GL(NM) quantum dynamical R -matrix based on solution of the associative Yang-Baxter equation
aa r X i v : . [ m a t h . QA ] J un GL(NM) quantum dynamical R -matrix based on solutionof the associative Yang-Baxter equation I. Sechin A. Zotov
Steklov Mathematical Institute of Russian Academy of Sciences,Gubkina str. 8, Moscow, 119991, Russia
E-mails: [email protected], [email protected]
Abstract
In this letter we construct GL NM -valued dynamical R -matrix by means of unitaryskew-symmetric solution of the associative Yang-Baxter equation in the fundamental rep-resentation of GL N . In N = 1 case the obtained answer reproduces the GL M -valuedFelder’s R -matrix, while in the M = 1 case it provides the GL N R -matrix of vertex typeincluding the Baxter-Belavin’s elliptic one and its degenerations. Yang-Baxter equations.
Consider a matrix-valued function R ~ ( z ) ∈ Mat( N, C ) ⊗ , whichsolves the associative Yang-Baxter equation [8, 18]: R ~ ( z ) R η ( z ) = R η ( z ) R ~ − η ( z ) + R η − ~ ( z ) R ~ ( z ) , z ab = z a − z b . (1)Here, following notations of the Quantum Inverse Scattering Method [20], an operator R ~ ab ( z )in (1) is considered as Mat( N, C ) ⊗ -valued. It acts non-trivially in the a -th and b -th tensorcomponents only. For example, R ~ ( z ) is of the form R ~ ( z ) = N X i,j,k,l =1 R ijkl ( ~ , z ) e ij ⊗ N ⊗ e kl , (2)where the set { e ij } is the standard basis in Mat( N, C ), 1 N – is the identity matrix in Mat( N, C )and R ijkl ( ~ , z ) are functions of complex variables ~ (the Planck constant) and z (the spectralparameter).Let the solution of (1) satisfies also the properties of the skew-symmetry R ~ ( z ) = − R − ~ ( − z ) = − P R − ~ ( − z ) P , P = N X i,j =1 e ij ⊗ e ji (3)1nd unitarity R ~ ( z ) R ~ ( − z ) = ( ℘ ( ~ ) − ℘ ( z )) 1 N ⊗ N , (4)where ℘ ( x ) – is the Weierstrass ℘ -function. We assume that it is equal to 1 / sinh ( x ) or 1 /x for trigonometric (hyperbolic) or rational R -matrices respectively. Notice that solution of (1)with the properties (3)-(4) is a true quantum R -matrix of vertex type, i.e. it satisfies thequantum (non-dynamical) Yang-Baxter equation : R ~ ( z ) R ~ ( z ) R ~ ( z ) = R ~ ( z ) R ~ ( z ) R ~ ( z ) . (5)Equation (1) can be view as matrix extension of the genus one Fay trisecant identity: φ ( ~ , z ) φ ( η, z ) = φ ( η, z ) φ ( ~ − η, z ) + φ ( η − ~ , z ) φ ( ~ , z ) , (6)which coincides with (1) in scalar ( N = 1) case. It plays a crucial role in the theory of classicaland quantum integrable systems [12, 21, 4]. Solution of (6) satisfying the (scalar versions of)properties (3)-(4) is the Kronecker function: φ ( ~ , z ) = ϑ ′ (0) ϑ ( ~ + z ) ϑ ( ~ ) ϑ ( z ) , ϑ ( x ) = X k ∈ Z exp (cid:18) πıτ ( k + 12 ) + 2 πı ( x + 12 )( k + 12 ) (cid:19) , (7)where Im( τ ) >
0. Its trigonometric and rational limits are given by coth( ~ ) + coth( z ) and ~ − + z − respectively. Similarly, the elliptic solution of (1) with properties (3)-(4) is known [18]to be given by the Baxter-Belavin’s R -matrix [3]. The trigonometric solutions were classified in[19]. They include the XXZ R -matrix, its 7-vertex deformation [5] and their GL N generalizations[1] (see a brief review in [11]). The rational solutions consist of the XXX R -matrix, its 11-vertexdeformation [5] and their GL N generalizations [22, 13] – deformations of the GL N Yang’s R -matrix R ~ ( z ) = ~ − N ⊗ N + z − P . Summarizing, we deal with the R -matrices consideredas matrix generalizations of the Kronecker function (including its trigonometric and rationalversions).To formulate the main result we also need the Felder’s dynamical GL M R -matrix [7]: R F ( ~ , z , z | q ) = R F ( ~ , z − z | q ) == φ ( ~ , z − z ) M X i =1 E ii ⊗ E ii + M X i = j E ij ⊗ E ji φ ( z − z , q ij ) + M X i = j E ii ⊗ E jj φ ( ~ , − q ij ) , (8)where q , ..., q M – are (free) dynamical parameters, q ij = q i − q j , (9)and the set { E ij } is the standard basis in Mat( M, C ).The R -matrix (8) is a solution of the quantum dynamical Yang-Baxter equation: R ~ ( z , z | q ) R ~ ( z , z | q − ~ (2) ) R ~ ( z , z | q ) == R ~ ( z , z | q − ~ (1) ) R ~ ( z , z | q ) R ~ ( z , z | q − ~ (3) ) , (10)where the shifts of the dynamical arguments { q i } are performed as follows: R ~ ( z , z | q + ~ (3) ) = P ~ R ~ ( z , z | q ) P − ~ , P ~ = M X k =1 M ⊗ M ⊗ E kk exp (cid:16) ~ ∂∂q k (cid:17) . (11) The latter statement is easily verified. See e.g. [14]. uantum dynamical GL NM R -matrix. Consider the following Mat(
N M, C )-valued expres-sion: R ~ ′ ′ ( z, w ) = M X i =1 1 ′ E ii ⊗ ′ E ii ⊗ R ~ ( z − w ) + M X i = j ′ E ij ⊗ ′ E ji ⊗ R q ij ( z − w )++ M X i = j ′ E ii ⊗ ′ E jj ⊗ N ⊗ N φ ( ~ , − q ij ) , (12)where the Mat( N M, C ) indices are represented in a way that the Mat( M, C )-valued tensorcomponents are numbered by the primed numbers, and the Mat( N, C )-valued components arethose without primes (as previously). Put it differently, the indices are arranged throughMat( N M, C ) ⊗ ∼ = Mat( M, C ) ⊗ ⊗ Mat( N, C ) ⊗ . The order of tensor components is, in fact,not important. It is chosen as in (12) just to emphasize its similarity with the Felder’s R -matrix(8). The latter is reproduced from (12) in the N = 1 case, when the GL N R -matrix entering(12) turns into the Kronecker function (7).The results of the paper are summarized in the following Theorem
Let R ~ ( z ) be some GL N quantum non-dynamical R -matrix satisfying the associativeYang-Baxter equation (1) and the properties (3)-(4). Then the expression (12) is a quantumdynamical R -matrix, i.e. it satisfies the quantum dynamical Yang-Baxter equation: R ~ ′ ′ ( z , z | q ) R ~ ′ ′ ( z , z | q − ~ (2) ) R ~ ′ ′ ( z , z | q ) == R ~ ′ ′ ( z , z | q − ~ (1) ) R ~ ′ ′ ( z , z | q ) R ~ ′ ′ ( z , z | q − ~ (3) ) , (13) where the shifts of arguments { q i } are performed similarly to (11): R ~ ′ ′ ( z , z | q + ~ (3) ) = P ~ ′ R ~ ′ ′ ( z , z | q ) P − ~ ′ , P ~ ′ = M X k =1 1 ′ M ⊗ ′ M ⊗ ′ E kk ⊗ N ⊗ N ⊗ N exp (cid:16) ~ ∂∂q k (cid:17) . (14) Proof:
It is useful to write (1) as R ~ ab ( z ab ) R ηbc ( z bc ) = R η − ~ bc ( z bc ) R ~ ac ( z ac ) + R ηac ( z ac ) R ~ − ηab ( z ab ) , (15)where a, b, c are distinct numbers from the set { , , } . Besides (15) and the properties (3)-(4)the proof of (13) uses the Yang-Baxter equation (5) for the GL N R -matrix and the followingcubic relation: R ~ ab ( z ab ) R ηac ( z ac ) R ~ bc ( z bc ) − R ηbc ( z bc ) R ~ ac ( z ac ) R ηab ( z ab ) = R ~ + ηac ( z ac )( ℘ ( ~ ) − ℘ ( η )) , (16)which is true under hypothesis of the theorem. If ~ = η it reduces to (5). In the general case(16) leads (due to skew-symmetry of its r.h.s.) to R ηab R ~ ac R ηbc + R ~ ab R ηac R ~ bc = R ηbc R ~ ac R ηab + R ~ bc R ηac R ~ ab , R ~ ab = R ~ ab ( z a − z b ) , (17)known as the Yang-Baxter equation with two Planck constants [15]. The verification of (13) isa straightforward but cumbersome calculation. Consider, for example, the equation arising in3he tensor component ′ E ij ⊗ ′ E kk ⊗ ′ E ji with i = j = k = i : R q ik ( z ) R q kj ( z ) R q ik ( z ) + φ ( ~ , q ik ) φ ( ~ , q ki ) R q ij ( z ) == R q kj ( z ) R q ik ( z ) R q kj ( z ) + φ ( ~ , q kj ) φ ( ~ , q jk ) R q ij ( z ) . (18)To prove it one should use (16) written in the form R q ik ( z ) R q kj ( z ) R q ik ( z ) − R q kj ( z ) R q ik ( z ) R q kj ( z ) == ( ℘ ( q ik ) − ℘ ( q kj )) R q ik + q kj ( z ) = ( ℘ ( q ik ) − ℘ ( q kj )) R q ij ( z ) (19)and the well-known property of the Kronecker function (scalar version of the unitarity) φ ( ~ , q ik ) φ ( ~ , q ki ) = ℘ ( ~ ) − ℘ ( q ik ) , φ ( ~ , q kj ) φ ( ~ , q jk ) = ℘ ( ~ ) − ℘ ( q kj ) . (20)The rest of the tensor components are verified similarly. (cid:4) In the elliptic case, when R ~ ( z ) is the Baxter-Belavin’s R -matrix, the result of the theoremis known [17]. Similar results for the classical r -matrices were obtained previously by P. Etingofand O. Schiffmann [6] and later in [16, 23], where the Hitchin type systems were described on theHiggs bundles with non-trivial characteristic classes. Recently, these type models appeared inthe context of R -matrix valued Lax pairs and quantum long-range spin chains [9, 10]. In [17] theanswer (12) was verified explicitly in the elliptic case without use of the associative Yang-Baxterequation. In this respect the approach of this paper provides much simpler proof. What ismore important, the answer (12) is also valid for all trigonometric and rational degenerations ofthe elliptic R -matrix (satisfying the properties required in the Theorem). In the light of resultsof [10] the R -matrix (12) is the one necessary for quantization of the (generalized) model ofinteracting tops. Classical gl NM r -matrix. As a by-product of the Theorem we also get the classical dynamicalYang-Baxter equation for the classical r -matrix of the generalized interacting tops [10]. Considerthe classical limit of the GL N R -matrix from the Theorem: R ~ ( z ) = ~ − N ⊗ N + r ( z ) + O ( ~ ) . (21)The coefficient r ( z ) is the classical r -matrix, and the quantum Yang-Baxter equation (5)reduces in the limit (21) to the classical (non-dynamical) Yang-Baxter equation:[ r ( z ) , r ( z )] + [ r ( z ) , r ( z )] + [ r ( z ) , r ( z )] = 0 . (22)Similarly, the classical dynamical r -matrix appears from (8) through (21). It satisfies theclassical dynamical Yang-Baxter equation:[ r ( z ) , r ( z )] + [ r ( z ) , r ( z )] + [ r ( z ) , r ( z )]+[ ˆ ∂ , r ( z )] − [ ˆ ∂ , r ( z )] + [ ˆ ∂ , r ( z )] = 0 , (23)which underlies the Poisson structure of the spin Calogero-Moser model [2]. Hereˆ ∂ = M X k =1 M ⊗ M ⊗ E kk ∂ q k , P ~ ) = 1 M ⊗ + ~ ˆ ∂ + O ( ~ ) . (24)4n the same way, starting from the quantum R -matrix (12) one gets the classical r -matrix r ′ ′ ( z ) = M X i =1 1 ′ E ii ⊗ ′ E ii ⊗ r ( z ) + M X i = j ′ E ij ⊗ ′ E ji ⊗ R q ij ( z ) , (25)and the classical dynamical Yang-Baxter equation follows from (13):[ r ′ ′ ( z ) , r ′ ′ ( z )] + [ r ′ ′ ( z ) , r ′ ′ ( z )] + [ r ′ ′ ( z ) , r ′ ′ ( z )]++[ ˆ ∂ ′ , r ′ ′ ( z )] − [ ˆ ∂ ′ , r ′ ′ ( z )] + [ ˆ ∂ ′ , r ′ ′ ( z )] = 0 . (26)withˆ ∂ ′ = M X k =1 1 ′ M ⊗ ′ M ⊗ ′ E kk ⊗ N ⊗ N ⊗ N ∂ q k , P ~ ′ ( ) = 1 MN ⊗ + ~ ˆ ∂ ′ + O ( ~ ) . (27) Acknowledgments.
The second author is a Young Russian Mathematics award winner. Thework was performed at the Steklov Mathematical Institute of Russian Academy of Sciences,Moscow. This work is supported by the Russian Science Foundation under grant 19-11-00062.
References [1] A. Antonov, K. Hasegawa, A. Zabrodin, Nucl. Phys. B503 (1997) 747–770; hep-th/9704074.[2] E. Billey, J. Avan, O. Babelon, Physics Letters A, 186 (1994) 114–118; hep-th/9312042.I. Krichever, O. Babelon, E. Billey, M. Talon, Amer. Math. Soc. Transl. (2) Vol. 170 (1995) 83–120.[3] R.J. Baxter, Ann. Phys. 70 (1972) 193–228.A.A. Belavin, Nucl. Phys. B, 180 (1981) 189–200.[4] V.M. Buchstaber, G. Felder, A.V. Veselov, Duke Math. J. 76 (1994) 885–911; hep-th/9403178.[5] I.V. Cherednik, Theor. Math. Phys., 47:2 (1981) 422–425.[6] P. Etingof, O. Schiffmann, Mathematical Research Letters, 6 (1999) 593-612; math/9908115 [math.QA].P. Etingof, O. Schiffmann, London Math. Soc., Lectrure Notes Series, 290 (2001) 89–129; math.QA/9908064.[7] G. Felder,
Elliptic quantum groups , Proc. of the XI-th ICMP (Paris, 1994), Int. Press, Cambridge, (1995)211–218; hep-th/9412207.[8] S. Fomin, A.N. Kirillov, Advances in geometry; Progress in Mathematics, Vol. 172 (1999) 147–182.Anatol N. Kirillov, SIGMA 12 (2016), 002; arXiv:1502.00426 [math.RT].[9] A. Grekov, A. Zotov, J. Phys. A: Math. Theor. 51 (2018) 315202; arXiv: 1801.00245.I. Sechin, A. Zotov, Phys. Lett. B, 781 (2018) 1-7 , arXiv: 1801.08908.[10] A. Grekov, I. Sechin, A. Zotov, arXiv:1905.07820 [math-ph].[11] T. Krasnov, A. Zotov, Annales Henri Poincare (2019), https://doi.org/10.1007/s00023-019-00815-1;arXiv:1812.04209 [math-ph].[12] I. Krichever, Funct. Anal. Appl., 14:4 (1980) 282–290.[13] A. Levin, M. Olshanetsky, A. Zotov, JHEP 07 (2014) 012; arXiv:1405.7523 [hep-th].
14] A. Levin, M. Olshanetsky, A. Zotov, JHEP 10 (2014) 109; arXiv:1408.6246 [hep-th].A.M. Levin, M.A. Olshanetsky, A.V. Zotov, Theoret. and Math. Phys. 184:1 (2015) 924–939;arXiv:1501.07351 [math-ph].A.V. Zotov, Theoret. and Math. Phys., 197:3 (2018), 1755-1770.[15] A. Levin, M. Olshanetsky, A. Zotov, J. Phys. A: Math. Theor., 49:1 (2016), 14003 , 19 pp., Exactly SolvedModels and Beyond: a special issue in honour of R. J. Baxter’s 75th birthday, arXiv: 1507.02617.[16] A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, Comm. Math. Phys., 316:1 (2012) 1-44; arXiv:1006.0702.A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, J. Geom. Phys., 62:8 (2012) 1810-1850 , arXiv:1007.4127.Andrey M. Levin, Mikhail A. Olshanetsky, Andrey V. Smirnov, Andrei V. Zotov, SIGMA, 8 (2012), 095;arXiv: 1207.4386.[17] A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, J. Phys. A: Math. Theor. 46:3 (2013) 035201;arXiv:1208.5750 [math-ph].[18] A. Polishchuk, Advances in Mathematics 168:1 (2002) 56-95.[19] T. Schedler, Mathematical Research Letters, 10:3 (2003) 301–321; arXiv:math/0212258 [math.QA].A. Polishchuk, Algebra, Arithmetic, and Geometry, Progress in Mathematics book series, Volume 270(2010) 573–617; arXiv:math/0612761 [math.QA].[20] E.K. Sklyanin, Journal of Soviet Mathematics, 19:5 (1982) 1546-1596.[21] E.K. Sklyanin, LOMI Preprint E-3-1979.E.K. Sklyanin, Funct. Anal. Appl., 16:4 (1982) 263–270.E.K. Sklyanin, Funct. Anal. Appl., 17:4 (1983) 273–284.[22] A. Smirnov, Central European Journal of Physics 8 (4), 542–554.[23] A.V. Zotov, A.V. Smirnov, Theoret. and Math. Phys., 177:1 (2013), 1281-1338.14] A. Levin, M. Olshanetsky, A. Zotov, JHEP 10 (2014) 109; arXiv:1408.6246 [hep-th].A.M. Levin, M.A. Olshanetsky, A.V. Zotov, Theoret. and Math. Phys. 184:1 (2015) 924–939;arXiv:1501.07351 [math-ph].A.V. Zotov, Theoret. and Math. Phys., 197:3 (2018), 1755-1770.[15] A. Levin, M. Olshanetsky, A. Zotov, J. Phys. A: Math. Theor., 49:1 (2016), 14003 , 19 pp., Exactly SolvedModels and Beyond: a special issue in honour of R. J. Baxter’s 75th birthday, arXiv: 1507.02617.[16] A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, Comm. Math. Phys., 316:1 (2012) 1-44; arXiv:1006.0702.A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, J. Geom. Phys., 62:8 (2012) 1810-1850 , arXiv:1007.4127.Andrey M. Levin, Mikhail A. Olshanetsky, Andrey V. Smirnov, Andrei V. Zotov, SIGMA, 8 (2012), 095;arXiv: 1207.4386.[17] A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, J. Phys. A: Math. Theor. 46:3 (2013) 035201;arXiv:1208.5750 [math-ph].[18] A. Polishchuk, Advances in Mathematics 168:1 (2002) 56-95.[19] T. Schedler, Mathematical Research Letters, 10:3 (2003) 301–321; arXiv:math/0212258 [math.QA].A. Polishchuk, Algebra, Arithmetic, and Geometry, Progress in Mathematics book series, Volume 270(2010) 573–617; arXiv:math/0612761 [math.QA].[20] E.K. Sklyanin, Journal of Soviet Mathematics, 19:5 (1982) 1546-1596.[21] E.K. Sklyanin, LOMI Preprint E-3-1979.E.K. Sklyanin, Funct. Anal. Appl., 16:4 (1982) 263–270.E.K. Sklyanin, Funct. Anal. Appl., 17:4 (1983) 273–284.[22] A. Smirnov, Central European Journal of Physics 8 (4), 542–554.[23] A.V. Zotov, A.V. Smirnov, Theoret. and Math. Phys., 177:1 (2013), 1281-1338.