Exceptional geometry and Borcherds superalgebras

We study generalized diffeomorphisms in exceptional geometry with U-duality group E_{n(n)} from an algebraic point of view. By extending the Lie algebra e_n to an infinite-dimensional Borcherds superalgebra, involving also the extension to e_{n+1}, the generalized Lie derivatives can be expressed in a simple way, and the expressions take the same form for any n less than 8. The closure of the transformations then follows from the Jacobi identity and the grading of e_{n+1} with respect to e_n.Comment: 19 pages. v2: Changes in the part of section 3.3 about generalized Jordan triple systems. v3: Typos corrected. Published version. v4: Infinitesimal change

Borcherds and Kac-Moody extensions of simple finite-dimensional Lie algebras

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 to either 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.Comment: 11 pages. v2: Published version. Minor corrections and clarifications. References update

Exceptional Lie algebras and M-theory

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.Comment: PhD thesis, defended on December 10, 200

Superalgebras, constraints and partition functions

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.Comment: 27 pages. v2: Explanations and references added. Published versio

Oxidizing Borcherds symmetries

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 E(11-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.Comment: 21 pages, 3 figures, 5 table

The octic E8 invariant

We give an explicit expression for the primitive E8-invariant tensor with eight symmetric indices. The result is presented in a manifestly Spin(16)/Z2-covariant notation.Comment: 10 pp, plain te

Generalized conformal realizations of Kac-Moody algebras

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 3x3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f4, e6, e7, e8 for n=2. Moreover, we obtain their infinite-dimensional extensions for n greater or equal to 3. In the case of 2x2 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).Comment: 24 pages, v2: abstract and introduction rewritten, contents rearranged, former section 4 moved to an appendi

Generators and relations for (generalised) Cartan type superalgebras

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, $A(n-1,0) = \mathfrak{sl}(1|n)$ can be constructed by adding a "gray" node to the Dynkin diagram of $A_{n-1} = \mathfrak{sl}(n)$, 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.Comment: 6 pages, talk presented at Group32, Prague, July 2018. v2: Minor change