nLab Borcherds algebra

Idea

The concept of Borcherds algebra or Borcherds-Kac-Moody algebra is a generalization of that of Kac-Moody algebra (hence also called generalized Kac-Moody algebra) obtained by allowing imaginary simple roots.

References

Original articles:

• Richard Borcherds, Generalized Kac-Moody algebras, J. Algebra 115 (1988), 501–512.

• Richard Borcherds, Central extensions of generalized Kac-Moody algebras, J. Algebra.140, 330-335 (1991).

• Victor Kac, Infinite dimensional Lie algebras, third edition, Cambridge University Press, 1990.

See also:

• Wikipedia, Generalized Kac-Moody algebra

Relation to U-duality and E11 (via mysterious duality) is discussed in

• Pierre Henry-Labordere, Bernard Julia, Louis Paulot, Borcherds symmetries in M-theory, JHEP 0204 (2002) 049 (arXiv:http://arxiv.org/abs/hep-th/0203070)

• Marc Henneaux, Bernard Julia, Jérôme Levie, $E_{11}$, Borcherds algebras and maximal supergravity (arxiv:1007.5241)

and specifically to tensor hierarchies in gauged supergravity

• Jakob Palmkvist, Tensor hierarchies, Borcherds algebras and $E_{11}$, JHEP 1202 (2012) 066 (arXiv:1110.4892)

• Jakob Palmkvist, The tensor hierarchy algebra, J. Math. Phys. 55, 011701 (2014) (arXiv:1305.0018)

and to exceptional generalized geometry

• Jakob Palmkvist, Exceptional geometry and Borcherds superalgebras (arXiv:1507.08828)