The evolution of the spectrum of a Frobenius Lie algebra under deformation

The category of Frobenius Lie algebras is stable under deformation, and here we examine explicit infinitesimal deformations of four and six dimensional Frobenius Lie algebras with the goal of understanding if the spectrum of a Frobenius Lie algebra can evolve under deformation. It can.

[1]  A. Makhlouf,et al.  Computing the index of Lie algebras , 2010 .

[2]  A. Ooms.  On frobenius lie algebras , 1980 .

[3]  Cohomology of Lie semidirect products and poset algebras , 2014, 1407.0428.

[4]  A. Ooms.  On Lie algebras having a primitive universal enveloping algebra , 1974 .

[5]  Elisabeth Remm,et al.  Contact and Frobeniusian forms on Lie groups , 2014 .

[6]  Nicholas W. Mayers,et al.  The Index and Spectrum of Lie Poset Algebras of Types B, C, and D , 2021, The Electronic Journal of Combinatorics.

[7]  A. A. Belavin,et al.  Solutions of the classical Yang - Baxter equation for simple Lie algebras , 1982 .

[8]  The Principal Element of a Frobenius Lie Algebra , 2008, 0801.4808.

[9]  A. Kirillov.  Lectures on the Orbit Method , 2004 .

[10]  M. Gerstenhaber,et al.  Boundary Solutions of the Classical Yang--Baxter Equation , 1996, q-alg/9609014.

[11]  D. Panyushev.  Inductive Formulas for the Index of Seaweed Lie Algebras , 2001 .

[12]  A. Joseph.  On semi-invariants and index for biparabolic (seaweed) algebras, II☆ , 2006 .

[13]  Vincent E. Coll,et al.  The index of Lie poset algebras , 2019, J. Comb. Theory, Ser. A.

[14]  Thomas F. Fox.  An introduction to algebraic deformation theory , 1993 .

[15]  A. Diatta,et al.  On properties of principal elements of Frobenius Lie algebras , 2012, 1212.5380.

[16]  A. Voronov.  Quantizing Deformation Theory II , 2018, Pure and Applied Mathematics Quarterly.

[17]  V. Dergachev,et al.  Index of Lie algebras of seaweed type. , 2000 .

[18]  Vincent E. Coll,et al.  The unbroken spectrum of Frobenius seaweeds II: type-B and type-C , 2019, 1907.08775.

[19]  L. Verhóczki.  Classification of Frobenius Lie algebras of dimension ≤ 6 , 2007 .

[20]  Vincent E. Coll,et al.  The index of nilpotent Lie poset algebras , 2020, 2004.08512.

[21]  A. Nijenhuis,et al.  Deformations of Lie Algebra Structures , 1967 .

[22]  Bialgebra actions, twists, and universal deformation formulas , 1994, hep-th/9411140.

[23]  Vincent E. Coll,et al.  The unbroken spectrum of type-A Frobenius seaweeds , 2016, 1606.05397.

[24]  Graphs, Frobenius functionals, and the classical Yang-Baxter equation , 2008, 0808.2423.