A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group
Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra 𝖌 = 𝖌(A) and (adjoint) Kac-Moody group 𝐺 = 𝐺(A) = ⟨exp(ad(teᵢ)),exp(ad(tfᵢ)) | t ∈ ℂ⟩ where ei and fi are the simple root vectors. Let (B⁺, B⁻, N) be the twin BN-pair naturally associated to 𝐺 and let (𝓑⁺, 𝓑⁻) be th...
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Carbone, Lisa Feingold, Alex J. Freyn, Walter 2025-12-15T15:25:15Z 2020 A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group. Lisa Carbone, Alex J. Feingold and Walter Freyn. SIGMA 16 (2020), 045, 47 pages 1815-0659 2020 Mathematics Subject Classification: 20G44; 20E42; 20F05; 51E24 arXiv:1606.05638 https://nasplib.isofts.kiev.ua/handle/123456789/210705 https://doi.org/10.3842/SIGMA.2020.045 Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra 𝖌 = 𝖌(A) and (adjoint) Kac-Moody group 𝐺 = 𝐺(A) = ⟨exp(ad(teᵢ)),exp(ad(tfᵢ)) | t ∈ ℂ⟩ where ei and fi are the simple root vectors. Let (B⁺, B⁻, N) be the twin BN-pair naturally associated to 𝐺 and let (𝓑⁺, 𝓑⁻) be the corresponding twin building with Weyl group W and natural 𝐺-action, which respects the usual W-valued distance and codistance functions. This work connects the twin building (𝓑⁺, 𝓑⁻) of 𝐺 and the Kac-Moody algebra 𝖌 = 𝖌(A) in a new geometrical way. The Cartan-Chevalley involution, ω, of 𝖌 has a fixed point real subalgebra, 𝔨, the 'compact' (unitary) real form of 𝖌, and 𝔨 contains the compact Cartan t = 𝔨 ∩ h. We show that a real bilinear form (⋅,⋅) is Lorentzian with signatures (1,∞) on 𝔨, and (1, n−1) on t. We define {k ∈ 𝔨 | (k,k) ≤0 } to be the lightcone of 𝔨, and similarly for t. Let K be the compact (unitary) real form of 𝐺, that is, the fixed point subgroup of the lifting of ω to 𝐺. We construct a K-equivariant embedding of the twin building of 𝐺 into the lightcone of the compact real form 𝔨 of 𝖌. Our embedding gives a geometric model of part of the twin building, where each half consists of infinitely many copies of a W-tessellated hyperbolic space glued together along hyperplanes of the faces. Locally, at each such face, we find an SU(2)-orbit of chambers stabilized by U(1), which is thus parametrized by a Riemann sphere SU(2)/U(1) ≅ S². For n = 2, the twin building is a twin tree. In this case, we construct our embedding explicitly, and we describe the action of the real root groups on the fundamental twin apartment. We also construct a spherical twin building at infinity, and construct an embedding of it into the set of rays on the boundary of the lightcone. This material is based upon work supported by the National Science Foundation under Grant No.1002477. The first author was supported in part by the Simons Foundation, Mathematics and Physical Sciences-Collaboration Grants for Mathematicians, Award Number: 422182. All authors wish to thank the IHES for support during various visits from 2013 to 2019. The second and third authors wish to thank the Max-Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam, Germany, for support during various visits during 2013-2019. The authors wish to thank Peter Abramenko for his helpful comments on an earlier draft of the manuscript and for his more recent comments on twin tree structures. They would also like to thank Victor Kac for helpful comments in May 2015 at IHES. The second author would like to thank Kai-Uwe Bux, Max Horn, Tobias Hartnick, Ralf Kohl, and Peter Abramenko for helpful discussions at the June 2015 conference on "Generalizations of Symmetric Spaces" in Israel. Finally, the authors wish to express their thanks to the anonymous referees for their many valuable suggestions to improve this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group |
| spellingShingle |
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group Carbone, Lisa Feingold, Alex J. Freyn, Walter |
| title_short |
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group |
| title_full |
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group |
| title_fullStr |
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group |
| title_full_unstemmed |
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group |
| title_sort |
lightcone embedding of the twin building of a hyperbolic kac-moody group |
| author |
Carbone, Lisa Feingold, Alex J. Freyn, Walter |
| author_facet |
Carbone, Lisa Feingold, Alex J. Freyn, Walter |
| publishDate |
2020 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra 𝖌 = 𝖌(A) and (adjoint) Kac-Moody group 𝐺 = 𝐺(A) = ⟨exp(ad(teᵢ)),exp(ad(tfᵢ)) | t ∈ ℂ⟩ where ei and fi are the simple root vectors. Let (B⁺, B⁻, N) be the twin BN-pair naturally associated to 𝐺 and let (𝓑⁺, 𝓑⁻) be the corresponding twin building with Weyl group W and natural 𝐺-action, which respects the usual W-valued distance and codistance functions. This work connects the twin building (𝓑⁺, 𝓑⁻) of 𝐺 and the Kac-Moody algebra 𝖌 = 𝖌(A) in a new geometrical way. The Cartan-Chevalley involution, ω, of 𝖌 has a fixed point real subalgebra, 𝔨, the 'compact' (unitary) real form of 𝖌, and 𝔨 contains the compact Cartan t = 𝔨 ∩ h. We show that a real bilinear form (⋅,⋅) is Lorentzian with signatures (1,∞) on 𝔨, and (1, n−1) on t. We define {k ∈ 𝔨 | (k,k) ≤0 } to be the lightcone of 𝔨, and similarly for t. Let K be the compact (unitary) real form of 𝐺, that is, the fixed point subgroup of the lifting of ω to 𝐺. We construct a K-equivariant embedding of the twin building of 𝐺 into the lightcone of the compact real form 𝔨 of 𝖌. Our embedding gives a geometric model of part of the twin building, where each half consists of infinitely many copies of a W-tessellated hyperbolic space glued together along hyperplanes of the faces. Locally, at each such face, we find an SU(2)-orbit of chambers stabilized by U(1), which is thus parametrized by a Riemann sphere SU(2)/U(1) ≅ S². For n = 2, the twin building is a twin tree. In this case, we construct our embedding explicitly, and we describe the action of the real root groups on the fundamental twin apartment. We also construct a spherical twin building at infinity, and construct an embedding of it into the set of rays on the boundary of the lightcone.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210705 |
| citation_txt |
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group. Lisa Carbone, Alex J. Feingold and Walter Freyn. SIGMA 16 (2020), 045, 47 pages |
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2025-12-17T12:04:32Z |
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2025-12-17T12:04:32Z |
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