Classical R-Operators and Integrable Generalizations of Thirring Equations

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 4 (2008), 011, 19 pages Classical R -Op erators and In tegr able Generalizations of Thirring Equations ⋆ T ar as V. SKR YPNYK †‡ † SISSA, via Beirut 2-4, 3401 4 T r ieste, Italy E-mail: skrypnyk@sissa.it † Bo golyub ov Institute for The or etic al Physics, 14-b Metr olo gichna Str., Kyiv 03680 , Ukr aine Received Octob er 31, 2007, in f inal form Ja nuary 18, 200 8; Published online F ebruary 0 1, 2 008 Original article is av ailable at http:/ /www. emis. de/journals/SIGMA/2008/011/ Abstract. W e cons truct dif ferent int egrable genera lizations of the ma s sive Thirring equa - tions corresp onding loop algebras e g σ in dif feren t gradings and asso cia ted “triangular” R - op erator s. W e consider the most interesting ca ses connected with the Coxeter automor - phisms, seco nd order automorphisms and with “ Kostant–Adler–Symes” R -op era tors. W e recov er a known ma trix generaliza tion of the complex Thirring equations a s a pa rtial case of our constr uction. Key wor ds: inf inite-dimensional Lie algebras ; classica l R -op era tors; hierarchies of integrable equations 2000 Mathematics Subje ct Classific ation: 17B70; 17 B 80; 37 K10; 37K 30; 70 H06 1 In tro duction A theory of hierarc hies of in te grable equatio ns in partial deriv ativ es is b ased on the p ossibilit y to r epresent eac h of th e equations of the hierarc h y in the so-calle d zero-curv ature form : ∂ U ( x, t, u ) ∂ t − ∂ V ( x, t, u ) ∂ x + [ U ( x, t, u ) , V ( x, t , u )] = 0 , (1) where U ( x, t, u ), V ( x, t, u ) are g -v al ued fu nctions with d ynamical v ariable co ef f icien ts, g is simp le (reductiv e) Lie algebra and u is an additional complex parameter u sually called sp ectral. In order for the equation (1) to b e consisten t it is necessary that U ( x, t, u ) and V ( x, t, u ) b elong to some closed inf in ite-dimensional Lie algebra e g of g -v al ued fun ctions of u . There are sev eral approac hes to constru ction of zero-curv ature equations (1) starting fr om Lie algebras e g . All of them are b ased on th e so-called Kostant –Adler–Symes sc heme. On e of the most simple and general app roac hes is the approac h of [3, 10] and [6, 7] that interprets equation (1) as a consistency condition for a tw o comm uting Hamilt onian f lo ws written in the Euler–Arnold or Lax f orm. In the framework of this appr oac h elemen ts U ( x, t, u ) and V ( x, t, u ) coincide with the algebra-v alued gradien ts of the comm uting Hamiltonians constructed with the help of the Kostan t–Adler–Symes sc heme, the cornerstone of whic h is a decomp osition of the Lie algebra e g (as a vec tor space) in a direct sum of its t w o subalgebras: e g = e g + + e g − . The algebra-v alued gradients of the co mm uting Hamiltonians coincide w ith the restrictions of the algebra-v alued gradien ts of Casimir fun ctions of e g on to the subalgebras e g ± . Hence, suc h ⋆ This p ap er is a contribution to t he Pro ceedings of the Seven th Internatio nal Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–3 0, 2007, Kyiv, Ukraine). The full collection is a v aila ble at http://w ww.emis.de/j ournals/SIGMA/symmetry2007.h tml 2 T.V. Skrypnyk approac h p ermits [14] to construct using Lie a lgebra e g the three types of int egrable equations: t w o typ es of equations with U - V pair belonging to the same Lie subalgebras e g ± and the third t yp e of equations w ith U -op erator b elonging to e g + and V -operator b elonging to e g − (or vise v erse). The latter equations are sometimes c alled “negativ e f lo ws” of in tegrable hierarc hies. Nev ertheless the approac h of [3, 6, 7] do es not co v er all integ rable equations. In particular, it w as not possible to p ro duce b y means of this approac h inte grable equations possessin g U - V pairs with U -op erator b elongi ng to e g + and V -op erato r b elonging to e g − in the case e g + ∩ e g − 6 = 0, i.e. in cases dropping out of the scop e of Kostant– Adler–Symes metho d. An example of suc h situation is the Th irring in tegrable equations with the s tand ard sl (2)-v alued U - V pair [9]. In th e presen t pap er we generalize the method of [3, 10] of pro du cing soliton e quations a nd their U - V pairs, f illing the gap describ ed abov e, i.e. making the method to include a ll kno wn soliton e quations, and among them Thirring equation. W e u se th e same idea a s in [3] and [6], i.e. w e inte rpret zero-curv ature condition as a compatibilit y condition for a set of commuting Hamiltonian f lo ws on e g ∗ but constructed not with the help of Kostan t–Adler–Symes metho d but with the help of its generalizatio n – metho d of the classical R -op erator, where R : e g → e g , satisf ies a mo dif ied Y ang–B axter equation [12 ]. In this case one can also def ine [12] t w o Lie subalgebras e g R ± suc h that e g R ± + e g R − = e g , but in this case t his s u m is not a d irect s um of t w o v ector spaces, i.e. e g R ± ∩ e g R − 6 = 0. Hence, in order to ac hiev e our goal it is n ecessary to construct w ith the help of the R - op erator an algebra of m utually comm uting functions. Contrary to the classical approac h of [12] they should comm ute not with r esp ect to the R -brac k et { , } R but with resp ect to the initial Lie–P oisson brac k et { , } on e g ∗ . In our pr evious p ap er [13] the corresp onding functions were constructed us ing the ring of Casimir functions I G ( ˜ g ∗ ). In more detail, we pro v ed that the func- tions I R ± k ( L ) ≡ I k (( R ∗ ± 1)( L )), where I k ( L ) , I l ( L ) ∈ I G ( e g ∗ ), constitute an Ab elian sub algebra in C ∞ ( e g ∗ ) with resp ect to t he standard Lie–P oi sson b rac k ets { , } on e g ∗ . The alg ebra-v alued gradien ts of functions the I R ± k ( L ) b elong to the su balgebras e g R ± corresp ondin gly . In th e ca se when the R -op er ator is of Kostant– Adler–Symes t yp e, i.e. R = P + − P − , w here P ± are p ro jection op erators on to sub algebras e g R ± = e g ± and e g + ∩ e g − = 0 we r e-obtain the r esu lts of [10] (see also [14]) as a partial case of our construction. In the cases of more complicated R -op erators our scheme is new and generalize s th e approac h of [10]. In particular, the imp ortan t class of U - V pairs sati sfying zero-curv ature equations that can b e obtained b y our metho d are connected with the s o-called “triangular” R -op erators. In more detail, if e g p ossess a “triangular” decomp osition: e g = e g + + g 0 + e g − , w here the sum is a direct sum of v ecto r spaces, e g ± and g 0 are closed Lie sub algebras and e g ± are g 0 -mo dules, R 0 is a solution of the mo d if i ed Y ang–Baxter equation on g 0 , P ± are the pro jection op erators on to the subalgebras e g ± then R = P + + R 0 − P − is a solution of the mo dif ied Y ang– Baxter equation on e g ( see [13] for t he deta iled pro of ). The Lie subalgebras e g R ± ha v e in this case the follo wing form: e g R ± = e g ± + g 0 , i.e. e g R + ∩ e g R − 6 = 0. Suc h R -op erators are connected with the Thirring-t yp e in tegrable mo dels. These R -op erators w ere f irst considered in [4] and in [5], where the usual sl (2)-Thirr ing equation was obtained using “geomet ric” tec hnique. In order to obtain the Thirring integ rable equation and its v arious generalizations in the framew ork of our algebraic approac h we consider the case when e g coincides w ith a lo op algebra e g = g ⊗ P ol( u − 1 , u ) or its “t wisted” w ith th e help of f i nite-order automorphism σ su balgeb- ra e g σ . Th e algebras e g σ p ossess the n atural “triangular” decomp osition with the algebra g 0 b eing a red uctiv e s u balgebra of g stable u nder the action of σ . F o r eac h algebra e g σ with a n atur al triangular d ecomp osition and for eac h classical R op erator R 0 on g 0 w e def ine an integ rable equation of the hyperb olic t yp e which w e call the “non-Ab elian generalized Thirrin g equation”. W e sho w that in the case g = sl (2), σ 2 = 1 and g 0 = h , w here h is a Cartan su balgebra of sl (2) our construction yields the usu al Thirrin g equation and its standard sl (2)-v alued U - V pair. W e consider in some detail the c ases of the ge neralized Thirr ing e quations that corresp ond to Classical R -Op erators and I n tegrable Generalizatio ns of Thirrin g Equations 3 the second order automorph ism of g . F or the case of suc h automorphisms and g = g l ( n ) and sp ecial choi ce of R 0 w e obtain the so-called matrix generalizat ion of complex Thirrin g equations obtained by other metho d in [15]. After a reduction to real su balgebra u ( n ) these equations read as f ollo ws: i∂ x − Ψ + =  1 κ − Ψ + (Ψ † − Ψ − ) + κ + Ψ −  , i∂ x + Ψ − = −  1 κ + (Ψ + Ψ † + )Ψ − + κ − Ψ +  , where Ψ ± ∈ Mat ( p, q ), κ ± ∈ R are constants, i.e. are exact m atrix analogues of the usual massive Thirring equations. W e also consider in detail t he case of the generalized Thirr ing equations that corresp ond to the Co x eter automorph isms of g . W e call the ge neralized Thirr ing equations corresp ondin g to the th e Co xeter automorphisms the “generalized Ab elian Thirr ing equations”. W e sho w that the num b er of in dep end en t f ields in the generalized Ab elian Thirr ing equation corr esp onding to Co xeter automorphism i s equal to 2(dim g − rank g ) and the ord er of the equations g ro ws with the growth of the rank of Lie algebra g . W e consider in d etail the generalized Ab elian Thirrin g equations corresp onding to the case g = sl (3). F or the sake of completeness we also consider n on -linear d if f er ential equ ations of hyp erb olic t yp e corresp onding to e g σ and th e K ostan t–Adler–Symes R -op erator. W e sh o w that the ob tained equations are in a sence inte rmediate b et w een the generalized Thirr ing and non-Ab elian T o da equations. The cases of the second order and Co xeter automorphisms are considered. The s l (2) and sl (3) examples are work ed out in detail. The structure of the presen t p ap er i s the follo wing: in the second sec tion w e describe com- m utativ e subalgebras asso ciated w ith the classical R op erator. In the third section w e obtain asso ciated zero-curv ature equations. A t last in the f ourth section w e consider in tegrable hierar- c hies asso ciat ed with lo op algebras in dif feren t gradings and dif feren t R -op erators. 2 Comm u tativ e subalgebras and classical R op erator Let e g b e a Lie algebra (f inite or inf inite-dimensional) with a Lie brac k et [ , ]. Let R : e g → e g b e s ome linear op erator on e g . The op erator R is called the classical R -op erator if it satisf ies mo dif ied Y ang–Baxter equation: R ([ R ( X ) , Y ] + [ X, R ( Y )]) − [ R ( X ) , R ( Y )] = [ X, Y ] ∀ X , Y ∈ e g . (2) W e will use, in add ition to the op erator R , the follo wing op erators: R ± ≡ R ± 1. As it is kno wn [2, 12] the map s R ± def i ne Lie su balgebras e g R ± ⊂ e g : e g R ± = Im R ± . It is easy to see from their def inition e g R + + e g R − = e g , but, in general, this su m is not a direct sum of vect or spaces, i.e., e g R + ∩ e g R − 6 = 0. Let e g ∗ b e the dual space to e g and h , i : e g ∗ × e g → C be a pairing b et w ee n e g ∗ and e g . Let { X i } b e a basis in th e Lie algebra e g , { X ∗ i } b e a basis in th e dual sp ace e g ∗ : h X ∗ j , X i i = δ ij , L = P i L i X ∗ i ∈ e g ∗ b e the generic e lemen t of e g ∗ , L i b e the co ord inate functions on e g ∗ . Let us consider the stand ard Lie–P oisson br ac k et betw een F 1 , F 2 ∈ C ∞ ( e g ∗ ): { F 1 ( L ) , F 2 ( L ) } = h L, [ ∇ F 1 , ∇ F 2 ] i , where ∇ F k ( L ) = X i ∂ F k ( L ) ∂ L i X i is a so-called algebra-v alued gradient of F k . The summations hereafter are imp lied o v er all b asic elemen ts of e g . 4 T.V. Skrypnyk Let R ∗ b e the op erator dual to R , acting in the space e g ∗ : h R ∗ ( Y ) , X i ≡ h Y , R ( X ) i , ∀ Y ∈ e g ∗ , X ∈ e g . Let I G ( e g ∗ ) b e the ring of in v arian ts of the coadjoin t represent ation of e g . W e will consider the f unctions I R ± k ( L ) on e g ∗ def i ned b y th e follo wing form ulas: I R ± k ( L ) ≡ I k (( R ∗ ± 1)( L )) ≡ I k ( R ∗ ± ( L )) . The follo w ing theorem holds true [13]: Theorem 1. F u nctions { I R + k ( L ) } and { I R − l ( L ) } , wher e I k ( L ) , I l ( L ) ∈ I G ( e g ∗ ) , g ener ate an Ab e lian sub algebr a in C ∞ ( e g ∗ ) with r esp e ct to the standar d Lie–Poisson br ackets { , } on e g ∗ : ( i ) { I R + k ( L ) , I R + l ( L ) } = 0 , ( ii ) { I R − k ( L ) , I R − l ( L ) } = 0 , ( iii ) { I R + k ( L ) , I R − l ( L ) } = 0 . Remark 1. Note that th e commuta tiv e subalgebras constructed in this theorem dif fer f rom the comm utativ e sub algebras constructed usin g standard R -matrix scheme [12]. Indeed, our theorem states comm u tativit y of fu n ctions { I R + k ( L ) } and { I R − l ( L ) } with resp ect to the initial Lie–P oisson br ac k et { , } on e g . The standard R -matrix sc heme states comm utativit y of the functions { I k ( L ) } w ith resp ect to the so-call ed R -brac k et { , } R , where: { F 1 ( L ) , F 2 ( L ) } R = h L, [ R ( ∇ F 1 ) , ∇ F 2 ] + [ ∇ F 1 , R ( ∇ F 2 )] i . Theorem 1 provides us a large Ab elian sub algebra in the sp ace ( C ∞ ( e g ∗ ) , { , } ). W e will con- sider the follo w ing t w o examples of the R -op er ators and the corresp onding Ab elian sub algebras. Example 1. Let us consider the case of the Lie algebras e g w ith the so-called “Kostan t– Adler– Symes” (K AS ) decomp osition in to a d irect sum of the t w o v ector subspaces e g ± : e g = e g + + e g − , where subsp aces e g ± are closed Lie subalgebras. Let P ± b e the pro jection op erators ont o the subalgebras e g ± resp ectiv ely . Then it is known [12] that in this case it is p ossible to def ine t he so-calle d K ostan t–Adler–Symes R -matrix: R = P + − P − . It is easy to see that R + = 1 + R = 2 P + , R − = R − 1 = − 2 P − , are p rop ortional to the pro jection op erators on to the subalgebras e g ± . It follo w s that e g R ± ≡ e g ± and e g R + ∩ e g R − = 0. The P oisson commuting fu nctions I R ± k ( L ) acquire the follo wing simple form: I R ± k ( L ) ≡ I ± k ( L ) ≡ I k ( L ± ) , where L ± ≡ P ∗ ± L, i.e. I ± k ( L ) are restrictions of the coadjoin t inv ariants onto the dual spaces e g ∗ ± . Example 2. L et us consider the case of Lie algebras e g w ith th e “triangular” d ecomp osition: e g = e g + + g 0 + e g − , where the sum is a direct sum of v ec tor spaces, e g ± and g 0 are closed subalgebras, and e g ± are g 0 -mo dules. As it is known [4 ] (see also [13] for the detailed pro of ), if R 0 is a solution of the mo dif ied Y ang–Baxter equation (2) on g 0 then R = P + + R 0 − P − (3) is a solution of the mo dif ie d Y ang–Baxter (2) equatio n on g if P ± are the pro jection op erators on to the su balgebras e g ± . Classical R -Op erators and I n tegrable Generalizatio ns of Thirrin g Equations 5 In th e case when R 0 = ± I d 0 (whic h are ob viously the solutions of the equation (2 ) on g 0 ) we obtain that R -matrix (3) passes to the standard Kostan t–Adler–Symes R -matrix. Nev ertheless in the considered “triangular cases” there are other p ossibilities. F or example, if a Lie sub algebra g 0 is Ab elian then (3) is a solution of (2) for an y tensor R 0 on g 0 . In the case of th e R -matrix (3) we h a v e: R + = 2 P + + ( P 0 + R 0 ), R − = − (2 P − + ( P 0 − R 0 )), e g R ± = e g ± + Im ( R 0 ) ± , w here ( R 0 ) ± = ( R 0 ± P 0 ) are the R ± -op erators on g 0 and e g R + ∩ e g R − = Im ( R 0 ) + ∩ Im ( R 0 ) − . The P oisson-comm utativ e functions I R ± k ( L ) acquire the follo wing form: I R ± k ( L ) ≡ I R 0 , ± k ( L ) = I k  L ± + (1 ± R ∗ 0 ) 2 ( L 0 )  , where L ± ≡ P ∗ ± L , L 0 ≡ P ∗ 0 L . W e will u se suc h fun ctions constru cting dif feren t generaliza tions of the Th irring mo del. The Th eorem 1 giv es u s a set of m utually comm utati v e functions on e g ∗ with resp ect to the brac k ets { , } that can b e used as an algebra of the integ rals of some Hamiltonian system on e g ∗ . F or the Hamiltonian f unction one ma y c hose one of the fun ctions I R ± k ( L ) or their linear com bination. Let us consider the corresp ondin g Hamiltonian equation: dL i dt ± k = { L i , I R ± k ( L ) } . (4) The follo w ing pr op osition is true [13]: Prop osition 1. The Hamiltonian e quations of motion (4) c an b e written in the E uler–Ar nold (gener alize d L ax) form: dL dt ± k = ad ∗ V ± k L, (5) wher e V ± k ≡ ∇ I R ± k ( L ) . Remark 2. In the case when e g ∗ can b e iden tif ied w ith e g and a coadjoin t representat ion can b e i dent if ied with an adj oin t one, equatio n (5 ) m ay b e w ritten in the usual Lax (comm utator) form. In the presen t pap er w e will n ot consider f in ite-dimensional Hamiltonian sys tems that could b e obtained in th e framewo rk of our constru ction bu t will use equation (5) in order to generate hierarc hies of soliton equ ations in 1 + 1 dimensions. 3 In tegra ble hierarc hies and classical R -op erators Let us remind one of the main Lie algebraic app roac hes to the theory of soliton equations [ 10]. It is based on the zero-curv ature c ondition and its in te rpretation as a consistency co ndition of t w o comm uting Lax f lo ws. The f ollo w ing Prop osition is true: Prop osition 2. L e t H 1 , H 2 ∈ C ∞ ( e g ∗ ) b e two Poisson c ommuting functions: { H 1 , H 2 } = 0 , wher e { , } is a standar d Lie–Poisson br ackets on e g ∗ . Then their e g -value d gr adients satisfy the “mo difie d” zer o-cu rvatur e e quation: ∂ ∇ H 1 ∂ t 2 − ∂ ∇ H 2 ∂ t 1 + [ ∇ H 1 , ∇ H 2 ] = k ∇ I , (6) wher e I is a Casimir function and t i ar e p ar am eters alo ng the tr aje ctor ies of Hamiltonian e qua- tions c orr esp ond ing to the Hamiltonians H i and k is an arbitr ary c onsta nt. 6 T.V. Skrypnyk F or the theory of soliton equations one needs the usu al zero-curv ature condition (case k = 0 in the ab o v e pr op osition) and the inf inite set of the comm uting Hamiltonians generating the cor- resp ond ing U - V pairs. Th is can b e ac hiev ed requ iring that e g is inf in ite-dimensional of a sp ecial t yp e. In more detail, the follo wing theorem is tru e: Theorem 2. L et e g b e an infinite-dimensional Lie algebr a of g - v alue d function of the one c omp lex variable u and a Lie algebr a g b e semisimple. L e t R b e a classic al R op er ato r on e g , L ( u ) b e the generic element of the dual sp ac e e g ∗ , I k ( L ( u )) b e Casimir functions on e g ∗ such that functions I R ± k ( L ( u )) ar e finite p olynomials on e g ∗ . Then the e g -value d functions ∇ I R ± k ( L ( u )) satisfy the zer o-curvatur e e quations: ∂ ∇ I R ± k ( L ( u )) ∂ t ± l − ∂ ∇ I R ± l ( L ( u )) ∂ t ± k + [ ∇ I R ± k ( L ( u )) , ∇ I R ± l ( L ( u ))] = 0 , (7) ∂ ∇ I R ± k ( L ( u )) ∂ t ∓ l − ∂ ∇ I R ∓ l ( L ( u )) ∂ t ± k + [ ∇ I R ± k ( L ( u )) , ∇ I R ∓ l ( L ( u ))] = 0 . (8) Pro of . As it f ollo w s from Prop osition 2 and T heorem 1 the algebra-v alued gradients ∇ I R ± k ( L ( u )), ∇ I R ± l ( L ( u )) and ∇ I R ± k ( L ( u )), ∇ I R ∓ l ( L ( u )) satisfy the “mo d if i ed” zero-curv ature equations (6). On the other hand, as it is not dif f icult to show, for the case of the Lie algebras e g describ ed in the theorem the algebra-v alued gradients of the Casimir functions are prop ortional to p o w ers of the generic elemen t of the dual space L ( u ), i.e. to th e f orm al p o w er series. O n the other hand, due to the condition that all I R ± k ( L ( u )) are f i nite p olynomials, their algebra-v alued gradien ts are finite linear com binations of the b asic elemen ts of the L ie algebra e g . Hence the corresp ondin g mo dif ied zero-curv ature equations are satisf ied if and o nly if the corresp ondin g co ef f icien ts k in these equations are equal to zero, i.e. wh en they are redu ced to the u sual zero-curv ature conditions. This p ro v es th e theorem.  Remark 3. Note that equations (7 ), (8) def ine three types of in tegrable hierarc hies: tw o “small” hierarc hies asso ciated with Lie subalgebras e g R ± def i ned b y equations (7) and one “large” hierar- c h y asso ciated with the whole Lie algebra e g that include b oth t yp es of equations (7) and (8). Equations (8) ha v e an in terpretation of the “negativ e f lows” of the integrable hierarch y asso- ciated with e g R ± . In the case e g R + ≃ e g R − the corresp onding “small ” h ierarc hies are equiv alent and the large h ierarc h y asso ciated with e g ma y b e called a “ double” of the hierarch y asso ciated with e g R ± . In the next subsection we w ill consider in detail a simple example of the ab o v e theorem when e g is a loop algebra and R -op erator is not of Kostan t–A dler–Symes type but a triangular one. W e will b e in terested in the “large” hierarc h y asso ciated with e g and, in more detail, in the simplest equation of this h ierarch y wh ic h will coincide with the generalizatio n of the Thirrin g equation. 4 Lo op algebras and general ized Th irring equations 4.1 Lo op algebras and c lassical R -op erators In this s u bsection we remin d sev eral imp ortan t facts from the theory of lo op algebras [8]. Let g b e semisimple (r ed uctiv e) Lie algebra. Let g = p − 1 P j =0 g j b e Z p = Z /p Z gradin g of g , i.e.: [ g i , g j ] ⊂ g i + j where j denotes the class of equiv alence of the elemen ts j ∈ Z mo d p Z . It is kno wn that the Z p -grading of g ma y b e def i ned with the help of some automorph ism σ of the order p , su c h that σ ( g i ) = e 2 π ik /p g i and g 0 is the algebra of σ -inv arian ts: σ ( g 0 ) = g 0 . Classical R -Op erators and I n tegrable Generalizatio ns of Thirrin g Equations 7 Let e g = g ⊗ Pol( u, u − 1 ) b e a lo op algebra. Let us consider the follo wing sub space in e g : e g σ = M j ∈ Z g j ⊗ u j . It is kn own [8 ] that this subspace is a closed Lie subalgebra and if we extend the automorphism σ to the map ˜ σ of the whole algebra e g , def in ing its action on the space g ⊗ P ol( u, u − 1 ) in the standard w a y [8]: ˜ σ ( X ⊗ u k ) = σ ( X ) ⊗ e − 2 π ik /p u k , then the sub algebra e g σ can b e def ined as th e subalgebra of ˜ σ -in v ariants in e g : e g σ = { X ⊗ p ( u ) ∈ e g | ˜ σ ( X ⊗ p ( u )) = X ⊗ p ( u ) } . W e will call the algebra e g σ the loop su balgebra “t wisted” with th e help of σ . The basis in e g σ consists of algebra-v alued fu nctions { X j α ≡ X j α u j } , where X j α ∈ g j . Let us def ine the pairing b et w een e g σ and ( e g σ ) ∗ in the standard wa y: h X, Y i = res u =0 u − 1 ( X ( u ) , Y ( u )) , where X ∈ e g σ , Y ∈ ( e g σ ) ∗ and ( , ) is a bilinear, in v arian t, nondegenerate form on g . It is ea sy to see th at with resp ect to such a pairing the d ual space ( e g σ ) ∗ ma y be iden tif ied with the Lie algebra e g σ itself. T he dual basis in ( e g σ ) ∗ has the form: { Y j α ≡ X − j ,α u − j } , where X − j ,α is a dual basis in the sp ace g − j . The Lie algebra e g σ p ossesses KAS decomp osition e g σ = e g σ + + e g σ − [11], where e g σ + = M j ≥ 0 g j ⊗ u j , e g σ − = M j < 0 g j ⊗ u j . It def i nes in a natural wa y the Kostant– Adler–Symes R -op erator: R = P + − P − , where P ± are pro jection op erators onto Lie algebra e g σ ± . The t wisted lo op algebra e g σ also p ossesses “triangular” d ecomp osition: e g σ = e g σ + + g 0 + e g σ − , where e g σ − ≡ e g σ − , e g σ + ≡ e g σ + + g 0 . It def i nes in a n atural wa y th e triangular R -op erator R = P + + R 0 − P − , where P ± are the pr o jection op erators onto Lie algebra e g σ ± and R 0 is an R -op erator on g 0 . The Lie alge bra g 0 in this decomp osition i s a reductiv e Lie subalgebra of the Lie algebra g . Due to the fact that a lot of solutions of the mo dif ied cla ssical Y ang–Baxter equations on reductiv e Lie algebras are kn o wn, one can construct explicitly R -op erators R 0 on g 0 , the Lie subalgebras e g R ± = e g ± + Im( R 0 ) ± and the Poisson-co mm utativ e functions constru cted in the Theorem 1. 4.2 Generalized Thirring mo dels As we ha v e seen in the previous sub section, eac h of the gradings of th e lo op algebras, corre- sp ond ing to the dif feren t automorph isms σ yields its o wn triangular decomp osition. Let us consider the simplest equations of in tegrable hierarc hies, corresp onding to dif ferent triangular 8 T.V. Skrypnyk decomp ositions in more d etail. The generic ele men ts of th e dual sp ace to the Lie algebras e g R ± are written as follo ws: L ± ( u ) = r ∗ ±   dim g ¯ 0 X α =1 L 0 α X 0 ,α   + ±∞ X j = ± 1 dim g ¯ j X α =1 L ( j ) α X − j ,α u − j = r ∗ ± ( L (0) ) + ±∞ X j = ± 1 L ( j ) u − j , where L ( − j ) ∈ g ¯ j and in ord er to simp lify th e notations we pu t r ± ≡ (1 ± R 0 ) 2 . Let ( , ) b e an in v arian t nondegenerated form o n the u nderlying semisimple (red uctiv e) Lie algebra g . Then I 2 ( L ) = 1 2 ( L, L ) i s a second o rder Casimir fun ction on g ∗ . The corresp onding generating function of the second order int egrals has the form: I ± 2 ( L ( u )) = I 2 ( L ± ( u )) = 1 2   r ∗ ± ( L (0) ) + ±∞ X j = ± 1 L ( j ) u − j , r ∗ ± ( L (0) ) + ±∞ X j = ± 1 L ( j ) u − j   = ±∞ X k =0 I pk 2 u − pk , where p is an order of σ . Th e simp lest of these integ rals are: I ± 0 2 = 1 2  r ∗ ± ( L (0) ) , r ∗ ± ( L (0) )  , I ± p 2 =  r ∗ ± ( L (0) ) , L ( ± p )  + 1 2 p − 1 X j =1  L ( ± j ) , L ( ± ( p − j ))  . The algebra-v alued gradien ts of functions I ± p 2 read as f ollo ws: ∇ I ± p 2 ( u ) = u ± p r ∗ ± ( L (0) ) + p − 1 X j =1 u ± ( p − j ) L ( ± j ) + r ± ( L ( ± p ) ) . (9) They coincide w ith the U - V p air of the generalized T hirring mo del. T he corresp ond ing zero- curv atur e condition: ∂ ∇ I + p 2 ( L ( u )) ∂ x − − ∂ ∇ I − p 2 ( L ( u )) ∂ x + + [ ∇ I + p 2 ( L ( u )) , ∇ I − p 2 ( L ( u ))] = 0 (10) yields the f ollo w ing system of dif feren tia l equations in partial deriv ative s: − ∂ x − r ∗ + ( L (0) ) = [ r ∗ + ( L (0) ) , r − ( L ( − p ) )] , (11a) ∂ x + r − ( L (0) ) = [ r + ( L ( p ) ) , r ∗ − ( L (0) )] , (11b) − ∂ x − L ( p − j ) = [ L ( p − j ) , r − ( L ( − p ) )] + p − 1 X i = j +1 [ L ( p − i ) , L ( i − j − p ) ] + [ r ∗ + ( L (0) ) , L ( − j ) ] , (11c) ∂ x + L ( j − p ) = [ r + ( L ( p ) ) , L ( j − p ) )] + p − 1 X i = j +1 [ L ( p + j − i ) , L ( j − p ) ] + [ L ( j ) , r ∗ − ( L (0) )] , j ∈ 1 , p − 1 , (11d) ∂ x + r − ( L ( − p ) ) − ∂ x − r + ( L ( p ) ) = [ r ∗ + ( L (0) ) , r ∗ − ( L (0) )] + p − 1 X i =1 [ L ( p − i ) , L ( i − p ) ] + [ r + ( L ( p ) ) , r − ( L ( − p ) )] . (11e) These equations are int e gr able gener alization of Thirring e quations corresp onding to Lie algeb- ra g , its Z p -grading def ined w ith the help of the automorph ism σ of the order p and the classical R -op erator R 0 on g ¯ 0 . W e will call this system of equations the non-Ab elian ge ner alize d Thirring system . In order to recognize in this complicated system of h yp erb olic equ ations generalization of Thirring equations we w ill consider several examples. Classical R -Op erators and I n tegrable Generalizatio ns of Thirrin g Equations 9 4.2.1 Case of the second-order automorphism Let us consider the case wh en automorphism σ is inv olutiv e, i.e. p = 2. Then g 1 = g − 1 and L ax matrices L ± ( u ) are th e follo wing: L ± ( u ) = r ∗ ± ( L (0) ) + u ∓ 1 L ( ± 1) + u ∓ 2 L ( ± 2) + · · · , and the U - V pair (9) acquire more simp le form: ∇ I ± 2 2 ( u ) = u ± 2 r ∗ ± ( L (0) ) + u ± 1 L ( ± 1) + r ± ( L ( ± 2) ) . (12) The corresp onding zero-curv ature condition yields the follo wing system of dif feren tial equations in partial d eriv ativ es: ∂ x − r ∗ + ( L (0) ) = [ r − ( L ( − 2) ) , r ∗ + ( L (0) )] , (13a) ∂ x + r ∗ − ( L (0) ) = [ r + ( L (2) ) , r ∗ − ( L (0) )] , (13b) ∂ x − L (1) = [ r − ( L ( − 2) ) , L (1) ] + [ L ( − 1) , r ∗ + ( L (0) )] , (13c) ∂ x + L ( − 1) = [ r + ( L (2) ) , L ( − 1) ] + [ L (1) , r ∗ − ( L (0) )] , (13d) ∂ x + r − ( L ( − 2) ) − ∂ x − r + ( L (2) ) = [ r ∗ + ( L (0) ) , r ∗ − ( L (0) )] + [ L (1) , L ( − 1) ] + [ r + ( L (2) ) , r − ( L ( − 2) )] . (13e) The system of equations (13) is still suf f icien tly complicated. In order to recognize in this system the u sual Thirring equ ations we hav e to consider the case g = sl (2). Example 3. Let g = sl (2) a nd σ b e the Cartan inv olution, i.e. sl (2) ¯ 0 = h = diag ( α, − α ) and sl (2) ¯ 1 consists of the matrices w ith zeros on the diagonal. The Lax matrices L ± ( u ) ha v e the follo wing form: L ± ( u ) = r ∗ ±  α (0) 0 0 − α (0)  + u ∓ 1  0 β ( ± 1) γ ( ± 1) 0  + u ∓ 2  α ( ± 2) 0 0 − α ( ± 2)  + · · · . Due to the fact that s l (2) ¯ 0 is Ab elian the one-dimensional linear maps r ± are written as follo ws: r ∗ ± ( L (0) ) = k ± L (0) where k ± are some constan ts, such that k + + k − = 1. The simplest Hamiltonians obtained in the fr amew ork of our sc heme are: I ± 0 2 = k ± ( α (0) ) 2 , I ± 2 2 = 2 k ± α (0) α ( ± 2) + β ( ± 1) γ ( ± 1) . As it follo ws fr om the Theorem 1 and, as it is also easy to ve rify by th e direct calculations, these Hamiltonians comm ute with r esp ect to the standard Lie–Po isson b r ac k et s on ] sl (2) σ : { α (2 k ) , β (2 l +1) } = β (2( k + l )+1) , { α (2 k ) , γ (2 l +1) } = − γ (2( k + l )+1) , { β (2 k +1) , γ (2 l +1) } = 2 α (2( k + l )+2) , { α (2 k ) , α (2 l ) } = { β (2 k +1) , β (2 l +1) } = { γ (2 k +1) , γ (2 l +1) } = 0 . As a consequence, th ey p ro duce th e correct U - V pair for zero-curv ature equations: ∇ I ± 2 2 = k ±  α ( ± 2) 0 0 − α ( ± 2)  + u ± 1  0 β ( ± 1) γ ( ± 1) 0  + k ± u ± 2  α (0) 0 0 − α (0)  , where U ≡ ∇ I +2 2 , V ≡ ∇ I − 2 2 . Due to the fact that I ± 0 2 are constan ts of motion w e ha v e that α (0) is also a constan t of motion and e quations (13a), (13b) are satisf ied automatically . The in tegrals I ± 2 2 are constan ts 10 T.V. Skrypnyk of motion to o, that is w hy α ( ± 2) are expressed via β ( ± 1) , γ ( ± 1) . Hence, equation (13e ) is n ot indep en d en t and follo ws from equations (13c), (13d). T hese equations are indep enden t. The last summand in th e equ ations (13c ), (13d) giv es the linear “massive term” in the Thir ring equation. The f irst summ and giv es the cubic non-linearit y . Let us w rite the equations (13c), (13d ) in the case at hand in m ore d etail taking in to accoun t the explicit f orm of th e matrices L ( k ) and linear op erators r ± : ∂ x − γ (1) = 2 k + α (0) γ ( − 1) − 2 k − α ( − 2) γ (1) , ∂ x − β (1) = − 2 k + α (0) β ( − 1) + 2 k − α ( − 2) β (1) , ∂ x + γ ( − 1) = 2 k − α (0) γ (1) − 2 k + α (2) γ ( − 1) , ∂ x + β ( − 1) = − 2 k − α (0) β (1) + 2 k + α (2) β ( − 1) , (14) where w e h a v e pu t I ± 2 2 = 0 and, hence, α ( ± 2) = − 1 2 k ± α (0) ( β ( ± 1) γ ( ± 1) ). The system of equations (1 4) admits a redu ction α (0) = ic , γ ( ± 1) = − ¯ β ( ± 1) ≡ − ¯ ψ ± 1 , whic h corresp onds to the restriction onto the real sub algebra su (2) of the complex Lie algebra sl (2). After suc h a r eduction th e system of equations (14) is simplif ied and acquires the form : ic∂ x − ψ 1 = 2 k + c 2 ψ − 1 + | ψ − 1 | 2 ψ 1 , ic∂ x + ψ − 1 = 2 k − c 2 ψ 1 + | ψ 1 | 2 ψ − 1 . In the case k + = k − = 1 2 these equations are u sual Thirring equ ations with a mass m = c 2 . 4.2.2 Matrix generalization of Thirring equation Let u s return to th e generalized Thirrin g mo del in the case of the higher ran k Lie algebra g , its automorphism of the second o rder and corr esp onding Z 2 -grading of g : g = g ¯ 0 + g ¯ 1 . W e are in terested in the cases that w ill b e maximally close to the case of the ordinary Thirrin g equation. In particular, w e w ish to h a v e r ∗ ± ( L (0) ) that enter into our U - V p air (12) near the second order of sp ectral parameter to b e constan t along all time f lo w s . As it follo ws from the equ ations (13a), (13b), this is not true for the general r -matrix R 0 on g ¯ 0 . F ortun ately there are v ery sp ecial cases when it is ind eed s o. The follo wing pr op osition h olds true: Prop osition 3. L et g ¯ 0 admit the de c omp osition into dir e ct sum of two r e ductive sub algebr as: g ¯ 0 = g + ¯ 0 ⊕ g − ¯ 0 . L et L (0) = L (0) + + L (0) − b e the c orr esp o nding de c omp osition of the element of the dual sp ac e. L et ζ ( g ± ¯ 0 ) b e a c enter of the sub algebr a g ± ¯ 0 , K ( L (0) ± ) b e a p art of L (0) ± dual to the c enter of the sub algebr a g ± ¯ 0 . Then R 0 = P + 0 − P − 0 is the R op er ator on g ¯ 0 , r + ( L (0) ) = L (0) + , r − ( L (0) ) = L (0) − ar e c onsta nt along al l time flows and the r e duction L (0) = K + + K − , wher e K ± is a c onst ant element of ζ ( g ± ¯ 0 ) ∗ , is c onsistent with al l e quations of the hier ar chy (7) , (8) c orr esp ond ing to the lo op algebr a e g σ and triangular R -op er ator on e g σ with the describ e d ab ove KAS R -op er ato r R 0 on g ¯ 0 . Pro of . Let us at f irst note that in the case u nder consideration r ∗ ± = r ± . Let u s sho w that r ∗ ± ( L (0) ) = r ± ( L (0) ) are constan t along all time f lo ws generated b y the second order Hamilto - nians I ± 2 k 2 . It is easy to sho w that th e corresp ondin g algebra-v alued gradients ha v e in this case the follo wing form: ∇ I ± 2 k 2 = P ± ( u 2 k L ± ( u )) + r ± ( L ( ± 2 k ) ) = u 2 k L ± ( u ) − P ∓ ( u 2 k L ± ( u )) − r ∓ ( L ( ± 2 k ) ) , where P ± are the p ro jection op erators on to the sub algebras e g σ ± in the triangular decomp osition of the Lie algebra e g σ and we to ok into account that r + + r − = 1. L et us subs titute this expression in to th e Lax equation: dL ( u ) dt ± 2 k = [ ∇ I ± 2 k 2 , L ( u )] , (15) Classical R -Op erators and I n tegrable Generalizatio ns of Thirrin g Equations 11 and tak e in to accoun t that the case un der consid er ation corresp onds to the K ostan t–Adler–Symes decomp osition and, hence, we h a v e correctly def i ned the decomp osition L ( u ) = L + ( u ) + L − ( u ), and eac h of t wo inf inite-co mp onent Lax equation (15) is correctly restricted to eac h of the subspaces L ± ( u ): dL ± ( u ) dt ± 2 k = [ − P ∓ ( u 2 k L ± ( u )) − r ∓ ( L ( ± 2 k ) ) , L ± ( u )] , dL ± ( u ) dt ∓ 2 k = [ P ∓ ( u 2 k L ∓ ( u )) + r ∓ ( L ( ∓ 2 k ) ) , L ± ( u )] . Making p ro jection onto the subalgebra g ¯ 0 in these equ ations w e obtain the equations: dr ± ( L (0) ) dt ± 2 k = − [ r ∓ ( L ( ± 2 k ) ) , r ± ( L (0) )] , dr ± ( L (0) ) dt ∓ 2 k = [ r ∓ ( L ( ∓ 2 k ) ) , r ± ( L (0) )] . Due to the fact that our r -matrix is of Kostan t–Adler–Symes t yp e, w e hav e [ r ∓ ( X ) , r ± ( Y )] = 0, ∀ X , Y ∈ g ¯ 0 and, hence, r + ( L (0) ) ≡ L (0) + , r − ( L (0) ) ≡ L (0) − are constant along all time f lo ws generated b y I ± 2 k 2 . In analogo us wa y it is sho wn that L (0) = L (0) + + L (0) − are constan t with resp ect to the time f lows generated by the h igher ord er integ rals I ± 2 k m . Hence, in this case comp onent s of L (0) b elong to the alg ebra of the in tegrals of moti on of our inf i nite-comp onen t Hamiltonian or Lax system. This a lgebra of int egrals is, generally sp eaking, non-co mm utativ e i.e. { l (0) α , l (0) β } = c γ αβ l (0) γ , where c γ αβ are structure constan ts of the Lie algebra g ¯ 0 . That is why in order to hav e a correct and consistent reduction with resp ect t o these integ rals w e ha v e to restrict the dynamics to the surface of zero lev el of the integrals b elonging to ([ g ¯ 0 , g ¯ 0 ]) ∗ . Oth er part of ( g ¯ 0 ) ∗ , namely the one b elonging to ( g ¯ 0 / [ g ¯ 0 , g ¯ 0 ]) ∗ = ( ζ ( g ¯ 0 )) ∗ = ( ζ ( g + ¯ 0 )) ∗ + ( ζ ( g − ¯ 0 )) ∗ , ma y b e put to b e equal to a constan t, i.e. correct reduction is: L (0) = K + + K − , where K ± is a constan t elemen t of ζ ( g ± ¯ 0 ) ∗ .  Hence, in this case w e ha v e the follo wing form of simplest U - V pair (12) for the zero-curv ature condition: ∇ I ± 2 2 ( u ) = u ± 2 K ± + u ± 1 L ( ± 1) + L ( ± 2) ± , (16) where L ( ± 2) ± = P ± 0 ( L ( ± 2) ) ∈ g ± ¯ 0 . The corresp onding equations (13a), (13b ) are satisf ied auto- matically and th e r est of equations of the system (13) are: ∂ x − L (1) = [ L ( − 2) − , L (1) ] + [ L ( − 1) , K + ] , (17a) ∂ x + L ( − 1) = [ L (2) + , L ( − 1) ] + [ L (1) , K − ] , (17b) ∂ x + L ( − 2) − − ∂ x − L (2) + = [ L (1) , L ( − 1) ] , (17c) where w e h a v e again us ed that in our case [ r + ( X ) , r − ( Y )] = 0, ∀ X, Y ∈ g ¯ 0 . The equations (17a), (17b) are the simplest p ossible g ener alizations of the Thirring e qu ations . The f irst term in the righ t- hand-side of th is equation is an analog of the cubic n on -linearity of Thirring equatio n. The second term is an analog of linear “massiv e” term of the Thirring equations. In all the cases one can expr ess L ( ± 2) ± via L ( ± 1) , and equation (17c ) will follo w from equations (17a), (17b). In ord er to sh o w this we will consider the follo wing example. Example 4. Let us co nsider the case g = g l ( n ), with the fol lo wing Z 2 -grading: g = g l ( n ) ¯ 0 + g l ( n ) ¯ 1 , where g l ( n ) ¯ 0 = g l ( p ) + g l ( q ) a nd g l ( n ) ¯ 1 = C 2 pq , i.e. g l ( n ) ¯ 0 =  ˆ α 0 0 ˆ δ  , gl ( n ) ¯ 1 =  0 ˆ β ˆ γ 0  , where ˆ α ∈ g l ( p ), ˆ δ ∈ g l ( q ), ˆ β ∈ Mat( p, q ), ˆ γ ∈ Mat( q , p ). 12 T.V. Skrypnyk In this case g + ¯ 0 = g l ( p ), g − ¯ 0 = g l ( q ), ζ ( g + ¯ 0 ) = k + 1 p , ζ ( g − ¯ 0 ) = k − 1 q . The corresp onding U - V pair (16) h as the form : ∇ I +2 2 ( u ) = k + u 2  1 p 0 0 0  + u  0 ˆ β + ˆ γ + 0  +  ˆ α + 0 0 0  , ∇ I − 2 2 ( u ) = k − u − 2  0 0 0 1 q  + u − 1  0 ˆ β − ˆ γ − 0  +  0 0 0 ˆ δ −  . The corresp onding zero-curv ature condition yields in this case the follo wing equations: ∂ x − ˆ β + = − ( ˆ β + ˆ δ − + k + ˆ β − ) , ∂ x − ˆ γ + = ( ˆ δ − ˆ γ + + k + ˆ γ − ) , (18a) ∂ x + ˆ β − = ( ˆ α + ˆ β − + k − ˆ β + ) , ∂ x + ˆ γ − = − ( ˆ γ − ˆ α + + k − ˆ γ + ) , (18b) ∂ x + ˆ δ − = ( ˆ γ + ˆ β − − ˆ γ − ˆ β + ) , ∂ x − ˆ α + = ( ˆ β − ˆ γ + − ˆ γ − ˆ β + ) . (18c) By direct v erif ication it is easy to sho w that the substitution of v ariables: ˆ δ − = − 1 k − ˆ γ − ˆ β − , ˆ α + = − 1 k + ˆ β + ˆ γ + solv es equation (18c) and yields the follo wing non-linear dif feren tial equations: ∂ x − ˆ β + = 1 k − ˆ β + ( ˆ γ − ˆ β − ) − k + ˆ β − , ∂ x − ˆ γ + = − 1 k − ( ˆ γ − ˆ β − ) γ + + k + ˆ γ − , ∂ x + ˆ β − = − 1 k + ( ˆ β + ˆ γ + ) ˆ β − + k − ˆ β + , ∂ x + ˆ γ − = 1 k + ˆ γ − ( ˆ β + ˆ γ + ) − k − ˆ γ + . These equations are matrix gener alizat ion of the c om plex Thirring system [15 ]. They admit sev eral r eductions (restrict ions to the dif ferent real f orm of th e algebra g l ( n )). F or the case of an u ( n ) r eduction we h a v e ˆ γ ± = − ˆ β † ± = − Ψ † ± , k ± = iκ ± , κ ± ∈ R and w e ob tain the follo wing non-linear matrix equations in p artial deriv at iv es: i∂ x − Ψ + = 1 κ − Ψ + (Ψ † − Ψ − ) + κ + Ψ − , i∂ x + Ψ − = − 1 κ + (Ψ + Ψ † + )Ψ − + κ − Ψ + . These equ ations are a matrix gener aliza tion of the massive Thirring e quations ( Ψ ± ∈ Mat( p, q )). In the case of p = n − 1, q = 1 or p = 1, q = n − 1 w e obtain a ve ctor ge ner alization of Thir ring e quations . In the sp ecial case n = 2, p = q = 1 and κ − = κ + w e reco v er the usual scalar massiv e Thirring equations with mass m = ( κ + ) 2 . 4.2.3 Case of Co xeter automorphism Let σ b e a Coxet er automorph ism. In this case p = h , w here h is a Coxet er num b er of g and algebra g 0 is Ab elian. That is why all tensors R 0 are solutions of the m YBE on g 0 and maps r ± are arbitrary (mo dulo the constraint r + + r − = 1). Moreo ver, in this case th e follo wing prop osition holds: Prop osition 4. L et σ b e a Coxeter automorphism of g and maps r ± on g b e nonde gener ate d. In this c ase L (0) is c onstant along al l time flows and c omp onents of L ( ± h ) ar e expr esse d as p olynomials of c omp onents of L ( ± 1) , . . . , L ( ± ( h − 1)) . Classical R -Op erators and I n tegrable Generalizatio ns of Thirrin g Equations 13 Pro of . In the considered case w e ha v e that dim g 0 = rank g . On the other h and th er e exist r = r ank g ind ep endent C asimir fu nctions I k ( L ), k ∈ 1 , h , on g ∗ . Hence there exist 2 r ank g in tegrals of the follo wing form: I ± 0 k = I k ( r ∗ ± ( L (0) )) . They are constant along all time f lo w s generated by all other inte grals I ± l k . That is why w e ma y put them to b e equal to constants. On the other hand, it is n ot d if f icult to see that all r indep en d en t comp onen ts of L (0) can b e functionally expressed via I ± 0 k . Hence, they are constan ts too. At last, w e ha ve 2 ran k g in te grals I ± h k , whic h are constan t alo ng all time f l o ws and we ma y put th em to b e equal to constants I ± h k = const ± h k . It easy to see that I ± h k are linear in L ( ± h ) and, on the surf ace of leve l of I ± h k , all comp onents of L ( ± h ) are expressed p olynomially via comp onen ts of L ( ± 1) , . . . , L ( ± ( h − 1)) if th e m ap s r ± are non-degenerate.  This prop osition has the follo wing imp ortan t corollary: Corollary 1. The numb er of indep endent fields in the gener alize d Thirring e quation (10 ) , c or- r esp onding to Coxeter automorphism, is e q u al to 2 h − 1 P j =1 dim g j = 2(dim g − rank g ) . Let us explic itly consider Thirrin g-type equations (11) in the case of the C o xeter automor- phisms. In th is case g 0 ≃ g h is Ab elian and equations (11a), (11b) b ecome trivial. Moreo v er, due to the fact that L ( ± h ) is expressed via L ( ± j ) where j < h , equation (11d) b ecomes a consequence of equations (11c), (11d). In the resulting system of equations (11) is simplif ied to the follo wing system: ∂ x − L ( h − j ) = [ r − ( L ( − h ) ) , L ( h − j ) )] + h − 1 X i = j +1 [ L ( i − j − h ) , L ( h − i ) ] + [ L ( − j ) , r ∗ + ( L (0) )] , ∂ x + L ( j − h ) = [ r + ( L ( h ) ) , L ( j − h ) )] + h − 1 X i = j +1 [ L ( h + j − i ) , L ( j − h ) ] + [ L ( j ) , r ∗ − ( L (0) )] , (19) where j ∈ { 1 , h − 1 } , L (0) is a constant matrix and L ( ± h ) is p olynomial in L ( ± j ) . W e call this system of equations the Ab elian gener alize d Thirring e quations . The last summand in equations (19) is an analog of the “massiv e term” in th e Th irring equation. The f ir s t s ummand is an analog of the cubic non-linearit y in the T hirring equation, but with the growth of the rank of g the d egree of this term is also growing. The other terms are of the second order in dynamical v ariables. They are absent in the ordinary Thirring system corresp on ding to the case e g = ] sl (2). Let us explicit ly consider the simp lest example, whic h already possesses all features of the generalized Th irring equation: Example 5. L et g = g l (3). Its Z 3 -grading corr esp onding to the Coxe ter automorphism has the follo wing form: g l (3) ¯ 0 =   α 1 0 0 0 α 2 0 0 0 α 3   , g l (3) ¯ 1 =   0 β 1 0 0 0 β 2 β 3 0 0   , g l (3) ¯ 2 =   0 0 γ 3 γ 1 0 0 0 γ 2 0   , g l (3) ¯ 3 = g l (3) ¯ 0 , g l (3) − 1 = g l (3) ¯ 2 , g l (3) − 2 = g l (3) ¯ 1 . The Lax op erators b elonging to the dual spaces to ] g l (3) R ± are: L ± ( u ) = r ∗ ± ( L (0) ) + L ( ± 1) u ∓ 1 + L ( ± 2) u ∓ 2 + L ( ± 3) u ∓ 3 + · · · , where L (0) , L ( ± 3) ∈ g l (3) ¯ 0 , L (1) , L ( − 2) ∈ g l (3) ¯ 2 , L (2) , L ( − 1) ∈ g l (3) ¯ 1 . 14 T.V. Skrypnyk In order to simp lify the form of the resulting soliton equations we will u se th e follo wing notations: L (0) =   α 1 0 0 0 α 2 0 0 0 α 3   , L (2) =   0 γ + 1 0 0 0 γ + 2 γ + 3 0 0   , L (1) =   0 0 β + 3 β + 1 0 0 0 β + 2 0   , L ( ± 3) =   δ ± 1 0 0 0 δ ± 2 0 0 0 δ ± 3   , L ( − 1) =   0 β − 1 0 0 0 β − 2 β − 3 0 0   , L ( − 2) =   0 0 γ − 3 γ − 1 0 0 0 γ − 2 0   . The generating functions of the Poisso n-comm uting integral s of the corresp ond ing integ rable system are: I k ( L ± ( u )) = 1 k tr ( L ± ( u )) k = ∞ X m =0 I ± m k u ∓ m . In particular, w e ha v e the follo wing inte grals: I ± 0 k = 1 k tr ( r ∗ ± ( L (0) )) k , k ∈ 1 , 3 . F rom the fact that they are f ixed along all the time f l o ws we obtain that all α i are constan ts of motion. W e also ha v e the follo wing integral s: I ± 3 1 = tr L ( ± 3) , I ± 3 2 = tr ( r ∗ ± ( L (0) ) L ( ± 3) ) + tr ( L ( ± 1) L ( ± 2) ) , I ± 3 3 = 1 3 tr ( L ( ± 1) ) 3 + tr  r ∗ ± ( L (0) )( L ( ± 1) L ( ± 2) + L ( ± 2) L ( ± 1) )  + tr  r ∗ ± ( L (0) ) 2 L ( ± 3)  . These inte grals p ermit u s to express comp onen ts of L ( ± 3) via comp on ents of L ( ± 1) and L ( ± 2) in the case of th e non-degenerated maps r ± . Let u s do this exp licitly . F or the sak e of simp licit y w e will p ut that r ∗ ± ( L (0) ) = k ± L (0) , where k ± are some constants and k + + k − = 1. In this case w e will ha v e the follo wing explicit form of the ab o v e in tegrals: I ± 3 1 = 3 X i =1 δ ± i , I ± 3 2 = k ± 3 X i =1 α i δ ± i + 3 X i =1 β ± i γ ± i , I ± 3 3 = k 2 ± 3 X i =1 α 2 i δ ± i + k ± 3 X i =1 α i ( β ± i γ ± i + β ± i − 1 γ ± i − 1 ) + β ± 1 β ± 2 β ± 3 , where w e h a v e implied in the last summ ation that β ± 0 ≡ β ± 3 , γ ± 0 ≡ γ ± 3 . The U - V pair corresp onding to the Hamiltonians I ± 3 2 ha v e the form: ∇ I ± 3 2 ( u ) = u ± 3 r ∗ ± ( L (0) ) + u ± 2 L ( ± 1) + u ± 1 L ( ± 2) + r ± ( L ( ± 3) ) . The corresp onding zero-curv ature equation reads as follo ws: ∂ x − L (1) = [ r − ( L ( − 3) ) , L (1) ] + [ L ( − 2) , r ∗ + ( L (0) )] , ∂ x + L ( − 1) = [ r + ( L (3) ) , L ( − 1) ] + [ L (2) , r ∗ − ( L (0) )] , ∂ x − L (2) = [ r − ( L ( − 3) ) , L (2) ] + [ L ( − 1) , r ∗ + ( L (0) )] + [ L (1) , L ( − 2) ] , ∂ x + L ( − 2) = [ r + ( L (3) ) , L ( − 2) ] + [ L (1) , r ∗ − ( L (0) )] + [ L (2) , L ( − 1) ] . In the comp onen t form we hav e the follo wing equations: ∂ x − β + i = k − ( δ − i − δ − i +1 ) β + i + k + ( c i +1 − c i ) γ − i , Classical R -Op erators and I n tegrable Generalizatio ns of Thirrin g Equations 15 ∂ x + β − i = k + ( δ + i − δ + i +1 ) β − i + k − ( c i +1 − c i ) γ + i , ∂ x − γ + i = k − ( δ − i +1 − δ − i ) γ + i + k + ( c i − c i +1 ) β − i + ( β + k γ − j − β + j γ − k ) , ∂ x + γ + i = k + ( δ − i +1 − δ − i ) γ + i + k − ( c i − c i +1 ) β − i + ( γ + j β − k − γ + k β − j ) , (20) where c i ≡ α i , ind ices i , j , k constitute a cyclic p ermutatio n of the ind ices 1, 2, 3, and it is implied that δ ± 3+1 ≡ δ ± 1 , c 3+1 ≡ c 1 . T aking in to accoun t th at all the integ rals I k ± 3 are constan ts of motion and putting th eir v alues to b e equal to zero w e can explicitly express v ariables δ ± i via α i = c i , β ± i , γ ± i : δ ± i = 1 k 2 ± ( c i − c j )( c i − c k ) × k ± ( c j + c k ) 3 X l =1 β ± l γ ± l − k ± 3 X l =1 c l ( β ± l γ ± l + β ± l − 1 γ ± l − 1 ) − β ± 1 β ± 2 β ± 3 ! , (21) where indices j and k are complemen tary to the ind ex i in the set { 1 , 2 , 3 } . Due to the fact that c i are constan ts of motion this expression is p olynomial of the third order in d y n amical v ariables. A t last, substituting (21) in to th e equ ations (20) w e obtain a system of n onlinear dif feren tial equations for the dyn amical v ariables β ± i and γ ± i . T hese equations are a g l (3) generalization of “complex” Thirring equations. Un fortunately , neither these equations nor more compli- cated g l ( n ) complex Th irring equations do not admit reductions to the real forms u (3) or u ( n ) resp ectiv ely . 4.3 The case of Kostan t–Adler–Symes R -op erators Let us no w consider in tegrable hierarc hies corresp onding to the case r + = 1, r − = 0 (or r − = 0, r − = 1), i.e. corresp onding to the Kostan t–Adler–Symes decomp osition of t lo op algebras. As it w as remind ed in Subsection 4.1, eac h of the gradings of the lo op algebras corresp ondin g to the dif feren t a utomorphisms σ of order p provides its o wn Kosta nt –Adler–Symes decomp o- sition: e g σ = e g σ + + e g σ − , and, hence, pro vid es c omm uting Hamilt onian f lo ws and hierarc hies of in tegrable equations. Let us consider the sim p lest equatio ns of in teg rable hierarc hies corresp onding to them. W e ha v e the follo wing generic elemen ts of the dual spaces ( e g σ ± ) ∗ : L + ( u ) = L (0) + ∞ X j =1 L ( j ) u − j , L − ( u ) = ∞ X j =1 L ( − j ) u j , where L ( − k ) ∈ g ¯ k , and the follo wing generating functions of the commutativ e in teg rals: I k ( L + ( u )) = ∞ X m =0 I m k u − m , I l ( L − ( u )) = ∞ X n =1 I − n l u n , where I k ( L ) are Casimir fun ctions of g . W e will b e interested in the follo wing Hamiltonians and their Hamiltonian f lo ws: I p 2 =  L (0) , L ( p )  + 1 2 p − 1 X j =1  L ( j ) , L ( p − j )  , I − p p = I p ( L ( − 1) ) . 16 T.V. Skrypnyk The algebra-v alued gradien ts of functions I p 2 , I − p p read as f ollo ws: ∇ I p 2 ( u ) = u p L (0) + p − 1 X j =1 u ( p − j ) L ( j ) + L ( p ) , ∇ I − p p ( u ) = u − 1 ˜ L (1) , where ˜ L (1) ≡ dim g ¯ 1 P i =1 ∂ I − p p ∂ L (1) α X − 1 α ∈ g − 1 . The corresp onding zero-curv ature condition yields the follo wing system of equations: ∂ x − L (0) = 0 , (22) ∂ x − L ( k ) = [ ˜ L (1) , L ( k − 1) ] , k ∈ { 1 , p } , (23) ∂ x + ˜ L (1) = [ L ( p ) , ˜ L (1) ] . (24) Remark 4. The gradien t ∇ I p 2 ( u ) is an analog of the U operator of the generalized Thirring hierarc h y . T h e other gradien t ∇ I − p p ( u ) is an analog of the V op erator of th e non -Ab elian T o d a equation. Hence the c orresp ond ing integ rable equations m ay b e considered as an in termed iate case b et w een generalized T h irring and (non-Ab elian) T o da equation. 4.3.1 Case of second order automorphism Let us consid er the case of the automorphism of the second order σ 2 = 1 ( p = 2), the corr esp on- ding Hamiltonians, their m atrix gradients and zero-curv ature conditions. W e will u s e commuting second order Hamiltonians of th e follo wing form: I 2 2 =  L (0) , L (2)  + 1 2  L (1) , L (1)  , I − 2 2 = 1 2 ( L ( − 1) , L ( − 1) ) . The algebra-v alued gradien ts of functions I 2 2 , I − 2 2 read as f ollo ws: ∇ I 2 2 ( u ) = u 2 L (0) + uL (1) + L (2) , ∇ I − p p ( u ) = u − 1 L ( − 1) . The corresp onding zero-curv ature condition yields the follo wing simple system of dif feren tial equations of h yp erb olic type: ∂ x − L (0) = 0 , ∂ x − L (1) = [ L ( − 1) , L (0) ] , ∂ x − L (2) = [ L ( − 1) , L (1) ] , ∂ x + L ( − 1) = [ L (2) , L ( − 1) ] , (25) where L (0) , L (2) ∈ g ¯ 0 , L ( ± 1) ∈ g ¯ 1 . Let us consider the follo wing example of these equations: Example 6. Let g = sl (2) and σ b e a Cartan inv olution, i.e . s l (2) ¯ 0 = diag ( α, − α ) and sl (2) ¯ 1 consists of the matrices with zeros on the diagonal. Th e simplest Hamiltonians obtained in the framew ork of K ostan t–Adler–Symes sc heme are: I 0 2 = ( α (0) ) 2 , I 2 2 = 2 α (0) α (2) + β (1) γ (1) , I 2 2 = β ( − 1) γ ( − 1) . They pro du ce the follo wing U - V pair for zero-curv ature equations: ∇ I 2 2 =  α (2) 0 0 − α (2)  + u  0 β (1) γ (1) 0  + u 2  α (0) 0 0 − α (0)  , ∇ I − 2 2 = u − 1  0 β ( − 1) γ ( − 1) 0  . Classical R -Op erators and I n tegrable Generalizatio ns of Thirrin g Equations 17 Due to the fact that I 0 2 are constan ts of motion we ha v e that α (0) is also a constan t of mo tion. Moreo v er, using the fact that I 2 2 is a constant of motion one can express α (2) via β (1) , γ (1) . Hence w e obtain fr om th e equations (25 ) the follo wing indep endent equations for the v ariab- les β ( ± 1) , γ ( ± 1) : ∂ x − γ (1) = 2 α (0) γ ( − 1) , (26a) ∂ x − β (1) = − 2 α (0) β ( − 1) , (26b) ∂ x + γ ( − 1) = − 2 α (2) γ ( − 1) , (26c) ∂ x + β ( − 1) = 2 α (2) β ( − 1) . (26d) F rom the equ ation (26b) it follo ws that β ( − 1) = − ∂ x − β (1) 2 α (0) . Substituting this into equation (26d) and taking in to acco unt that α (0) is a constant along all time f lo ws we obtain the follo win g equation: ∂ 2 x + x − β (1) = 2 α (2) ∂ x − β (1) . (27) T aking in to ac coun t that α (2) = − 1 2 α (0) ( β (1) γ (1) ), where w e hav e put that I 2 2 = 0, and making the reduction to the Lie algebra su (2): α (0) = ic , γ (1) = − ¯ β (1) ≡ − ¯ ψ w e obtain the follo w in g in tegrable equation in p artial deriv ative s: ∂ 2 x + x − ψ + i c | ψ | 2 ∂ x − ψ = 0 . This equation is (in a some sense) in termediate b etw een Th irring and sine-Gordon equations. 4.3.2 Case of Co xeter automorphism (principal gradation) Let u s consider again the case of the prin cipal gradation. In this case p = h and a subalgebra g ¯ 0 is Ab elian. In the same wa y as it w as done in the case of the “generalized Ab elian T hirring mo dels” it is p ossible to sh o w that L (0) is constan t along all time f lo ws and comp onents of L ( h ) are expressed p olynomially via the comp onent s of L ( k ) , k < h . L et us assume that constant s of motion L (0) are s u c h that the op erator ad L (0) is nondegenerate. In such a case we may solv e the f ir st of the equations (23) in the follo w ing wa y: ˜ L (1) = − ad − 1 L (0) ( ∂ x − L (1) ) . Substituting this expression in to equation (24) and taking int o accoun t comm utativit y of g ¯ 0 and, hence, operators ad − 1 L (0) and ad L ( h ) , we f inally obtain the follo w in g matrix dif feren tial equation in partial d eriv ativ es: ∂ x + x − L (1) =  L ( h ) ( L (1) , . . . , L ( h − 1) ) , ∂ x − L (1)  , where L ( k ) , k ∈ { 2 , h − 1 } satisfy the follo wing set of ordinary dif feren tia l equations: ∂ x − L ( k ) = [ L ( k − 1) , ad − 1 L (0) ( ∂ x − L (1) )] . Let us consider the follo wing example. Example 7. L et g = g l (3), h = 3. In this case we hav e the follo wing d if fer ential equations: ∂ x + x − L (1) = [ L (3) ( L (1) , L (2) ) , ∂ x − L (1) ] , (28) ∂ x − L (2) = [ L (1) , ad − 1 L (0) ( ∂ x − L (1) )] . 18 T.V. Skrypnyk Let L (0) , L (1) , L (2) , L (3) are parametrized a s in the Examp le 5, i.e. L (0) =   α 1 0 0 0 α 2 0 0 0 α 3   , L (1) =   0 0 β 3 β 1 0 0 0 β 2 0   , L (2) =   0 γ 1 0 0 0 γ 2 γ 3 0 0   , L (3) =   δ 1 0 0 0 δ 2 0 0 0 δ 3   . In suc h co ord inates equation (28) acquires th e follo wing form: ∂ 2 x − x + β i = ( δ i +1 ( β , γ ) − δ i ( β , γ )) ∂ x − β i , i ∈ { 1 , 3 } . (29) where, lik e in the Examp le 5, δ i ( β , γ ) are expressed via β j , γ k and constant s α i ≡ c i : δ i ( β , γ ) = 1 ( c i − c j )( c i − c k ) ( c j + c k ) 3 X l =1 β l γ l − 3 X l =1 c l ( β l γ l + β l − 1 γ l − 1 ) − β 1 β 2 β 3 ! . (30) This equ ation is an exact an alog of the equation (27). 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