ALMOST ABELIAN LIE GROUPS, SUBGROUPS AND QUOTIENTS
An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media—cosmology, crystallography etc. In theoretical physics and d...
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| Published in | Journal of mathematical sciences (New York, N.Y.) Vol. 266; no. 1; pp. 42 - 65 |
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| Main Authors | , , , , , , , |
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| Language | English |
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01.09.2022
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| Abstract | An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media—cosmology, crystallography etc. In theoretical physics and differential geometry, almost Abelian Lie groups and their homogeneous spaces provide some of the simplest solvmanifolds on which a variety of geometric structures, such as symplectic, Kähler, spin etc., are currently studied in explicit terms. Recently, almost Abelian Lie algebras were classified and studied in details. However, a systematic investigation of almost Abelian Lie groups has not been carried out yet, and the present paper is devoted to an explicit description of properties of this wide and diverse class of groups. The subject of investigation are real almost Abelian Lie groups with their Lie group theoretical aspects, such as the exponential map, faithful matrix representations, discrete and connected subgroups, quotients and automorphisms. The emphasis is put on explicit description of all technical details. |
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| AbstractList | An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media—cosmology, crystallography etc. In theoretical physics and differential geometry, almost Abelian Lie groups and their homogeneous spaces provide some of the simplest solvmanifolds on which a variety of geometric structures, such as symplectic, Kähler, spin etc., are currently studied in explicit terms. Recently, almost Abelian Lie algebras were classified and studied in details. However, a systematic investigation of almost Abelian Lie groups has not been carried out yet, and the present paper is devoted to an explicit description of properties of this wide and diverse class of groups. The subject of investigation are real almost Abelian Lie groups with their Lie group theoretical aspects, such as the exponential map, faithful matrix representations, discrete and connected subgroups, quotients and automorphisms. The emphasis is put on explicit description of all technical details. An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media—cosmology, crystallography etc. In theoretical physics and differential geometry, almost Abelian Lie groups and their homogeneous spaces provide some of the simplest solvmanifolds on which a variety of geometric structures, such as symplectic, Kähler, spin etc., are currently studied in explicit terms. Recently, almost Abelian Lie algebras were classified and studied in details. However, a systematic investigation of almost Abelian Lie groups has not been carried out yet, and the present paper is devoted to an explicit description of properties of this wide and diverse class of groups. The subject of investigation are real almost Abelian Lie groups with their Lie group theoretical aspects, such as the exponential map, faithful matrix representations, discrete and connected subgroups, quotients and automorphisms. The emphasis is put on explicit description of all technical details. |
| Author | Rakholia, Gautam Zhang, Hanwen Zhao, Zishuo Avetisyan, Zhirayr Berlow, Katalin Yang, Kelley Rios, Marcelo Almora Martin, Isaac |
| Author_xml | – sequence: 1 givenname: Marcelo Almora surname: Rios fullname: Rios, Marcelo Almora organization: Department of Mathematics, University of Montana – sequence: 2 givenname: Zhirayr surname: Avetisyan fullname: Avetisyan, Zhirayr email: jirayrag@gmail.com organization: Department of Mathematics, UC Santa Barbara, Regional Mathematical Center of Southern Federal University – sequence: 3 givenname: Katalin surname: Berlow fullname: Berlow, Katalin organization: Department of Mathematics, UC Berkeley – sequence: 4 givenname: Isaac surname: Martin fullname: Martin, Isaac organization: Department of Mathematics, University of Cambridge – sequence: 5 givenname: Gautam surname: Rakholia fullname: Rakholia, Gautam organization: Department of Mathematics, UC Santa Barbara – sequence: 6 givenname: Kelley surname: Yang fullname: Yang, Kelley organization: University of South California – sequence: 7 givenname: Hanwen surname: Zhang fullname: Zhang, Hanwen organization: Department of Mathematics, UC Santa Barbara – sequence: 8 givenname: Zishuo surname: Zhao fullname: Zhao, Zishuo organization: Department of Mathematics, UC Santa Barbara |
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| References | A. Fino and F. Paradiso. Generalized Kähler almost Abelian Lie groups. Annali di Mathematica, 200:1781–1812, 2021. M. Osinovsky. Bianchi universes admitting full groups of motions. Ann. Inst. Henri Poincaré (Section A: Physique théorique), 19:197—210, 1973. Gerald B. Folland. Harmonic Analysis in Phase Space. (AM-122). Princeton University Press, 1989. V. Fischer and M. Ruzhansky. Quantization on Nilpotent Lie Groups. Birkhüser, 2016. M. Freibert, A. Swann. The shear construction. Geometriae Dedicata, 198:71—101, 2018. D. Dummit, R. Foote. Abstract algebra. John Wiley & Sons, 2004. A. Fino, F. Paradiso. Balanced Hermitian structures on almost Abelian Lie algebras. Journal of Pure and Applied Algebra, 227(2):107186, 2023. M. Freibert, L. Schiemanowski and H. Weiss. Homogeneous spinor flow. The Quarterly Journal of Mathematics, 71(1):21—51, 2019. D. Andriot, E. Goi, R. Minasian and M. Petrini. Supersymmetry breaking branes on solvmanifolds and de Sitter vacua in string theory. Journal of High Energy Physics, 28, 2011. G. F. R. Ellis and H. van Elst. Cosmological models. (Cargese lectures 1998). Theoretical and Observational Cosmology, edited by M. Lachieze-Rey, 1998. Sundaram Thangavelu. Harmonic Analysis on the Heisenberg Group (Progress in Mathematics). Birkhäuser, 1 edition, 1998. B. C. Hall. Lie groups, Lie algebras and representations: an elementary introduction. Springer, 2015. Marco Freibert. Cocalibrated structures on Lie algebras with a codimension one Abelian ideal. Ann Glob Anal Geom, 42(4):537—563, 2012. Z. Avetisyan. Jordanable almost Abelian Lie agebras. ArXiv:1811.01252, 2018. L. Bagaglini and A. Fino. The Laplacian coflow on almost-abelian Lie groups. Annali di Matematica, 197:1855–1873, 2018. A. Petrov, V. Kaygorodov and V. Abdullin. Classification of gravitational fields of general form by motion groups. Izvestiia Visshikh Uchebnikh Zavedeniy. Matematika, 6(13):118—130, 1959. Z. Avetisyan and R. Verch. Explicit harmonic and spectral analysis in Bianchi I-VII-type cosmologies. Classical and Quantum Gravity, 30(15):155006, 2013. G. Rudolph, M. Schmidt. Differential geometry and mathematical physics. Part I. Springer, 2013. S. Console and M. Macrí. Lattices, cohomology and models of six dimensional almost abelian solvmanifolds. ArXiv, 1206.5977, 2012. Christoph Bock. On low dimensional solvmanifolds. Asian Journal of Mathematics, 20(2):199–262, 2016. G. Parry. Discrete structures in continuum description of defective crystals. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374(2066), 2016. M. Ryan and L. Shepley. Homogeneous relativistic cosmologies. Princeton University Press, 1975. J. Stanfield. Positive Hermitian curvature flow on nilpotent and almost-Abelian complex Lie groups. Annals of Global Analysis and Geometry, 60:401-429, 2021. I. Stewart, D. Tall. Algebraic number theory and Fermat’s last theorem. A. K. Peters, 2002. F. Paradiso. Locally conformally balanced metrics on almost Abelian Lie algebras. Complex Manifolds, 8(1):196–207, 2021. A. Andrada and M. Origlia. Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures. Manuscripta Mathematica, 155:389-417, 2017. Z. Avetisyan. The structure of almost Abelian Lie algebras. International Journal of Mathematics, 33(08):2250057, 2022. J. Lauret and C. Will. The Ricci pinching functional on solvmanifolds. The Quarterly Journal of Mathematics, 70(4):1281–1304, 2019. D. Burde, K. Dekimpe, B. Verbeke. Almost inner derivations of Lie algebras II. International Journal of Algebra and Computation, 31(02):341–364, 2021. L. Fuchs. Infinite Abelian groups. Vol 1. Academic Press, 1970. D. Burde, K. Dekimpe, B. Verbeke. Almost inner derivations of Lie algebras. Journal of Algebra and Its Applications, 17(11):1850214, 2018. 5872_CR22 5872_CR23 5872_CR24 5872_CR25 5872_CR26 5872_CR27 5872_CR28 5872_CR29 5872_CR20 5872_CR21 5872_CR2 5872_CR11 5872_CR3 5872_CR12 5872_CR13 5872_CR1 5872_CR14 5872_CR6 5872_CR15 5872_CR7 5872_CR16 5872_CR4 5872_CR17 5872_CR5 5872_CR18 5872_CR19 5872_CR8 5872_CR9 5872_CR30 5872_CR31 5872_CR10 |
| References_xml | – reference: Marco Freibert. Cocalibrated structures on Lie algebras with a codimension one Abelian ideal. Ann Glob Anal Geom, 42(4):537—563, 2012. – reference: A. Fino and F. Paradiso. Generalized Kähler almost Abelian Lie groups. Annali di Mathematica, 200:1781–1812, 2021. – reference: B. C. Hall. Lie groups, Lie algebras and representations: an elementary introduction. Springer, 2015. – reference: Z. Avetisyan. Jordanable almost Abelian Lie agebras. ArXiv:1811.01252, 2018. – reference: I. Stewart, D. Tall. Algebraic number theory and Fermat’s last theorem. A. K. Peters, 2002. – reference: D. Dummit, R. Foote. Abstract algebra. John Wiley & Sons, 2004. – reference: S. Console and M. Macrí. Lattices, cohomology and models of six dimensional almost abelian solvmanifolds. ArXiv, 1206.5977, 2012. – reference: M. Freibert, L. Schiemanowski and H. Weiss. Homogeneous spinor flow. The Quarterly Journal of Mathematics, 71(1):21—51, 2019. – reference: D. Andriot, E. Goi, R. Minasian and M. Petrini. Supersymmetry breaking branes on solvmanifolds and de Sitter vacua in string theory. Journal of High Energy Physics, 28, 2011. – reference: D. Burde, K. Dekimpe, B. Verbeke. Almost inner derivations of Lie algebras II. International Journal of Algebra and Computation, 31(02):341–364, 2021. – reference: V. Fischer and M. Ruzhansky. Quantization on Nilpotent Lie Groups. Birkhüser, 2016. – reference: J. Stanfield. Positive Hermitian curvature flow on nilpotent and almost-Abelian complex Lie groups. Annals of Global Analysis and Geometry, 60:401-429, 2021. – reference: Z. Avetisyan. The structure of almost Abelian Lie algebras. International Journal of Mathematics, 33(08):2250057, 2022. – reference: L. Fuchs. Infinite Abelian groups. Vol 1. Academic Press, 1970. – reference: A. Andrada and M. Origlia. Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures. Manuscripta Mathematica, 155:389-417, 2017. – reference: M. Ryan and L. Shepley. Homogeneous relativistic cosmologies. Princeton University Press, 1975. – reference: G. Rudolph, M. Schmidt. Differential geometry and mathematical physics. Part I. Springer, 2013. – reference: Gerald B. Folland. Harmonic Analysis in Phase Space. (AM-122). Princeton University Press, 1989. – reference: A. Fino, F. Paradiso. Balanced Hermitian structures on almost Abelian Lie algebras. Journal of Pure and Applied Algebra, 227(2):107186, 2023. – reference: Christoph Bock. On low dimensional solvmanifolds. Asian Journal of Mathematics, 20(2):199–262, 2016. – reference: G. Parry. Discrete structures in continuum description of defective crystals. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374(2066), 2016. – reference: L. Bagaglini and A. Fino. The Laplacian coflow on almost-abelian Lie groups. Annali di Matematica, 197:1855–1873, 2018. – reference: A. Petrov, V. Kaygorodov and V. Abdullin. Classification of gravitational fields of general form by motion groups. Izvestiia Visshikh Uchebnikh Zavedeniy. Matematika, 6(13):118—130, 1959. – reference: F. Paradiso. Locally conformally balanced metrics on almost Abelian Lie algebras. Complex Manifolds, 8(1):196–207, 2021. – reference: G. F. R. Ellis and H. van Elst. Cosmological models. (Cargese lectures 1998). Theoretical and Observational Cosmology, edited by M. Lachieze-Rey, 1998. – reference: J. Lauret and C. Will. The Ricci pinching functional on solvmanifolds. The Quarterly Journal of Mathematics, 70(4):1281–1304, 2019. – reference: M. Osinovsky. Bianchi universes admitting full groups of motions. Ann. Inst. Henri Poincaré (Section A: Physique théorique), 19:197—210, 1973. – reference: M. Freibert, A. Swann. The shear construction. Geometriae Dedicata, 198:71—101, 2018. – reference: D. Burde, K. Dekimpe, B. Verbeke. Almost inner derivations of Lie algebras. Journal of Algebra and Its Applications, 17(11):1850214, 2018. – reference: Sundaram Thangavelu. Harmonic Analysis on the Heisenberg Group (Progress in Mathematics). Birkhäuser, 1 edition, 1998. – reference: Z. Avetisyan and R. Verch. Explicit harmonic and spectral analysis in Bianchi I-VII-type cosmologies. 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| SubjectTerms | Anisotropic media Automorphisms Cosmology Crystallography Differential geometry Lie groups Mathematics Mathematics and Statistics Matrix representation Quotients Subgroups Theoretical physics |
| Title | ALMOST ABELIAN LIE GROUPS, SUBGROUPS AND QUOTIENTS |
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