Satake diagrams of affine Kac–Moody algebras

Tripathy, L K; Pati, K. C.

doi:10.1088/0305-4470/39/6/012

Cited by 6 publications

(7 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The affine extension in D = 2 yields a real form of d7 ✶ û(1). We will show that this real form, obtained from projecting from e 9|10 all charged states, builds a so(8, 6) ✶ û|1 (1), where, by so(8, 6), we mean the affine real form described by the D = 2 Satake diagram of Table 7 as determined in [82]. The proof requires working in a basis of g inv in which the Cartan subalgebra is chosen compact.…”

Section: The Non-split Real Invariant Subalgebra In D =mentioning

confidence: 93%

Hidden Borcherds symmetries in Bbb Z_norbifolds of M-theory and magnetized D-branes in type 0' orientifolds

Bagnoud

Carlevaro

2006

J. High Energy Phys.

View full text Add to dashboard Cite

We study T 11−D−q × T q / n orbifold compactifications of eleven-dimensional supergravity and M-theory using a purely algebraic method. Given the description of maximal supergravities reduced on square tori as non-linear coset σ-models, we exploit the mapping between scalar fields of the reduced theory and directions in the tangent space over the coset to construct the orbifold action as a non-Cartan preserving finite order inner automorphism of the complexified U-duality algebra. Focusing on the exceptional serie of Cremmer-Julia groups, we compute the residual U-duality symmetry after orbifold projection and determine the reality properties of their corresponding Lie algebras. We carry out this analysis as far as the hyperbolic e 10 algebra, conjectured to be a symmetry of M-theory. In this case the residual subalgebras are shown to be described by a special class of Borcherds and Kac-Moody algebras, modded out by their centres and derivations. Furthermore, we construct an alternative description of the orbifold action in terms of equivalence classes of shift vectors, and, in D = 1, we show that a root of e 10 can always be chosen as the class representative. Then, in the framework of the E 10|10 /K(E 10|10 ) effective σ-model approach to M-theory near a spacelike singularity, we identify these roots with brane configurations stabilizing the corresponding orbifolds. In the particular case of 2 orbifolds of M-theory descending to type 0' orientifolds, we argue that these roots can be interpreted as pairs of magnetized D9-and D9'-branes, carrying the lower-dimensional brane charges required for tadpole cancellation. More generally, we provide a classification of all such roots generating n product orbifolds for n 6, and hint at their possible interpretation. 12 Conclusion 95 A Highest roots, weights and the Matrix R 99 B The U-duality group for 11D supergravity 100 C Conventions and involutive automorphisms for the real form so(8, 6) 101

show abstract

Section: The Non-split Real Invariant Subalgebra In D =mentioning

confidence: 93%

Hidden Borcherds symmetries in Bbb Z_norbifolds of M-theory and magnetized D-branes in type 0' orientifolds

Bagnoud

Carlevaro

2006

J. High Energy Phys.

View full text Add to dashboard Cite

show abstract

“…The Tits-Satake diagrams with these properties are a subclass of the Tits-Satake diagrams describing all the split real forms of the complex affine extensions g + and double extensions g ++ , given in [36,72].…”

Section: A5 Iwasawa Decompositionmentioning

confidence: 99%

“…B Tits-Satake diagrams of g + , g ++ and g +++ In this appendix, we list the Tits-Satake diagrams for the extensions of all the real forms of all the finite-dimensional simple complex Lie algebras g. Given what has been stated above, the affine, over-extended and very extended roots are always white roots with no arrow connecting them to any other root. The Tits-Satake diagrams with these properties are a subclass of the Tits-Satake diagrams describing all the split real forms of the complex affine extensions g + and double extensions g ++ , given in [36,72].…”

Section: Notations and Conventionsmentioning

confidence: 99%

Real forms of extended Kac–Moody symmetries and higher spin gauge theories

2012

View full text Add to dashboard Cite

We consider the relation between higher spin gauge fields and real Kac-Moody Lie algebras. These algebras are obtained by double and triple extensions of real forms g 0 of the finite-dimensional simple algebras g arising in dimensional reductions of gravity and supergravity theories. Besides providing an exhaustive list of all such algebras, together with their associated involutions and restricted root diagrams, we are able to prove general properties of their spectrum of generators with respect to a decomposition of the triple extension of g 0 under its gravity subalgebra gl(D, R). These results are then combined with known consistent models of higher spin gauge theory to prove that all but finitely many generators correspond to nonpropagating fields and there are no higher spin fields contained in the Kac-Moody algebra.

show abstract

“…We will see in chapter 5 that they correspond to Kac-Moody symmetric spaces of type 1. A paper of Tripathy and Pati gives a list of Satake diagrams [TP06].…”

Section: Orthogonal Symmetric Affine Kac-moody Algebrasmentioning

confidence: 99%

Affine Kac-Moody symmetric spaces

Freyn

2011

Preprint

View full text Add to dashboard Cite

show abstract

Satake diagrams of affine Kac–Moody algebras

Cited by 6 publications

References 23 publications

Hidden Borcherds symmetries in Bbb Z_norbifolds of M-theory and magnetized D-branes in type 0' orientifolds

Hidden Borcherds symmetries in Bbb Z_norbifolds of M-theory and magnetized D-branes in type 0' orientifolds

Real forms of extended Kac–Moody symmetries and higher spin gauge theories

Affine Kac-Moody symmetric spaces

Contact Info

Product

Resources

About

Satake diagrams of affine Kac–Moody algebras

Cited by 6 publications

References 23 publications

Hidden Borcherds symmetries in Bbb Znorbifolds of M-theory and magnetized D-branes in type 0' orientifolds

Hidden Borcherds symmetries in Bbb Znorbifolds of M-theory and magnetized D-branes in type 0' orientifolds

Real forms of extended Kac–Moody symmetries and higher spin gauge theories

Affine Kac-Moody symmetric spaces

Contact Info

Product

Resources

About

Hidden Borcherds symmetries in Bbb Z_norbifolds of M-theory and magnetized D-branes in type 0' orientifolds

Hidden Borcherds symmetries in Bbb Z_norbifolds of M-theory and magnetized D-branes in type 0' orientifolds