{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:34:57Z","timestamp":1760236497010,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T00:00:00Z","timestamp":1638316800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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 Poincar\u00e9 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.<\/jats:p>","DOI":"10.3390\/sym13122289","type":"journal-article","created":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T02:56:14Z","timestamp":1638413774000},"page":"2289","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Space, Matter and Interactions in a Quantum Early Universe. Part II: Superalgebras and Vertex Algebras"],"prefix":"10.3390","volume":"13","author":[{"given":"Piero","family":"Truini","sequence":"first","affiliation":[{"name":"Istituto Nazionale Fisica Nucleare (INFN), Sezione di Genova, Via Dodecaneso 33, I-16146 Genova, Italy"},{"name":"Quantum Gravity Research (QGR), 101 S. Topanga Canyon Rd., Los Angeles, CA 90290, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7597-1050","authenticated-orcid":false,"given":"Alessio","family":"Marrani","sequence":"additional","affiliation":[{"name":"Centro Studi e Ricerche Enrico Fermi, Via Panisperna 89A, I-00184 Roma, Italy"}]},{"given":"Michael","family":"Rios","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research (QGR), 101 S. Topanga Canyon Rd., Los Angeles, CA 90290, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2938-3941","authenticated-orcid":false,"given":"Klee","family":"Irwin","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research (QGR), 101 S. Topanga Canyon Rd., Los Angeles, CA 90290, USA"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Truini, P., Marrani, A., Rios, M., and Irwin, K. (2020). Space, Matter and Interactions in a Quantum Early Universe. Part I: Kac-Moody and Borcherds Algebras. arXiv.","DOI":"10.3390\/sym13122342"},{"key":"ref_2","unstructured":"Kac, V.G. (1995). Infinite Dimensional Lie Algebras, Cambridge University Press. Third Edition Reprinted with Corrections."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1016\/0021-8693(88)90275-X","article-title":"Generalized Kac-Moody algebras","volume":"115","author":"Borcherds","year":"1988","journal-title":"J. Algebra"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/BF01232032","article-title":"Monstrous moonshine and monstrous Lie superalgebras. Invent. Math","volume":"109","author":"Borcherds","year":"1992","journal-title":"Invent. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Truini, P. (2018, January 5\u20136). Vertex operators for an expanding universe. 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