Artículo Original
Publicado 2009-11-05
Palabras clave
- Fusion algebras,
- Sk-orbits of Zk N,
- Young diagrams
Cómo citar
Saldarriaga, O. (2009). Type A Fusion Rules. Revista Integración, Temas De matemáticas, 27(2), 69–88. Recuperado a partir de https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/743
Resumen
In this paper we will define fusion algebras and give the general construction to obtain them from affine lie algebras. We also give several known methods to compute the structure constants for fusion algebras of type A.
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Referencias
[1] L. Begin, P. Mathieu and M. A. Walton. “su(3)k fusion coefficients”, Mod. Phys. Let. A, 7, no 35, 3255–3265 (1995).
[2] L. Begin, A. N. Kirillov, P. Mathieu and M. A. Walton “Berenstein-Zelevinsky triangles, elementary couplings and fusion rules”, Lett. Math. Phys., 28, no 4, 257–268 (1998).
[3] A. D. Berenstein and A. V. Zelevinsky. “Tensor product multiplicities and convex polytopes in partition space”, J. Geom. Phys., 5 (1988), no. 3, 453–472.
[4] A. Bertram, I. Ciocan-Fontaine, W. Fulton. “Quantum multiplication of Schur polynomials”, J. Alg., 219, no 2, 728–746 (1999).
[5] A. Feingold . “Fusion rules for affine Kac-Moody algebras”. Kac-Moody Lie Algebras and Related Topics, Ramanujan International Symposium on Kac-Moody Algebras and Applications, Jan. 28-31, 2002, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India, N. Sthanumoorthy, Kailash Misra, Editors, Contemporary Mathematics, 343, American Mathematical Society, Providence, RI, 2004, 53–96.
[6] A. Feingold and M. Weiner. “Type A Fusion rules from elementary group theory”. Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory, Proceedings of an International Conference on Infinite-Dimensional Lie Theory and Conformal Field Theory, May 23-27, 2000, University of Virginia, Charlottesville, Virginia, S. Berman, P. Fendley, Y.-Z. Huang, K. Misra, B. Parshall, Editors, Contemporary Mathematics, 297, American Mathematical Society, Providence, RI, 2002, 97–115.
[7] I. B. Frenkel, J. Lepowsky, A. Meurman. “Vertex Operator Algebras and the Monster”,Pure and Applied Math.,
134, Academic Press, Boston, 1988.
[8] I. B. Frenkel, Y. Zhu. “Vertex operator algebras associated to representations of affine and Virasoro algebras”, Duke Math. J., 66, 123–168 (1992).
[9] J. Fuchs. “Fusion rules in conformal field theory”, Fortsch. Phys., 42 (1994), 1–48.
[10] J. Fuchs. Affine Lie algebras and quantum groups, Cambridge monographs in mathematical physics, Cambridge University Press, 1992.
[11] W. Fulton. Young Tableaux, London Mathematical Society Students Texts 35, Cambridge University Press, 1997.
[12] D. Gepner, E. Witten. “String theory on group manifolds”, Nuclear Phys., B278, no 1, 111–130 (1986).
[13] F. M. Goodman, H. Wenzl. “Littlewood-Richardson coefficients for Hecke algebras at roots of unity”, Adv. Math., 82 (1990), 244–265.
[14] V. Kac. Introduction to Kac-Moody algebras, Cambridge University Press, 1991.
[15] I. G. Macdonald. Symmetric functions and Hall polynomials, Oxford University Press, 1979.
[16] O. Saldarriaga. “Fusion algebras, symmetric polynomials, and Sk-orbits of Zk N ”, Journal of algebra, 312-1, 257-293, 2007.
[17] G. Tudose. On the combinatorics of sl(n)-fusion algebra, Ph.D. Thesis, York University, North York, Ontario, Canada, October 2001.
[18] E. B. Vinberg. “Discrete linear groups generated by reflections”, Izvestija AN USSR (ser. mat.), 35 (1971), 1072–1112. English trans.: Math. USSR-Izvestija, 5 (1971), 1083–1119.
[19] M. A. Walton. “Algorithm for WZW fusion rules: a proof”, Phys. Lett. B, 241 (1990), no. 3, 365–368.
[20] M. A. Walton. “Tensor products and fusion rules”, Canadian Journal of Physics, 72 (1994), 527–536.
[21] Zhe-xian Wan. “Introduction to Kac-Moody algebras”, Pure and Applied Math., 134, Word scientific, New Jersey, 1991.
[2] L. Begin, A. N. Kirillov, P. Mathieu and M. A. Walton “Berenstein-Zelevinsky triangles, elementary couplings and fusion rules”, Lett. Math. Phys., 28, no 4, 257–268 (1998).
[3] A. D. Berenstein and A. V. Zelevinsky. “Tensor product multiplicities and convex polytopes in partition space”, J. Geom. Phys., 5 (1988), no. 3, 453–472.
[4] A. Bertram, I. Ciocan-Fontaine, W. Fulton. “Quantum multiplication of Schur polynomials”, J. Alg., 219, no 2, 728–746 (1999).
[5] A. Feingold . “Fusion rules for affine Kac-Moody algebras”. Kac-Moody Lie Algebras and Related Topics, Ramanujan International Symposium on Kac-Moody Algebras and Applications, Jan. 28-31, 2002, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India, N. Sthanumoorthy, Kailash Misra, Editors, Contemporary Mathematics, 343, American Mathematical Society, Providence, RI, 2004, 53–96.
[6] A. Feingold and M. Weiner. “Type A Fusion rules from elementary group theory”. Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory, Proceedings of an International Conference on Infinite-Dimensional Lie Theory and Conformal Field Theory, May 23-27, 2000, University of Virginia, Charlottesville, Virginia, S. Berman, P. Fendley, Y.-Z. Huang, K. Misra, B. Parshall, Editors, Contemporary Mathematics, 297, American Mathematical Society, Providence, RI, 2002, 97–115.
[7] I. B. Frenkel, J. Lepowsky, A. Meurman. “Vertex Operator Algebras and the Monster”,Pure and Applied Math.,
134, Academic Press, Boston, 1988.
[8] I. B. Frenkel, Y. Zhu. “Vertex operator algebras associated to representations of affine and Virasoro algebras”, Duke Math. J., 66, 123–168 (1992).
[9] J. Fuchs. “Fusion rules in conformal field theory”, Fortsch. Phys., 42 (1994), 1–48.
[10] J. Fuchs. Affine Lie algebras and quantum groups, Cambridge monographs in mathematical physics, Cambridge University Press, 1992.
[11] W. Fulton. Young Tableaux, London Mathematical Society Students Texts 35, Cambridge University Press, 1997.
[12] D. Gepner, E. Witten. “String theory on group manifolds”, Nuclear Phys., B278, no 1, 111–130 (1986).
[13] F. M. Goodman, H. Wenzl. “Littlewood-Richardson coefficients for Hecke algebras at roots of unity”, Adv. Math., 82 (1990), 244–265.
[14] V. Kac. Introduction to Kac-Moody algebras, Cambridge University Press, 1991.
[15] I. G. Macdonald. Symmetric functions and Hall polynomials, Oxford University Press, 1979.
[16] O. Saldarriaga. “Fusion algebras, symmetric polynomials, and Sk-orbits of Zk N ”, Journal of algebra, 312-1, 257-293, 2007.
[17] G. Tudose. On the combinatorics of sl(n)-fusion algebra, Ph.D. Thesis, York University, North York, Ontario, Canada, October 2001.
[18] E. B. Vinberg. “Discrete linear groups generated by reflections”, Izvestija AN USSR (ser. mat.), 35 (1971), 1072–1112. English trans.: Math. USSR-Izvestija, 5 (1971), 1083–1119.
[19] M. A. Walton. “Algorithm for WZW fusion rules: a proof”, Phys. Lett. B, 241 (1990), no. 3, 365–368.
[20] M. A. Walton. “Tensor products and fusion rules”, Canadian Journal of Physics, 72 (1994), 527–536.
[21] Zhe-xian Wan. “Introduction to Kac-Moody algebras”, Pure and Applied Math., 134, Word scientific, New Jersey, 1991.