Finite-Dimensional Irreducible Modules for the Three-Point 2 Loop Algebra

REFERENCES

• Benkart , G. , Terwilliger , P. ( 2007 ). The universal central extension of the three-point 2 loop algebra . Proc. Amer. Math. Soc. 135 : 1659 – 1668 .
   Web of Science ®Google Scholar
• Benkart , G. , Terwilliger , P. ( 2008 ). The equitable basis for 2 . Transformation Groups , submitted October 16 , 2008 .
   Google Scholar
• Bremner , M. ( 1994 ). Generalized affine Kac–Moody Lie algebras over localizations of the polynomial ring in one variable . Canad. Math. Bull. 37 : 21 – 28 .
   Web of Science ®Google Scholar
• Chari , V. ( 1986 ). Integrable representations of affine Lie algebras . Invent. Math. 85 : 317 – 335 .
   Web of Science ®Google Scholar
• Chari , V. , Pressley , A. ( 1991 ). Quantum affine algebras . Comm. Math. Phys. 142 : 261 – 283 .
   Web of Science ®Google Scholar
• Conway , J. ( 1973 ). Functions of One Complex Variable . Graduate texts in Mathematics, 11 . New York-Heidelberg : Springer-Verlag .
   Google Scholar
• Curtis , C. , Reiner , I. ( 1962 ). Representation Theory of Finite Groups and Associative Algebras . Pure and Applied Mathematics . Vol. XI . New York–London : Interscience Publishers, John Wiley & Sons .
   Google Scholar
• Date , E. , Roan , S. S. ( 2000 ). The structure of quotients of the Onsager algebra by closed ideals . J. Phys. A: Math. Gen. 33 : 3275 – 3296 .
   Google Scholar
• Davies , B. ( 1990 ). Onsager's algebra and superintegrability . J. Phys. A: Math. Gen. 23 : 2245 – 2261 .
   Google Scholar
• Davies , B. ( 1991 ). Onsager's algebra and the Dolan–Grady condition in the non-self-dual case . J. Math. Phys. 32 : 2945 – 2950 .
   Web of Science ®Google Scholar
• Dolan , L. , Grady , M. ( 1982 ). Conserved charges from self-duality . Phys. Rev. D (3) 25 : 1587 – 1604 .
   Web of Science ®Google Scholar
• Elduque , A. ( 2007 ). The S 4-action on the tetrahedron algebra . Proc. Roy. Soc. Edinburgh Sect. A 137 : 1227 – 1248 .
   Google Scholar
• Fialowski , A. , Schlichenmaier , M. ( 2005 ). Global geometric deformations of current algebras as Krichever–Novikov type algebras . Comm. Math. Phys. 260 : 579 – 612 .
   Google Scholar
• Hartwig , B. ( 2007 ). The tetrahedron algebra and its finite-dimensional irreducible modules . Linear Algebra Appl. 422 : 219 – 235 .
   Web of Science ®Google Scholar
• Hartwig , B. , Terwilliger , P. ( 2007 ). The Tetrahedron algebra, the Onsager algebra, and the 2 loop algebra . J. Algebra 308 : 840 – 863 .
   Web of Science ®Google Scholar
• Humphreys , J. ( 1978 ). Introduction to Lie Algebras and Representation Theory. Second Printing, Revised . Graduate Texts in Mathematics, 9. New York–Berlin : Springer-Verlag .
   Google Scholar
• Humphreys , J. ( 1990 ). Reflection Groups and Coxeter Groups . Cambridge Studies in Advanced Mathematics 29 . Cambridge : Cambridge University Press .
   Google Scholar
• Ito , T. , Terwilliger , P. ( 2007a ). Tridiagonal pairs and the quantum affine algebra U q ( ) . Ramanujan J. 13 : 39 – 62 .
   Web of Science ®Google Scholar
• Ito , T. , Terwilliger , P. ( 2007b ). Two non-nilpotent linear transformations that satisfy the cubic q-Serre relations . J. Algebra Appl. 6 : 477 – 503 .
   Web of Science ®Google Scholar
• Ito , T. , Tanabe , K. , Terwilliger , P. ( 2001 ). Some algebra related to P- and Q-polynomial association schemes . In: Codes and Association Schemes (Piscataway NJ, 1999) . Providence , RI : Amer. Math. Soc. , pp. 167 – 192 .
   Google Scholar
• Ito , T. , Terwilliger , P. , Weng , C. W. ( 2006 ). The quantum algebra U q (2) and its equitable presentation . J. Algebra 298 : 284 – 301 .
   Web of Science ®Google Scholar
• Koekoek , R. , Swarttouw , R. F. ( 1998 ). The Askey Scheme of Hypergeometric Orthogonal Polyomials and its q-Analog, report 98-17, Delft University of Technology, The Netherlands. Available at http://aw.twi.tudelft.nl/~koekoek/ research.html. Accessed July, 2007 .
   Google Scholar
• Nishino , A. , Deguchi , T. ( 2006 ). The L(2) symmetry of the Bazhanov–Stroganov model associated with the superintegrable chiral Potts model . Phys. Lett. A 356 : 366 – 370 .
   Google Scholar
• Onsager , L. ( 1944 ). Crystal statistics. I. A two-dimensional model with an order-disorder transition . Phys. Rev. (2) 65 : 117 – 149 .
   Google Scholar
• Pascasio , A. A. , Terwilliger , P. The Tetrahedron algebra and the Hamming graphs. In preparation.
   Google Scholar
• Perk , J. H. H. ( 1989 ). Star-triangle relations, quantum Lax pairs, and higher genus curves . Proceedings of Symposia in Pure Mathematics 49 : 341 – 354 . Providence , RI : Amer. Math. Soc.
   Google Scholar
• Schlichenmaier , M. ( 2003a ). Higher genus affine algebras of Krichever–Novikov type . Moscow Math. J. 3 : 1395 – 1427 .
   Web of Science ®Google Scholar
• Schlichenmaier , M. ( 2003b ). Local cocyles and central extensions for multipoint algebras of Krichever–Novikov type . J. Reine Angew. Math. 559 : 53 – 94 .
   Web of Science ®Google Scholar
• Terwilliger , P. ( 2001 ). Two linear transformations each tridiagonal with respect to an eigenbasis of the other . Linear Algebra Appl. 330 : 149 – 203 .
   Web of Science ®Google Scholar
• Terwilliger , P. ( 2004 ). Leonard pairs and the q-Racah polynomials . Linear Algebra Appl. 387 : 235 – 276 .
   Web of Science ®Google Scholar
• Terwilliger , P. ( 2005 ). Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array . Des. Codes Cryptogr. 34 : 307 – 332 .
   Web of Science ®Google Scholar
• Terwilliger , P. ( 2006 ). The equitable presentation for the quantum group U q () associated with a symmetrizable Kac–Moody algebra  . J. Algebra 298 : 302 – 319 .
   Web of Science ®Google Scholar
• Terwilliger , P. , Vidunas , R. ( 2004 ). Leonard pairs and the Askey–Wilson relations . J. Algebra Appl. 3 : 411 – 426 .
   Google Scholar
• Communicated by A. Elduque.
   Google Scholar