Higher-Dimensional Automorphic Lie Algebras

The paper presents the complete classification of Automorphic Lie Algebras based on $${{\mathfrak {sl}}}_{n}(\mathbb {C})$$sln(C), where the symmetry group G is finite and acts on $${{\mathfrak {sl}}}_n(\mathbb {C})$$sln(C) by inner automorphisms, $${{\mathfrak {sl}}}_n(\mathbb {C})$$sln(C) has no trivial summands, and where the poles are in any of the exceptional G-orbits in $$\overline{\mathbb {C}}$$C¯. A key feature of the classification is the study of the algebras in the context of classical invariant theory. This provides on the one hand a powerful tool from the computational point of view; on the other, it opens new questions from an algebraic perspective (e.g. structure theory), which suggest further applications of these algebras, beyond the context of integrable systems. In particular, the research shows that this class of Automorphic Lie Algebras associated with the $$\mathbb {T}\mathbb {O}\mathbb {Y}$$TOY groups (tetrahedral, octahedral and icosahedral groups) depend on the group through the automorphic functions only; thus, they are group independent as Lie algebras. This can be established by defining a Chevalley normal form for these algebras, generalising this classical notion to the case of Lie algebras over a polynomial ring.

[1]  E. Zhmud.  Kernels of projective representations of finite groups , 1992 .

[2]  Higher Genus Affine Lie Algebras of Krichever -- Novikov Type , 2005, math/0510440.

[3]  M. Schlichenmaier,et al.  From the Virasoro Algebra to Krichever–Novikov Type Algebras and Beyond , 2013, 1301.7725.

[4]  Eckhard Meinrenken,et al.  LIE GROUPS AND LIE ALGEBRAS , 2021, Lie Groups, Lie Algebras, and Cohomology. (MN-34), Volume 34.

[5]  Felix Klein,et al.  Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade , 1884 .

[6]  J. Schur,et al.  Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. , 1911 .

[7]  Richard P. Stanley,et al.  Invariants of finite groups and their applications to combinatorics , 1979 .

[8]  J. Sanders,et al.  Automorphic lie algebras and cohomology of root systems , 2015, 1512.07020.

[9]  Joe Harris,et al.  Representation Theory: A First Course , 1991 .

[10]  Takahiro Ueda,et al.  FORM version 4.0 , 2012, Comput. Phys. Commun..

[11]  A. Mikhailov,et al.  Reductions of integrable equations: dihedral group , 2004, nlin/0404013.

[12]  A. Mikhailov,et al.  The reduction problem and the inverse scattering method , 1981 .

[13]  J. Humphreys.  Introduction to Lie Algebras and Representation Theory , 1973 .

[14]  Larry Smith,et al.  Polynomial Invariants of Finite Groups , 1995 .

[15]  S. Konstantinou-Rizos,et al.  Darboux transformations, finite reduction groups and related Yang–Baxter maps , 2012, 1205.4910.

[16]  Darboux polynomial matrices: the classical Massive Thirring Model as study case , 2014, 1411.7965.

[17]  Higher genus affine algebras of Krichever - Novikov type , 2002, math/0210360.

[18]  Max Neunhöffer,et al.  LIE Λ-ALGEBRAS , 2009 .

[19]  W. Marsden.  I and J , 2012 .

[20]  A. W. Knapp.  Lie groups beyond an introduction , 1988 .

[21]  J. Sanders,et al.  Automorphic Lie algebras with dihedral symmetry , 2014, 1410.2914.

[22]  M. Reid.  McKay correspondence , 1997, alg-geom/9702016.

[23]  V. Knibbeler.  Invariants of Automorphic Lie Algebras , 2015, 1504.03616.

[24]  I. Stewart,et al.  Infinite-dimensional Lie algebras , 1974 .

[25]  G. Greuel,et al.  A Singular Introduction to Commutative Algebra , 2002 .

[26]  Fredrik Meyer,et al.  Representation theory , 2015 .

[27]  G. Lusztig.  Homomorphisms of the alternating group A5 into reductive groups , 2003 .

[28]  T. A. Springer.  Poincaré series of binary polyhedral groups and McKay's correspondence , 1987 .

[29]  J. Sanders,et al.  On the Classification of Automorphic Lie Algebras , 2009, 0912.1697.

[30]  F. Klein,et al.  Die Hauptgleichungen vom fünften Grade , 1993 .

[31]  Darboux transformation with dihedral reduction group , 2014, 1402.5660.

[32]  K. Lamotke.  Regular solids and isolated singularities , 1986 .

[33]  Randall R. Holmes.  Linear Representations of Finite Groups , 2008 .

[34]  J. Schur,et al.  Über die Darstellung der endlichen Gruppen durch gebrochen lineare Substitutionen. , 1904 .

[35]  Quantum affine Cartan matrices, Poincaré series of binary polyhedral groups, and reflection representations , 2005, math/0503542.

[36]  Reduction Groups and Automorphic Lie Algebras , 2004, math-ph/0407048.

[37]  F. Klein.  Lectures On The Icosahedron And The Solution Of Equations Of The Fifth Degree , 2007 .