A Magnus group construction for a class of Borcherds algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911378993840128 |
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| author | Carbone, Lisa Jurisich, Elizabeth |
| author_facet | Carbone, Lisa Jurisich, Elizabeth |
| contents | We construct a group associated to a class of Borcherds algebras that admit a direct sum decomposition into a Kac--Moody (or semi-simple) subalgebra and a pair of free Lie subalgebras. Such Borcherds algebras have no mutually orthogonal imaginary simple roots.Our group is a semi-direct product of a Kac--Moody (or semi-simple) group and a Magnus group of invertible formal power series corresponding to a basis of a certain highest weight module determined by the simple imaginary roots. We show that our group is independent of this choice of basis, up to isomorphism. We apply our construction to a number of concrete examples, such as certain Borcherds algebras formed using root lattices of hyperbolic Kac--Moody algebras, the Monster Lie algebra, Monstrous Lie algebras of Fricke type and the gnome Lie algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10886 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Magnus group construction for a class of Borcherds algebras Carbone, Lisa Jurisich, Elizabeth Quantum Algebra High Energy Physics - Theory Representation Theory We construct a group associated to a class of Borcherds algebras that admit a direct sum decomposition into a Kac--Moody (or semi-simple) subalgebra and a pair of free Lie subalgebras. Such Borcherds algebras have no mutually orthogonal imaginary simple roots.Our group is a semi-direct product of a Kac--Moody (or semi-simple) group and a Magnus group of invertible formal power series corresponding to a basis of a certain highest weight module determined by the simple imaginary roots. We show that our group is independent of this choice of basis, up to isomorphism. We apply our construction to a number of concrete examples, such as certain Borcherds algebras formed using root lattices of hyperbolic Kac--Moody algebras, the Monster Lie algebra, Monstrous Lie algebras of Fricke type and the gnome Lie algebra. |
| title | A Magnus group construction for a class of Borcherds algebras |
| topic | Quantum Algebra High Energy Physics - Theory Representation Theory |
| url | https://arxiv.org/abs/2601.10886 |