalized Jordan triple systems. Also Paper IV deals with generalized Jordan triple systems, but in the context of superconformal M2-branes. We show that the recently proposed theories with six or eight supersymmetries can be expressed in terms of a graded Lie algebra. In Paper V we return to the bosonic E10 coset model, and apply it to gauged maximal supergravity in three dimensions.
We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n=1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of Hermitian 3 x 3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f(4), e(6), e(7), e(8) for n=2. Moreover, we obtain their infinite-dimensional extensions for n >= 3. In the case of 2 x 2 matrices, the resulting Lie algebras are of the form so(p+n, q+n) and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature (p,q).
For any decomposition of a Lie superalgebra G into a direct sum G=H circle plus E of a subalgebra H and a subspace E, without any further resctrictions on H and E, we construct a nonlinear realization of G on E. The result generalizes a theorem by Kantor from Lie algebras to Lie superalgebras. When G is a differential graded Lie algebra, we show that it gives a construction of an associated L-infinity-algebra.
Gauge deformations of maximal supergravity in D = 11 - n dimensions generically give rise to a tensor hierarchy of p-form fields that transform in specific representations of the global symmetry group E-n. We derive the formulas defining the hierarchy from a Borcherds superalgebra corresponding to E-n. Tiffs explains why the E-n representations in the tensor hierarchies also appear in the level decomposition of the Borcherds superalgebra. We show that the indefinite Kac-Moody algebra E-11 can be used equivalently to determine these representations, up to p = D, and for arbitrarily large p if E-11 is replaced by E-r with sufficiently large rank r.
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 <= D <= 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super) algebra. Instead the Hodge duality relations between level p and D - 2 - p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
The three-algebras used by Bagger and Lambert in N = 6 theories of ABJM type are in one-to-one correspondence with a certain type of Lie superalgebras. We show that the description of three-algebras as generalized Jordan triple systems naturally leads to this correspondence. Furthermore, we show that simple three-algebras correspond to simple Lie superalgebras, and vice versa. This gives a classification of simple three-algebras from the well-known classification of simple Lie superalgebras.
We write the Lagrangian of the general N = 5 three-dimensional superconformal Chern-Simons theory, based on a basic Lie superalgebra, in terms of our recently introduced N = 5 three-algebras. These include N = 6 and N = 8 three-algebras as special cases. When we impose an antisymmetry condition on the triple product, the supersymmetry automatically enhances, and the N = 5 Lagrangian reduces to that of the well known N = 6 theory, including the ABJM and ABJ models.