Abstract
We prove the equivalence of two presentations of the Yangian \(Y(\mathfrak {g})\) of a simple Lie algebra \(\mathfrak {g}\), and we also show the equivalence with a third presentation when \(\mathfrak {g}\) is either an orthogonal or a symplectic Lie algebra. As an application, we obtain an explicit correspondence between two versions of the classification theorem of finite-dimensional irreducible modules for orthogonal and symplectic Yangians.
Similar content being viewed by others
References
Akasaka, T., Kashiwara, M.: Finite-dimensional representations of quantum affine algebras. Publ. Res. Inst. Math. Sci. 33(5), 839–867 (1997). arXiv:math/9703028
Arnaudon, D., Avan, J., Crampé, N., Frappat, L., Ragoucy, E.: \(R\)-matrix presentation for super-Yangians \(Y{\rm osp}(m|2n))\). J. Math. Phys. 44(1), 302–308 (2003). arXiv:math/0111325
Arnaudon, D., Molev, A., Ragoucy, E.: On the \(R\)-matrix realization of Yangians and their representations. Ann. Henri Poincaré 7(7–8), 1269–1325 (2006). arXiv:math/0511481
Belliard, S., Regelskis, V.: Drinfeld J presentation of twisted Yangians. SIGMA 13, 011 (2017). arXiv:1401.2143
Belliard, S., Crampé, N.: Coideal algebras from twisted Manin triples. J. Geom. Phys. 62(10), 2009–2023 (2012)
Bernard, D.: An Introduction to Yangian Symmetries. Int. J. Mod. Phys. B 7(20–21), 3517–3530 (1993). arXiv:hep-th/9211133
Brundan, J., Kleshchev, A.: Parabolic presentations of the Yangian \(Y(\mathfrak{gl}_n)\). Commun. Math. Phys. 254(1), 191–220 (2005)
Chari, V., Pressley, A.: Fundamental representations of Yangians and singularities of R-matrices. J. Reine Angew. Math. 417, 87–128 (1991)
Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)
Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Birkhäuser, Boston (1997)
Crampé, N.: Hopf structure of the Yangian \(Y(\mathfrak{sl}_n)\) in the Drinfeld realization. J. Math. Phys. 45(1), 434–447 (2004)
Damiani, I.: Drinfeld realization of affine quantum algebras: the relations. Publ. Res. Inst. Math. Sci. 48(3), 661–733 (2012)
Damiani, I.: From the Drinfeld realization to the Drinfeld-Jimbo presentation of affine quantum algebras: injectivity. Publ. Res. Inst. Math. Sci. 51(1), 131–171 (2015)
Drinfel’d, V.G.: Hopf algebras and the quantum Yang–Baxter equation. Soviet Math. Dokl. 32, 254–258 (1985)
Drinfel’d, V.G.: Quantum groups. In: Gleason, A.M. (ed.) Proceedings of the International Congress of Mathematicians, Berkeley. Amer. Math. Soc., Providence, RI, pp. 798–820 (1986)
Drinfel’d, V.G.: A new realization of Yangians and quantum affine algebras. Sov. Math. Doklady 36(2), 212–216 (1988)
Etingof, P., Frenkel, I., Kirillov, A.: Lectures on representation theory and Knizhnik-Zamolodchikov equations. Mathematical Surveys and Monographs, vol. 58. American Mathematical Society, Providence, RI (1998)
Faddeev, L.D., Reshetikhin, N.Y., Takhtajan, L.A.: Quantization of Lie Groups and Lie Algebras. Leningrad Math. J. 1(1), 193–225 (1990)
Frenkel, I., Reshetikhin, N.: Quantum affine algebras and holonomic difference equations. Commun. Math. Phys. 146(1), 1–60 (1992)
Fulton, W., Harris, J.: Representation Theory-A First Course. Graduate Texts in Mathematics, Readings in Mathematics, vol. 129. Springer, New York (1991)
Gautam, S., Toledano-Laredo, V.: Yangians and quantum loop algebras. Sel. Math. New Ser. 19, 271–336 (2013). arXiv:1012.3687
Gautam, S., Toledano-Laredo, V.: Yangians, quantum loop algebras and abelian difference equations. J. Am. Math. Soc. 29(3), 775–824 (2016). arXiv:1310.7318
Gautam, S., Toledano-Laredo, V.: Meromorphic tensor equivalence for Yangians and quantum loop algebras. Publ. Math. Inst. Hautes Études Sci. 125, 267–337 (2017). arXiv:1403.5251
Guay, N., Nakajima, H., Wendlandt, C.: Coproduct for Yangians of affine Kac–Moody algebras. arXiv:1701.05288v3
Guay, N., Regelskis, V.: Twisted Yangians for symmetric pairs of types B, C, D. Math. Z. 284(1–2), 131–166 (2016). arXiv:1407.5247
Guay, N., Regelskis, V., Wendlandt, C.: Twisted Yangians of small rank. J. Math. Phys. 57(4), 041703 (2016)
Guay, N., Regelskis, V., Wendlandt, C.: Representations of Twisted Yangians of types B-C-D: I. Sel. Math. New Ser. 23(3), 2071–2156 (2017). arXiv:1605.06733
Guay, N., Ma, X.: From quantum loop algebras to Yangians. J. Lond. Math. Soc. (2) 86(3), 683–700 (2012)
Guay, N., Tan, Y.: Local Weyl modules and cyclicity of tensor products for Yangians. J. Algebra 432, 228–251 (2015). arXiv:1503.06510
Jimbo, M.: Quantum R matrix for the generalized Toda system. Commun. Math. Phys. 102(4), 537–547 (1986)
Jing, N., Liu, M.: Isomorphism between two realizations of the Yangian \(Y(\mathfrak{so}_3)\). J. Phys. A 46(7), 075201 (2013). arXiv:1301.3962
Jing, N., Liu, M., Molev, A.: Isomorphism between the \(R\)-matrix and Drinfeld presentations of Yangian in types \(B\), \(C\) and \(D\). arXiv:1705.08155
Kashiwara, M.: On level-zero representations of quantized affine algebras. Duke Math. J. 112(1), 117–175 (2002)
Kassel, C.: Quantum Groups. Graduate Texts in Mathematics, vol. 155. Springer, New York (1995)
Kazhdan, D., Soibelman, Y.: Representations of quantum affine algebras. Sel. Math. (N.S.) 1(3), 537–595 (1995)
Knapp, A.: Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 140, 2nd edn. Birkhäuser, Boston (2002)
Kulish, P.P., Sklyanin, E.K.: Solutions of the Yang–Baxter equation. J. Sov. Math. 19, 1596–1620 (1982)
Kumar, S.: Tensor product decomposition. In: Proceedings of the International Congress of Mathematicians, vol. II. Hindustan Book Agency, New Delhi, pp. 1226–1261 (2010)
Levendorskii, S.Z.: On PBW bases for Yangians. Lett. Math. Phys. 27(1), 37–42 (1993)
Levendorskii, S.Z.: On generators and defining relations of Yangians. J. Geom. Phys. 12(1), 1–11 (1993)
Loebbert, F.: Lectures on Yangian Symmetry. J. Phys. A 49(32), 323002 (2016). arXiv:1606.02947
Mackay, N.J.: Introduction to Yangian symmetry in integrable field theory. Int. J. Modern Phys. A 20(30), 7189–7217 (2005)
Molev, A.: Irreducibility criterion for tensor products of Yangian evaluation modules. Duke Math. J. 112(2), 307–341 (2002)
Molev, A.: Yangians and Their Applications. Handbook of Algebra, vol. 3. Elsevier, Amsterdam (2003)
Molev, A.: Yangians and Classical Lie Algebras. Mathematical Surveys and Monographs, vol. 143. American Mathematical Society, Providence, RI (2007)
Molev, A.: Feigin-Frenkel center in types \(B\), \(C\) and \(D\). Invent. Math. 191(1), 1–34 (2013)
Molev, A., Mukhin, E.: Yangian characters and classical \(\cal{W} \)-algebras. In: Conformal Field Theory, Automorphic Forms and Related Topics, Contrib. Math. Comput. Sci., vol. 8. Springer, Heidelberg, pp. 287–334 (2014)
Molev, A., Mukhin, E.: Eigenvalues of Bethe vectors in the Gaudin model. arXiv:1506.01884
Nazarov, M., Tarasov, V.: On irreducibility of tensor products of Yangian modules associated with skew Young diagrams. Duke Math. J. 112(2), 343–378 (2002)
Rej, A., Spill F.: The Yangian of \(\mathfrak{sl}(n|m)\) and its quantum \(R\)-matrices. J. High Energy Phys. 012 (2011). arXiv:1008.0872
Tan, Y.: Braid group actions and tensor products of Yangians. arXiv:1510.01533
Zamolodchikov, A.B., Zamolodchikov, A.B.: Relativistic factorized \(S\)-matrix in two dimensions having \(O(N)\) isotropic symmetry. Nucl. Phys. B 133(3), 525–535 (1978)
Acknowledgements
We thank V. Toledano Laredo for some helpful pointers and H. Nakajima for indicating that the results in Section A.8 of [21] can be used to prove Proposition 2.2. Additionally, we thank S. Gautam for pointing out an inaccuracy in an earlier version of the paper. The first and third named authors gratefully acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada provided via the Discovery Grant Program and the Alexander Graham Bell Canada Graduate Scholarships—Doctoral Program, respectively. Part of this work was done during the second named author’s visits to the University of Alberta; he thanks the University of Alberta for the hospitality. The second named author was supported in part by the Engineering and Physical Sciences Research Council of the UK, Grant Number EP/K031805/1; he gratefully acknowledges the financial support.
Author information
Authors and Affiliations
Corresponding author
A Proof of Theorem 2.6 when \(\mathfrak {g}=\mathfrak {s}\mathfrak {l}_2\)
A Proof of Theorem 2.6 when \(\mathfrak {g}=\mathfrak {s}\mathfrak {l}_2\)
In this appendix, we complete the proof of Theorem 2.6 in the case where \(\mathfrak {g}=\mathfrak {s}\mathfrak {l}_2\). The extra difficulty when \(\mathfrak {g}= \mathfrak {s}\mathfrak {l}_2\) is to check that \(\varPhi _{\mathrm {cr},J}\) preserves the relation \([{\tilde{h}}_{i1},[x_{i1}^+,x_{i1}^-]]=0\) of \(Y_\zeta ^\mathrm{cr}(\mathfrak {g})\).
We normalize our symmetric invariant non-degenerate bilinear form \((\cdot , \cdot )\) on \(\mathfrak {s}\mathfrak {l}_2\) so that it is given by \((X,Y)=\mathrm {Tr}(XY)\) for all \(X,Y\in \mathfrak {s}\mathfrak {l}_2\). With this choice, the single positive root \(\alpha \) has length 2 and \(\{x_{i0}^+,x_{i0}^-,h\}\) coincides with the standard basis \(\{e,f,h\}\) of \(\mathfrak {s}\mathfrak {l}_2\). We will primarily employ the latter notation throughout the remainder of this proof, but when dealing with equations involving e and f simultaneously, we will sometimes replace them with \(x^+\) and \(x^-\), respectively. An orthonormal basis with respect to \((\cdot ,\cdot )\) is \(\mathcal {B}=\{E,F,H\}\), where
In particular, we have
Step 1: Reducing the defining relations.
We begin by showing that the relation
is satisfied in \(Y_\zeta (\mathfrak {g})\). This relation is well known throughout the literature (see for instance (4.64) in [5] and Definition 2.22 in [44]), but to our knowledge the relevant computations have never appeared explicitly, so we have decided to include them.
Consider equation (2.4) with \(X_1=E=X_3\) and \(X_2=F=X_4\). Since \([J(E),J(F)]=\sqrt{-1}[J(e),J(f)]\) and \([E,J(F)]=\sqrt{-1}J(h)\), the left-hand side is equal to \(-2\left[ [J(e),J(f)],J(h)\right] \). Thus, it suffices to show that the right-hand side of (2.4) with \(X_1=E=X_3\) and \(X_2=F=X_4\) is equal to \(-2\zeta ^2\left( fJ(e)-J(f)\,e\right) h\). By definition, it is equal to
As \([E,F]=\sqrt{-1}h\) and \((\cdot ,\cdot )\) is invariant and symmetric, we have
Thus, we can rewrite (A.1) as
multiplied by \(2\sqrt{-1}\zeta ^2\). Consider the different nonzero possibilities for \(\big [[E,X_\lambda ],[F,X_\mu ] \big ]\). They are
Both \(([H,E],\cdot )\) and \(([H,F],\cdot )\) applied to the first of these is zero, while \(([H,E],\cdot )\) applied to [[E, H], [F, E]] is zero, and \(([H,F],\cdot )\) applied to [[E, F], [F, H]] is zero. Therefore, (A.2) reduces to
By invariance, \(\left( [h,F],\left[ [E,H],[F,E] \right] \right) =-\left( [h,E],\left[ [E,F],[F,H] \right] \right) \), and moreover
Hence, (A.2) multiplied by \(2\sqrt{-1}\zeta ^2\), which is the right-hand side of (2.4) with \(X_1=E=X_3\) and \(X_2=F=X_4\), is equal to \(8i\sqrt{2}\zeta ^2\left( \{F,H,J(E)\}-\{H,E,J(F)\}\right) \). Let us rewrite this in terms of e, f and h:
Expanding \(\{h,f,J(e)\}-\{e,h,J(f)\}\) and using the defining relations of \(\mathfrak {s}\mathfrak {l}_2\) to rewrite it only in terms of \(fJ(e)\,h\) and \(J(f)\,eh\), we find that it is equal to \(\frac{1}{4}\big (J(f)\,eh-fJ(e)\,h\big )\), which gives the desired result.
Step 2: Checking the relation \(\big [\tilde{h}_1,[x_1^-,x_1^+]\big ]=0\) is preserved.
By definition of \(\varPhi _{\mathrm {cr},J}\), this amounts to checking that
where
Expanding the left-hand side, we obtain
Step 2.1: \(-\zeta ^3 \left[ v,[w^-,w^+]\right] =0\).
We have
Now, since \([v,h]=0\) and \([\{e,f\},h]=0\), we have that \([v,\{e,f\}h+h\{e,f\}]=0\). Hence,
Using the commutator relations of \(\mathfrak {s}\mathfrak {l}_2\), we can write \(e\{f,h\}+\{f,h\}e=4efh-2h-2h^2\). Note that \([h,efh]=0\). Thus,
Step 2.2: The terms in (A.3) involving \(\zeta ^2\).
After rewriting \(\left[ J(h),[J(f),J(e)]\right] =\zeta ^2(fJ(e)-J(f)e)h\), the sum of the terms involving \(\zeta ^2\) in (A.3) is
Consider first \(\left[ J(h),[w^-,w^+]\right] \). Recall that (A.4) and \(\{e,f\}=2v+h^2\). Hence, since J(h) commutes with h and v, we have
Observe that \(e\{f,h\}+\{f,h\}e=4efh-2h-2h^2\) and \([J(h),\{e,\{f,h\}\}]=8(J(e)fh-eJ(f)h)\). Therefore, we have
Let us now turn to the last term in (A.5). The identity
implies that we can expand \([J(f),w^+]+[w^-,J(e)]\) in the form
Hence,
We would like to simplify the last term. Since \([h,x^\pm J(x^\mp )+J(x^\mp )x^\pm ]=0\), we can replace v by \(\tilde{v}\) (\(= \nu + \frac{1}{2} h^2 = \frac{1}{2}\{e,f\}\)). Next, the sequence of equalities
allow us to express \(2[{\tilde{v}},\{e,J(f)\}+\{f,J(e)\}]\) as
The terms involving J(h) all cancel out, and we are left with
We would like to express the right-hand side in terms of fJ(e)h and J(f)eh only. Using the defining relations of \(\mathfrak {s}\mathfrak {l}_2\), we find that
Multiplying by \(\frac{1}{4}\) and substituting the result back into (A.7), we obtain
Substituting this and (A.6) back into (A.5) gives
Since \([v,J(h)]\,h=\frac{1}{2}[ef+fe,J(h)]\,h=\left( eJ(f)-J(e)f+J(f)e-fJ(e)\right) h=2\left( eJ(f)-J(e)f\right) h\), the above expression is zero, hence (A.5) is also zero.
Step 2.2: The terms involving \(\zeta \).
In order to see that (A.3) vanishes, it remains to see that
First observe that \([h^2,[J(f),J(e)]]=0\), so we may replace v by \({\tilde{v}}\) in (A.8). We then have
Since \(2w^+=[{\tilde{v}}, e]\) and \(-2w^-=[{\tilde{v}},f]\), we can rewrite the left-hand side of (A.8) as
Consider the last two terms. Since \([J(h),[J(x^\pm ),[\tilde{v},x^\mp ]]]=[J(h),[[J(x^\pm ),{\tilde{v}}],x^\mp ]]\pm [J(h),[\tilde{v},J(h)]]\), we have
Substituting these new expression back into (A.9), we obtain that the left-hand side of (A.8) is equal to
Let us show that this vanishes. Since \({\tilde{v}}=\frac{1}{2}\{e,f\}\), we have
Thus, \([J(h),[J(f),{\tilde{v}}]]=J(f)J(h)+J(h)J(f)\) and, similarly, \([J(h),[{\tilde{v}},J(e)]]=-J(e)J(h)-J(h)J(e)\). Substituting these back into (A.10), we see that it vanishes, and hence (A.8) holds. This completes the proof that \(\varPhi _{\mathrm {cr},J}\) preserves the relation \([h_1,[x_1^-,x_1^+]]=0\). \(\square \)
Rights and permissions
About this article
Cite this article
Guay, N., Regelskis, V. & Wendlandt, C. Equivalences between three presentations of orthogonal and symplectic Yangians. Lett Math Phys 109, 327–379 (2019). https://doi.org/10.1007/s11005-018-1108-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-018-1108-6