Noncommutative Pfaffians associated with the orthogonal algebra

D. V. Artamonov   ; V. A. Golubeva

Sbornik. Mathematics, Tome 203 (2012) no. 12, pp. 1685-1714

Résumé

Commutators of Pfaffians associated with the orthogonal algebra are found in skew-symmetric and root realizations of $\mathfrak{o}_N$. A generating function of Pfaffians is proved to satisfy the reflection equation. A relation between Pfaffians in skew-symmetric and root realizations of $\mathfrak{o}_N$ is established. Using these results we construct an integrable equation of Knizhnik-Zamolodchikov type using the Capelli central elements in $U(\mathfrak{o}_N)$, which are sums of squares of the considered Pfaffians. A classical limit of the obtained Knizhnik-Zamolodchikov type equation turns out to be a very specific system of equations of isomonodromic deformations. Bibliography: 18 titles.

Détail

Keywords: noncommutative Pfaffians, central elements, the Knizhnik-Zamolodchikov equation.
Mots-clés : orthogonal algebra, Capelli elements

Bibliographie

[1] A. A. Belavin, V. G. Drinfel'd, “Solutions of the classical Yang–Baxter equation for simple Lie algebras”, Funct. Anal. Appl., 16:3 (1982), 159–180 | DOI | MR | Zbl | Zbl

[2] N. J. MacKay, “Rational $R$-matrices in irreducible representations”, J. Phys. A, 24:17 (1991), 4017–4026 | DOI | MR | Zbl

[3] N. J. MacKay, “Rational $K$-matrices and representations of twisted Yangians”, J. Phys. A, 35:37 (2002), 7865–7876 | DOI | MR | Zbl

[4] A. Molev, Yangians and classical Lie algebras, Math. Surveys Monogr., 143, Amer. Math. Soc., Providence, RI, 2007 | MR | Zbl

[5] M. Itoh, T. Umeda, “On central elements in the universal enveloping algebras of the orthogonal Lie algebras”, Compositio Math., 127:3 (2001), 333–359 | DOI | MR | Zbl

[6] A. W. M. Dress, W. Wenzel, “A simple proof of an identity concerning Pfaffians of skew symmetric matrices”, Adv. Math., 112:1 (1995), 120–134 | DOI | MR | Zbl

[7] M. Ishikawa, S. Okada, M. Wakayama, “Applications of minor-summation formula. I. Littlewood's formulas”, J. Algebra, 183:1 (1996), 193–216 | DOI | MR | Zbl

[8] M. Ishikawa, M. Wakayama, “Applications of minor-summation formula. II. Pfaffians and Schur polynomials”, J. Combin. Theory Ser. A, 88:1 (1999), 136–157 | DOI | MR | Zbl

[9] M. Ishikawa, M. Wakayama, “Applications of minor summation formula. III. Plücker relations, lattice paths and Pfaffian identities”, J. Combin. Theory Ser. A, 113:1 (2006), 113–155 | DOI | MR | Zbl

[10] J. R. Stembridge, “Nonintersecting paths, Pfaffians, and plane partitions”, Adv. Math., 83:1 (1990), 96–131 | DOI | MR | Zbl

[11] K. Chen, “Iterated integrals of differential forms and loop space homology”, Ann. of Math. (2), 97 (1973), 217–246 | DOI | MR | Zbl

[12] J.-G. Luque, J.-Y. Thibon, “Pfaffian and Hafnian identities in shuffle algebras”, Adv. in Appl. Math., 29:4 (2002), 620–646 | DOI | MR | Zbl

[13] A. I. Molev, “Gelfand–Tsetlin bases for classical Lie algebras”, Handbook of algebra, Handb. Algebr., 4, Elsevier, Amsterdam, 2006, 109–170 | MR | Zbl

[14] I. V. Cherednik, “Factorizing particles on a half-line and root systems”, Theoret. and Math. Phys., 61:1 (1984), 977–983 | DOI | MR | Zbl

[15] N. J. MacKay, “Boundary integrability of nonlinear sigma models”, Theoret. and Math. Phys., 142:2 (2005), 270–274 | DOI | MR | Zbl

[16] N. J. MacKay, “Introduction to Yangian symmetry in integrable field theory”, Internat. J. Modern Phys. A, 20:30 (2005), 7189–7217 | DOI | MR | Zbl

[17] N. Reshetikhin, “The Knizhnik–Zamolodchikov system as a deformation of the isomonodromy problem”, Lett. Math. Phys., 26:3 (1992), 167–177 | DOI | MR | Zbl

[18] J. Harnad, “Quantum isomonodromic deformations and the Knizhnik–Zamolodchikov equations”, Symmetries and integrability of difference equations (Estérel, PQ, 1994), CRM Proc. Lecture Notes, 9, Amer. Math. Soc., Providence, RI, 1996, 155–161 | MR | Zbl