The SU(3) ⊃ SO(3) missing label problem and the analytical Bethe Ansatz

The missing label for basis vectors of SU(3) representations corresponding to the reduction SU(3) ⊃ SO(3) can be provided by the eigenvalues of SO(3) scalars in the enveloping algebra of su(3). There are only two such independent elements of degree three and four. It is shown how the one of degree four can be diagonalized using the analytical Bethe ansatz.

[1]  J. Hughes.  SU(3) in an O(3) basis. I. Properties of shift operators , 1973 .

[2]  N. Reshetikhin.  A method of functional equations in the theory of exactly solvable quantum systems , 1983 .

[3]  Jiri Patera,et al.  Complete sets of commuting operators and O (3) scalars in the enveloping algebra of SU (3) , 1974 .

[4]  L. Vinet,et al.  A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$ , 2020, 2005.13444.

[5]  N. Reshetikhin.  The spectrum of the transfer matrices connected with Kac-Moody algebras , 1987 .

[6]  R. T. Sharp,et al.  Everything you always wanted to know about SU(3) ⊃ 0(3) , 1975 .

[7]  L. Frappat,et al.  Analytical Bethe ansatz for closed and open gl(𝒩)-spin chains in any representation , 2004, math-ph/0411021.

[8]  Analytical Bethe Ansatz for open spin chains with soliton non preserving boundary conditions , 2005, math-ph/0503014.

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

[10]  R. Campoamor-Stursberg.  Internal labelling operators and contractions of Lie algebras , 2007, 0706.2581.

[11]  V. Drinfeld.  A New realization of Yangians and quantized affine algebras , 1987 .

[12]  Rafael I. Nepomechie,et al.  Analytical Bethe Ansatz for quantum algebra invariant spin chains , 1991, hep-th/9110050.

[13]  Resolution of the GL(3) ⊃ O(3) state labelling problem via the O(3)-invariant Bethe subalgebra of the twisted Yangian , 2004, hep-th/0411026.

[14]  R. Baxter.  Asymptotically degenerate maximum eigenvalues of the eight-vertex model transfer matrix and interfacial tension , 1973 .

[15]  P. Winternitz,et al.  A new basis for the representations of the rotation group. Lamé and Heun polynomials , 1973 .

[16]  P. Alam.  ‘G’ , 2021, Composites Engineering: An A–Z Guide.

[17]  G. Olshanskii,et al.  Yangians and Classical Lie Algebras , 1994, hep-th/9409025.

[18]  L. Vinet,et al.  Heun operator of Lie type and the modified algebraic Bethe ansatz , 2020, 2011.11659.

[19]  Jerry P. Draayer,et al.  A new procedure for constructing basis vectors of SU (3)⊃ SO (3) , 2016 .

[20]  J. Hughes.  SU(3) in an O(3) basis. II. Solution of the state labelling problem , 1973 .

[21]  Yunfeng Jiang,et al.  Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE , 2017, 1710.04693.

[22]  R. Baxter.  Partition function of the eight vertex lattice model , 1972 .

[23]  Alexander Varchenko,et al.  Bethe eigenvectors of higher transfer matrices , 2006 .

[24]  R. Baxter.  Exactly solved models in statistical mechanics , 1982 .

[25]  P. Alam.  ‘W’ , 2021, Composites Engineering.

[26]  N. Reshetikhin,et al.  Excitation spectrum of the anisotropic generalization of an SU3 magnet , 1983 .

[27]  P. Alam.  ‘L’ , 2021, Composites Engineering: An A–Z Guide.

[28]  Rafael I. Nepomechie,et al.  SPECTRUM OF THE TRANSFER MATRIX FOR THE Uq(Bn)-INVARIANT $A_{2n}^{(2)}$ OPEN SPIN CHAIN , 1994, hep-th/9409130.

[29]  Arun Ram.  Quantization of Lie groups and Lie algebras , 2016 .

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

[31]  H. De Meyer,et al.  On the spectra of SO(3) scalars in the enveloping algebra of SU(3) , 1985 .

[32]  W. Hsiang,et al.  Lectures on Lie Groups , 1998 .

[33]  V. Bargmann,et al.  Group theory of harmonic oscillators (II). The integrals of Motion for the quadrupole-quadrupole interaction , 1961 .

[34]  L. Vinet,et al.  The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type , 2018, Annales Henri Poincaré.

[35]  V. Tolstoy.  SU(3) symmetry for orbital angular momentum and method of extremal projection operators , 2006 .