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In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra gu that extends e9. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn gu into a Lie superalgebra sgu with no superpartners, in order to comply with the Pauli exclusion principle. There is a natural action of the Poincaré group on sgu, which is an automorphism in the massive sector. We introduce a mechanism for scattering that includes decays as particular resonant scattering. Finally, we complete the model by merging the local sgu into a vertex-type algebra.

We introduce a quantum model for the universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of the Kac–Moody Lie algebra e9. We investigate Kac–Moody and Borcherds algebras, and we propose a generalization that meets further requirements that we regard as fundamental in quantum gravity.

We introduce a quantum model for the universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of the Kac–Moody Lie algebra e 9 . We investigate Kac–Moody and Borcherds algebras, and we propose a generalization that meets further requirements that we regard as fundamental in quantum gravity. [ABSTRACT FROM AUTHOR]

In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra g u that extends e 9 . It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn g u into a Lie superalgebra sg u with no superpartners, in order to comply with the Pauli exclusion principle. There is a natural action of the Poincaré group on sg u , which is an automorphism in the massive sector. We introduce a mechanism for scattering that includes decays as particular resonant scattering. Finally, we complete the model by merging the local sg u into a vertex-type algebra. [ABSTRACT FROM AUTHOR]

The Weyl denominator identity has interesting combinatorial properties for several classes of Lie algebras. Along these lines, we prove that given a finite graph G , the chromatic symmetric function X G can be recovered from the Weyl denominator identity of a Borcherds-Kac-Moody Lie algebra g whose associated graph is G. This gives a connection between (a) the coefficients appearing when the chromatic symmetric function X G is expressed in terms of the power sum symmetric functions, and (b) the root multiplicities of the Borcherds algebra g. From this result, we deduce a Lie theoretic proof of various alternate expressions of the chromatic symmetric function obtained by Stanley. Examples using small rank Lie algebras are provided to illustrate our results. The absolute value of the linear coefficient of the chromatic polynomial of G is known as the chromatic discriminant of G. As an application of our main theorem, we identify a coefficient appearing in X G , which equals the chromatic discriminant. We also find a connection between the Weyl denominator and the G -elementary symmetric functions. Using this connection, we give a Lie theoretic proof of the non-negativity of coefficients of G -power sum symmetric functions. [ABSTRACT FROM AUTHOR]