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13c99b87eaaaf7766f99616b4bed90648e141fa0 | subsection | 17 | 54 | Notations, setting of the problem and preliminary results | \squareLemma 3
Assume that (f,\xi ,L) and (f^{\prime },\xi ^{\prime },L^{\prime }) are two triplets satisfying Assumptions (H). Suppose that (Y,Z,K) is a solution of the RBSDE (f,\xi ,L) and (Y^{\prime },Z^{\prime },K^{\prime }) is a solution of the RBSDE (f^{\prime },\xi ^{\prime },L^{\prime }). Let us set:\begin{arr... | {
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"Said Hamadene",
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1d15f88508ee042e6a27a8c1af0213cdc3ad4a4c | subsection | 18 | 54 | Notations, setting of the problem and preliminary results | Using Corollary REF , we have for all 0 \le t \le T:&& |Y̥_t|^p + c(p)\int _t^T |Y̥_s|^{p-2} \mathbf {1}_{Y̥_s \ne 0} |Z̥_s|^2 ds \le ||^p \\ && \qquad + p \int _t^T |Y̥_s|^{p-1} \mbox{sgn}(Y̥_s) (f(s,Y_s,Z_s)-f^{\prime }(s,Y^{\prime }_s,Z^{\prime }_s)) ds \\ && \qquad + p \int _t^T \alpha |Y̥_s|^{p-1} \mbox{sgn}(\Delt... | {
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} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
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5a63cd7b4ee03604115a07e67177019cd1082f9f | subsection | 19 | 54 | Notations, setting of the problem and preliminary results | \end{array}In the same way dealing with the other term as previously to obtain:\int _t^T |Y̥_s|^{p-1} \mbox{sgn}(\Delta Y_s) d(\Delta K_s) & = & \int _t^T |L_s-Y^{\prime }_s|^{p-2} \mathbf {1}_{L_s-Y^{\prime }_s \ne 0} ( L_s - Y^{\prime }_s) dK_s \\
&& - \int _t^T | Y_s - L^{\prime }_s|^{p-2} \mathbf {1}_{ Y_s - L^{\pr... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
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ba8fcd4be77153c79415ec5a5d58b8be534f6504 | subsection | 20 | 54 | Notations, setting of the problem and preliminary results | With t=0 and taking the expectation in (REF ) we have&& \frac{c(p)}{2} \mathbb {E}\int _0^T |Y̥_s|^{p-2} \mathbf {1}_{Y̥_s \ne 0} |Z̥_s|^2 ds \le \mathbb {E}||^p \\
&& \qquad + \left( p \kappa + \frac{p \kappa ^2}{(p-1)} \right) \mathbb {E}\int _0^T |Y̥_s|^{p} ds \\
&& \qquad + p\mathbb {E}\int _0^T |Y̥_s|^{p-1} |f̥(s,... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
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3c89f06d5678d57953c7cae03af7e6c4ecebd951 | subsection | 21 | 54 | Notations, setting of the problem and preliminary results | Next with (REF ), BDG inequality, and the two previous inequalities, we obtain after having chosen \rho small enough:\mathbb {E}\sup _{s \in [0,T]} |Y̥_s|^{p} & \le & C \mathbb {E}\left( ||^p + \left\lbrace \int _0^T |f̥(s,Y_s,Z_s)| ds \right\rbrace ^p \right)\\
&& + C \left( \mathbb {E}\sup _{s \in [0,T]}|\Delta L_s|^... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
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073bd1de6f689175655ac6cb90e71da89b13b75a | subsection | 22 | 54 | Existence via the Snell Envelope Method | We now focus on the issue of existence. To begin with let us first assume that the function f does not depend on (y,z).Theorem 2
The reflected BSDE associated with (f(t),\xi , L) has a unique L^p-solution.Proof. We are going to proof the existence of a solution in using the Snell envelope of processes. The Snell envel... | {
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"Said Hamadene",
"Alexandre Popier"
] | [
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720c45b8725010d119c5a2c0601e3581dbfcec12 | subsection | 23 | 54 | Existence via the Snell Envelope Method | So let \tau \le T be a stopping time and let us set L^\xi _t:=L_t \mathbf {1}_{[t<T]}+\xi \mathbf {1}_{[t=T]} and D_\tau the following stopping time:D_\tau =\inf \left\lbrace s\ge \tau , \tilde{Y}_s=\int _0^sf(u)du+L^\xi _s \right\rbrace \wedge T.Since the process L is continuous on [0,T[ and may have a positive jump a... | {
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9df8ec81930d51f0778812b15df87de73c39a3eb | subsection | 24 | 54 | Existence via the Snell Envelope Method | So for (U,V) \in \mathcal {B}^p we define (Y,Z,K) = \Phi (U,V) where (Y,Z) is the L^p-solution of the BSDE associated with (f(t,U_t,V_t),\xi ,L), i.e.,&& (Y,Z)\in {\cal B}^p,\,\, K\in \mathcal {S}^p\\&&Y_t = \xi + \int _t^T f(s,U_s,V_s)ds + K_T - K_t - \int _t^T Z_s dB_s,\,\,t\le T \\
&&Y_t \ge L_t \mbox{ and }(Y_t-L_t... | {
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} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
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12607df57ad7df8b2f09208ff2c3adba854ba49e | subsection | 25 | 54 | Existence via the Snell Envelope Method | Using Corollary REF , we have for all 0 \le t \le u \le T:&& e^{\alpha p t} |Y̥_t|^p + c(p) \int _t^u e^{\alpha ps } |Y̥_s|^{p-2} \mathbf {1}_{Y̥_s \ne 0} |Z̥_s|^2 ds \\ && \quad \le e^{\alpha p u} |Y̥_u|^p + p \int _t^u e^{\alpha p s} |Y̥_s|^{p-1} \mbox{sgn}(Y̥_s) f̥_s ds - p \int _t^u \alpha e^{\alpha p s } |Y̥_s|^p ... | {
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} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
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135a365371d0b8bd80b579c31e838f486cce0854 | subsection | 26 | 54 | Existence via the Snell Envelope Method | \\
&& \qquad \left. + \left( \int _t^u e^{2 \alpha s} |V̥_s|^{2} ds
\right)^{p/2} \right].Moreover using Fatou's Lemma&& \int _t^u e^{\alpha p s} |Y̥_s|^{p-1} \mbox{sgn}(Y̥_s) d(K̥_s) = \int _t^u e^{\alpha p s}|Y̥_s|^{p-2} \mathbf {1}_{Y̥_s \ne 0} ( Y_s - L_s) dK_s \\
&& \qquad + \int _t^u e^{\alpha p s}|Y̥_s|^{p-2} \m... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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a2defc6097b2844d7591bbbd6a61e363ec962daf | subsection | 27 | 54 | Existence via the Snell Envelope Method | Coming back to
(REF ) we obtain:&& e^{\alpha p t} |Y̥_t|^p + c(p) \int _t^u e^{\alpha ps } |Y̥_s|^{p-2}
\mathbf {1}_{Y̥_s \ne 0} |Z̥_s|^2 ds \\ && \quad \le e^{\alpha p u} |Y̥_u|^p + \left( \varepsilon ^{-\frac{p}{p-1}}
\frac{p-1}{p} - p \alpha \right) \int _t^u e^{\alpha p s} |Y̥_s|^{p} ds
\\ && \qquad + \frac{\kappa ... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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89f1a6045c21d6ebc6b48afd511fa476c92688b1 | subsection | 28 | 54 | Existence via the Snell Envelope Method | \\ && \qquad \left. + \left(
\int _t^u e^{2 \alpha s} |V̥_s|^{2} ds \right)^{p/2} \right]and\mathbb {E}\left[ \sup _{t \in [0,T]} e^{\alpha p t} |Y̥_t|^p \right] & \le &
\frac{\kappa ^p 2^{p-1}T\varepsilon ^p}{p} \mathbb {E}\left[ \left( \sup _{s \in [t,u]} e^{\alpha p s} |U̥_s|^{p} \right) \right. \\ &&
\left. + \left... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
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33b2076212768f382eda6baae28234b1543f1e65 | subsection | 29 | 54 | Existence via the Snell Envelope Method | \\ && \hspace{199.16928pt} \left. + \left( \int _t^u e^{2 \alpha s} |V̥_s|^{2} ds \right)^{p/2} \right] \\ &&\qquad + \frac{p\kappa ^p 2^{p-1}T\varepsilon ^p}{2c(p)} \mathbb {E}\left[
\left( \sup _{s \in [0,T]} e^{\alpha p s} |U̥_s|^{p} \right) +
\left( \int _t^u e^{2 \alpha s} |V̥_s|^{2} ds \right)^{p/2} \right].Final... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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8d36771b3780b051a72c61574bb5186d3d3187d5 | subsection | 30 | 54 | Existence via the Snell Envelope Method | On the
other hand since f is a Lipschitz function then for every \nu > 0|Y̥_0|^2 + \int _0^{\tau _n} e^{\beta s} |Z̥_s|^2 ds & \le & e^{\beta \tau _n} |Y̥_{\tau _n}|^2 + \left( \frac{\kappa ^2}{\nu } - \beta \right) \int _0^{\tau _n} e^{\beta s} |Y̥_s|^2 ds \\
& + & \nu \int _0^{\tau _n} e^{\beta s} (|U̥_s|^2 + |V̥_s|^... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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d31ace3e527dff7acedf2c97feca8f5771be018b | subsection | 31 | 54 | Existence via the Snell Envelope Method | \nu ^{p/2} \left( \int _0^{\tau _n} e^{\beta s} |V̥_s|^2 ds \right)^{p/2} + 2^{p/2} \left| \int _0^{\tau _n} e^{\beta s}
Y̥_s Z̥_s dB_s \right|^{p/2} \right\rbrace .But by the BDG inequality we have:&& \mathbb {E}\left| \int _0^{\tau _n} e^{\beta s} Y̥_s Z̥_s dB_s \right|^{p/2} \le \bar{c}_p\mathbb {E}\left[ \left( \in... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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a00e06dc5ae3fbd8a42aec1410f18316e893ab3c | subsection | 32 | 54 | Existence via the Snell Envelope Method | \\
& + & \left. \left( \int _0^{T} e^{\beta s} |V̥_s|^2 ds
\right)^{p/2} \right\rbrace + 2^{3p-1} \bar{c}_p2 \mathbb {E}\left[ \sup _{t \in [0,T]} e^{ \beta p/2 s} |Y̥_s|^p \right]Finally choosing \beta great enough (recall that \beta > 0) and
using Lemma REF , to obtain :\mathbb {E}\left( \int _0^{T} e^{\beta s} |Z̥_s... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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7aaa3c5711a404c4cd6ffee27f82404840163802 | subsection | 33 | 54 | Existence via the Snell Envelope Method | Recall that in the proofs of Lemmas REF and
REF we have required that the constants
\varepsilon , \alpha , \nu and \beta should satisfy:\varepsilon ^{-\frac{p}{p-1}} \frac{p-1}{p} \le p \alpha , \ \frac{\kappa ^2}{\nu } \le \betaC_{\alpha } = \frac{2\kappa ^p 2^{p-1}T\varepsilon ^p}{p} \left( 1 + \frac{p2}{4 c(p)} \rig... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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e3bdcaa7b0a632f0e8d3f13cbfbd8cb719bf2e2a | subsection | 34 | 54 | Existence via Penalization | We are going now to deal with the issue of existence of the
L^p-solution for the reflected BSDE associated with
(f(t,y,z),\xi ,L) in using the penalization method. Actually for
n\ge 1 let us consider (Y^n,Z^n) \in \mathcal {B}^p the unique
solution of the following BSDE:\forall t \in [0,T], \, Y^n_t = \xi + \int _t^T f... | {
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"doi": "",
"end": 477,
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"raw": "Briand, Ph., Delyon, B., Hu, Y., Pardoux, E. and Stoica, L. (2003). L^{p} solutions of backward stochastic differential equations, Stochastic Process. Appl., 108, 109–129.",
"source_ref_id": "... | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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50fe7e08717bba54f5dc876700e79df01872e1fc | subsection | 35 | 54 | Existence via Penalization | For any n\ge 0 and t\le T, we have:Y^n_t-Y^0_t=\int _t^T\lbrace a^n(s)(Y^n_s-Y^0_s)+b^n(s)(Z^n_s-Z^0_s)\rbrace ds+(K^n_T-K^n_t) -\int _t^T(Z^n_s-Z^0_s)dB_swhere the processes (a^n(s))_{s\le T} and (b^n(s))_{s\le T} are {\cal P}-measurable and uniformly bounded by the Lipschitz constant of f. But through Proposition
REF... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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61813d9905b3a91ff3749518d6780dac836ec354 | subsection | 36 | 54 | Existence via Penalization | Henceforth thanks to the monotonic limit of S.Peng (, Lemma 2.2, pp.481) the processes Y-Y^0 and K are RCLL and so is Y since Y^0 is continuous.Next from E[(K^n_T)^p]\le C for any n\ge 0 we deduce, in taking the limit as n\rightarrow \infty , that:\mathbb {E}\int _0^T(L_s-Y_s)^-ds]=0and then P-a.s., Y_t\ge L_t for any ... | {
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{
"arxiv_id": "",
"doi": "",
"end": 144,
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"raw": "Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type. Probability Theory and Related Fields 113, 473-499.",
"source_ref_id": "643b8ce23f2f1... | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
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44a73bc982bff11091f95579e0985648a6529770 | subsection | 37 | 54 | Existence via Penalization | Next for any
k\ge 0 and n\ge 0 we have:Y^n_{t\wedge \tau _k}=Y^n_{\tau _k}+\int _{t\wedge \tau _k}^{\tau _k}f(s,Y^n_s,Z^n_s)ds+K^n_{\tau _k}-K^n_{t\wedge \tau _k}-\int _{t\wedge \tau _k}^{\tau _k}Z^n_sdB_s,\,\,\forall t\le T.Then for any n,m and t\le T, it holds true
that:&&Y^n_{t\wedge \tau _k}-Y^m_{t\wedge \tau _k}=(... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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c00ad58351c6beef47a77365929a4962dcc63a18 | subsection | 38 | 54 | Existence via Penalization | Using now
Itô's formula to obtain:&&(Y^n_{t\wedge \tau _k}-Y^m_{t\wedge \tau _k})2+\int _{t\wedge \tau _k}^{\tau _k}|Z^n_s-Z^m_s|^2ds=(Y^n_{\tau _k}-Y^m_{\tau _k})2\\&&
\qquad \qquad +2\int _{t\wedge \tau _k}^{\tau _k}\lbrace a^{n,m}(s)(Y^n_s-Y^m_s)2+b^{n,m}(s)(Y^n_s-Y^m_s)(Z^n_s-Z^m_s)\rbrace ds\\&&\qquad \qquad +2\in... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
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3bea6337c56d35269167bf540a86ce18f8598ee8 | subsection | 39 | 54 | Existence via Penalization | Next using dominated convergence
theorem and Proposition REF to deduce that:\mathbb {E}\int _{0}^{\tau _k}|Z^n_s-Z^m_s|^2ds\rightarrow 0\mbox{ as }n,m\rightarrow \infty .Now thanks to Lemma REF , there exists a constant C
such that\mathbb {E}\lbrace \int _0^T|Z^n_s|^pds\rbrace \le C.Therefore there exists a subsequence... | {
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... | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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c15598cf44bde6b838518b6f48055d7c5022d2aa | subsection | 40 | 54 | Existence via Penalization | Now going back to
(REF ) taking the limit as n\rightarrow \infty to obtain
that:Y_{t\wedge \tau _k}=Y_{\tau _k}+\int _{t\wedge \tau _k}^{\tau _k}f(s,Y_s,Z_s)ds+K_{\tau _k}-K_{t\wedge \tau _k}-\int _{t\wedge \tau _k}^{\tau _k}Z_sdB_s,\,\,\forall t\le T.Additionally we can argue as in to obtain that:\int _0^{T\wedge \tau... | {
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... | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
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9ba7ae509324d1489f63289c5be26c5e8225f87c | subsection | 41 | 54 | Viscosity solutions | Let b : \mathbb {R}_+ \times \mathbb {R}^d \rightarrow \mathbb {R}^d, \sigma : \mathbb {R}_+ \times \mathbb {R}^d \rightarrow \mathbb {R}^{d \times d} be two globally Lipschitz functions and let us consider the following SDE:dX_t = b(t,X_t)dt + \sigma (t,X_t) dB_t, t\le T.We denote by (X^{t,x}_s)_{s \ge t} the unique s... | {
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} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
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8a65cc43ffa5992502eca800d503acdbda40a345 | subsection | 42 | 54 | Viscosity solutions | Note that if we have stronger conditions on b or \sigma , we can have weaker growth ones on f, g and h.From now on we assume that 1<p<2 and that for every (t,x) \in [0,T] \times \mathbb {R}^d let us define (Y^{t,x}_s,Z^{t,s}_x,K^{t,x}_s)_{s \in [t,T]} the unique solution of the reflected BSDEY^{t,x}_s = g(X^{t,x}_T) + ... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
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48b7fafb9986c4edf392a9f347c73907d4f70405 | subsection | 43 | 54 | Viscosity solutions | It is said to be:(i) a viscosity subsolution of (REF ) if u(T,x) \le g(x), x \in \mathbb {R}^d, and for any function \phi \in C^{1,2}((0,T) \times \mathbb {R}^d), if u-\phi has a local maximum at (t,x) then\min (u(t,x)-h(t,x),-\frac{\partial \phi }{\partial t} - \mathcal {L} \phi (t,x) - f(t,x,u(t,x),\sigma \nabla \phi... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
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3f303b396fc8a46d3391a158344581458147c20d | subsection | 44 | 54 | Continuity and viscosity solution | We have the following result:Proposition 3 For every (t,x), Y^{t,x}_t is deterministic and the functionu(t,x) = Y^{t,x}_tis continuous and satisfies\lim _{|x| \rightarrow + \infty } |u(t,x)| \exp ( -A (\ln |x|)^2) =0.Proof. It suffices to show that whenever (t_n,x_n) \rightarrow (t,x),\mathbb {E}\left( \sup _{s \in [0,... | {
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} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
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dc9c59dc915e72ad3e4b0ca5b7fcae1d617b0250 | subsection | 45 | 54 | Continuity and viscosity solution | For \tilde{\xi }, \tilde{f} and
\tilde{L}, let us denote by (\tilde{Y},\tilde{Z},\tilde{K}) the
unique solution of\tilde{Y}_t = \tilde{\xi } + \int _t^T \tilde{f}(s, \tilde{Y}_s,\tilde{Z}_s) ds +\tilde{K}_T-\tilde{K}_t- \int _t^T \tilde{Z}_s dB_swithP-a.s. \ \forall s \in [0,T], \ \tilde{L}_s \le \tilde{Y}_s \mbox{ and... | {
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"Said Hamadene",
"Alexandre Popier"
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dd71f68c53d49e09ff7dab2b1ef5ff383078b9c4 | subsection | 46 | 54 | Continuity and viscosity solution | For each (t,x) \in [0,T] \times \mathbb {R}^d, let (Y^{t,x,n}, Z^{t,x,n}) denote the solution
of the BSDE\begin{array}{l}\forall s \in [t,T], \ Y^{t,x,n}_s =
g(X^{t,x}_T)+ \int _s^T f(u,Y^{t,x,n}_u,Z^{t,x,n}_u) du
\\\qquad \qquad \qquad \qquad \qquad \qquad \qquad + n \int _s^T (Y^{t,x,n}_u - h(u,X^{t,x}_u))^- du -
\in... | {
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... | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
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d062d34e9446a167cd6c0905243203eca911aece | subsection | 47 | 54 | Continuity and viscosity solution | From Lemma 6.1 in , there exists sequences n_j \rightarrow + \infty , (t_j,x_j) \rightarrow (t,x) such that\frac{\partial \phi }{\partial t} (t_j,x_j) + \mathcal {L} \phi (t_j,x_j) + f_{n_j}(t_j,x_j,u_{n_j}(t_j,x_j),\sigma (t_j,x_j) \nabla \phi (t_j,x_j)) \le 0.From the assumption that u(t,x) > h(t,x) and the uniform c... | {
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"Said Hamadene",
"Alexandre Popier"
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"math.PR"
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0d19f119226b68fe6aaeb42bba57001be8f9346e | subsection | 48 | 54 | Uniqueness of the solution | In order to establish the uniqueness of the solution of equation
result (REF ), we need to impose the following
additional assumption on f. For each R > 0, there exists a
continuous function m_R : \mathbb {R}_+ \rightarrow \mathbb {R}_+ such that m_R(0) = 0 and|f(t,x,r,p) - f(t,y,r,p)| \le m_R(|x-y|(1+|p|)),for all t \... | {
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"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
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555196311ce143828bff71c6d3316a3123afe7ae | subsection | 49 | 54 | Uniqueness of the solution | Since (t_0,x_0) is a strict global maximum point of
u-v-\phi , there exists a sequence
(\bar{t},\bar{x},\bar{s},\bar{y}) such that(\bar{t},\bar{x},\bar{s},\bar{y}) is a global maximum point of \psi _{\varepsilon ,\alpha } in ([0,T] \times \bar{B_R})2 where B_R is a ball with a large radius R;
(\bar{t},\bar{x}), (\bar{... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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8bb2a370756f6c390009ac39bd7c8360ddf35e50 | subsection | 50 | 54 | Uniqueness of the solution | Then from the Lipschitz continuity of \sigma and b
we obtain:\mbox{Tr } (\sigma \sigma ^*(\bar{t},\bar{x}) X) - \mbox{Tr}
(\sigma \sigma ^*(\bar{s},\bar{y}) Y) \le C
\frac{|\bar{x}-\bar{y}|^2 + |\bar{t}-\bar{s}|^2}{\varepsilon ^2} + \mbox{Tr
} (\sigma \sigma ^*(\bar{t},\bar{x}) D^2 \phi (\bar{t},\bar{x}));and|\langle b... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
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"math.PR"
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7c63983ee34296049fd3fe454f2e91fe752b1deb | subsection | 51 | 54 | Uniqueness of the solution | If not, there exists a subsequence such
that u(\bar{t},\bar{x}) - h(\bar{t},\bar{x}) \le 0. Passing to
the limit we get u(t_0,x_0) - h(t_0, x_0) \le 0. But from the
assumption u(t_0,x_0) - v(t_0, x_0) > 0, we deduce that 0 \ge u(t_0,x_0) - h(t_0, x_0) > v(t_0,x_0) - h(t_0, x_0). Therefore we
have v(\bar{s},\bar{y}) - h... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
"math.PR"
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89df622c1aea4cca4bc48f3f713cd67d0cb92285 | subsection | 52 | 54 | Uniqueness of the solution | We remove the first term and the
term |\bar{t}-\bar{s}|^2 of the right-hand side above. Then we let
\varepsilon \rightarrow 0 and since (\bar{t},\bar{x}) \rightarrow (t_0,x_0) we finally
have-\frac{\partial \phi }{\partial t} (t_0,x_0) - \mathcal {L} \phi (t_0,x_0) - \kappa |w(t_0,x_0)| - \kappa |\sigma (t_0,x_0) \nabl... | {
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"sour... | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
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"math.PR"
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17263f8d1344d870e1e414f1a839a62b8b5f03a7 | subsection | 53 | 54 | Uniqueness of the solution | This means that the function w-\phi has a global maximum point at (t_0,x_0), where
\phi (t,x) = \alpha \chi (t,x) + (w-\alpha \chi )(t_0,x_0) e^{K(t-t_0)}.
We use the fact that w is a subsolution of
(REF ), i.e.,
-\frac{\partial \phi }{\partial t}(t_0,x_0) - \mathcal {L} \phi (t_0,x_0) - \kappa |w(t_0,x_0)| - \kappa |\... | {
"cite_spans": []
} | 0807.1846 | Lp-Solutions for Reected Backward Stochastic Differential Equations | [
"Said Hamadene",
"Alexandre Popier"
] | [
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435e94fc8a4242ae14e767c2618fbf39da95482a | abstract | 0 | 98 | Abstract | We derive exactly scalar products and form factors for integrable higher-spin
XXZ chains through the algebraic Bethe-ansatz method. Here spin values are
arbitrary and different spins can be mixed. We show the affine quantum-group
symmetry, $U_q(\hat{sl_2})$, for the monodromy matrix of the XXZ spin chain,
and then obta... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
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"hep-th",
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722da7f0bf12d95a41b63fdfa3f1b820d6dc1fb4 | subsection | 1 | 98 | Introduction | Correlation functions of the spin-1/2 XXZ spin chain have
attracted much attention in mathematical physics for more than a decade
, , .
The multiple-integral representations of XXZ correlation functions
were first derived in terms of the q-vertex operators
. Based on the algebraic Bethe ansatz method,
the determinant e... | {
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quantum-group symmetry | [
"Tetsuo Deguchi",
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] | [
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7c40ad74d24cd1fb38a01eef5a4091d4bb6516d9 | subsection | 2 | 98 | Introduction | We derive projection operators from the asymmetric R-matrices ,
and construct integrable higher-spin
XXZ spin chains by the fusion method similarly as the case of
the XXX spin chain .
Here we make an extensive use of
the q-analogues of Young's projection operators, which play a central role
in the q-analogue of the Sch... | {
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"source_ref_i... | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
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] | [
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d96dcbb6a3959c483ce0594228331dd9eb9f3bcf | subsection | 3 | 98 | Introduction | However, the method for the XXX case does not hold
for the XXZ spin chain which has no SU(2) symmetry.The derivation of the affine quantum-group symmetry
of the monodromy matrix should be
not only theoretically interesting but also practically useful
for calculation.
Here we remark that the infinite-dimensional symmetr... | {
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quantum-group symmetry | [
"Tetsuo Deguchi",
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] | [
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d3972dc4dd789d19ea2c263a3af92bb4a79b42fb | subsection | 4 | 98 | Introduction | In section 4 we construct the R matrices of
integrable higher-spin XXZ spin chains
with projection operators of U_q(sl_2) by the fusion method.
We also discuss the case of mixed spins.
In section 5 we formulate an explicit derivation of the
pseudo-diagonalized forms of the B-operators.
We also show it for the C-operato... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
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58d048ab89778af0dd7093ed9931e412482f28ee | subsection | 5 | 98 | Symmetric | We shall introduce the R-matrix for the XXZ spin chain .
We consider two types of R-matrices,
R_{ab}(u) and R_{ab}(\lambda , \mu ).
The R-matrix with a single rapidity argument,
R_{ab}(u), acts on the tensor product of
two vector spaces V_a and V_b,
i.e. R_{ab}(u) \in End(V_a \otimes V_b),
where parameter u is independ... | {
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e0b3adf26e1944102d6b2a523c40bad79139f121 | subsection | 6 | 98 | Symmetric | For a given set of matrix elements
A^{a, \, b}_{c, \, d}(\lambda _j, \lambda _k) (a,b,c,d=1,2)
we define operators A_{j, k}(\lambda _j, \lambda _k) and
A_{k,j}(\lambda _k, \lambda _j) byA_{j, k}(\lambda _j, \lambda _k) & = & \sum _{a,b,\alpha , \beta =1,2}
A^{a, \, \alpha }_{b, \, \beta }(\lambda _j, \lambda _k)
I_1 \o... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
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5cd40ec28be798d685f7089415329be3ada25060 | subsection | 7 | 98 | Symmetric | In terms of matrices we express operators A_{j, k} and
A_{k, j} for j < k byA_{j, k} =
\left(
\begin{array}{cccc}
A^{11}_{11} & A^{11}_{12} & A^{11}_{21} & A^{11}_{22} \\
A^{12}_{11} & A^{12}_{12} & A^{12}_{21} & A^{12}_{22} \\
A^{21}_{11} & A^{21}_{12} & A^{21}_{21} & A^{21}_{22} \\
A^{22}_{11} & A^{22}_{12} & A^{22}_... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
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ec2b9c7634f4f6e45b7d44a873e211cbb637844a | subsection | 8 | 98 | Symmetric | For instance, setting u=\lambda _1 - \lambda _2,
we have explicitlyR_{12} (\lambda _1, \lambda _2)
= \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & b(u) & c(u) & 0 \\
0 & c(u) & b(u) & 0 \\
0 & 0 & 0 & 1
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\right)_{[1,2]} \, .The R-matrices satisfy the Yang-Baxter equations:R_{12}(\lambda _1, \lambda _2) R_{... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
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48b05dce2ee7be80a3808c10768ac818d20875dc | subsection | 9 | 98 | Body | Let us introduce parameters \xi _1, \xi _2, \ldots , \xi _L,
which we call the inhomogeneous parameters.
In the case of the monodromy matrix, we
assume that parameters \lambda _j of the tensor product
V(\lambda _1) \otimes \cdots \otimes V(\lambda _L)
are given by the inhomogeneous parameters, i.e.
\lambda _j = \xi _j ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
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92e0f8021d8347b2a69ed78385dfcd7a0a5839fb | subsection | 10 | 98 | Body | It is clear that they
satisfy the following Yang-Baxter equations.R_{ab}(\lambda _a, \lambda _b) T_a(\lambda _a) T_b(\lambda _b)
= T_b(\lambda _b) T_a(\lambda _a) R_{ab}(\lambda _a, \lambda _b) \, .Let us introduce operator A_j acting on the jth site byA_j = \sum _{a,b=1,2} A^{a}_{b} I_0 \otimes \cdots \otimes I_{j-1} ... | {
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b9c1c2880fe2f78f7a0c934a214dd4737bc009ed | subsection | 11 | 98 | Body | We define permutation operator \Pi _{ab} which maps elements of
V_a \otimes V_b to those of V_b \otimes V_a as follows.\Pi _{ab} \, v_a \otimes v_b = v_b \otimes v_a \, , \quad v_a \in V_a \, , \quad v_b \in V_b \, .We define {\check{R}}_{ab}(u) by{\check{R}}_{ab}(u) = \Pi _{ab} R_{ab}(u)The operators {\check{R}}_{ab} ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
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1f7804bfb1e9ea19326d6c461fa8b848eb63ab5b | subsection | 12 | 98 | Products of | Let us consider the symmetric group {\cal S}_{n} of
n integers, 1, 2, \ldots , n.
We denote by \sigma an element of {\cal S}_{n}.
Then \sigma maps j to \sigma (j) for j=1, 2, \ldots , nDefinition 1 Let p be a sequence of n integers, 1, 2, \ldots , n,
and \sigma an element of the symmetric group {\cal S}_n.
We define th... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
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11c001964d31911dc4892651d86dc565511479f9 | subsection | 13 | 98 | Products of | We define R^{s_j}_{p} byR^{s_j}_{p} = R_{p_j, p_{j+1}}(\lambda _{p_j}, \lambda _{p_{j+1}}) \, .For the unit element e of {\cal S}_{n},
we define R^{e}_{p} by R^{e}_{p}=1. For a given element
\sigma of {\cal S}_n,
we define R^{\sigma }_{p} recursively by the following:R_{p}^{\sigma _A \sigma _B} = R_{\sigma _A(p)}^{\sig... | {
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quantum-group symmetry | [
"Tetsuo Deguchi",
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] | [
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e6673ffe2a4bd40c71f3df69e43de2693f39b011 | subsection | 14 | 98 | The quantum group invariance | We shall show that the monodromy matrix,
T_{0, 1 2 \cdots L}(\lambda ; \xi _1, \ldots , \xi _L),
has the symmetry of the affine quantum group, U_q(\widehat{sl_2}). | {
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} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
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0171e97644613f8eef47b5c16f3698f6e9f8741b | subsection | 15 | 98 | Quantum group | The quantum algebra U_q(sl_2)
is an associative algebra over {\bf C} generated by
X^{\pm }, K^{\pm } with the following relations: K K^{-1} & = & K K^{-1} = 1 \, , \quad K X^{\pm } K^{-1} = q^{\pm 2} X^{\pm } \, , \quad \, , \\
{[} X^{+}, X^{-} {]} & = &
{\frac{K - K^{-1}}{q- q^{-1}} } \, .The algebra U_q(sl_2) is also... | {
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quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
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fae4d7510c4d5555691266059733d42f466d8181 | subsection | 16 | 98 | Quantum group | We
define \Delta ^{(n)}(x) recursively by\Delta ^{(n)}(x) = \left(\Delta ^{(n-1)} \otimes id \right) \Delta (x) \quad {\rm for} \, \, x \in {\cal A} .Let us now introduce the following asymmetric R-matrices:R^{\pm }(u)
= \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & b(u) & c^{\mp }(u) & 0 \\
0 & c^{\pm }(u) & b(u) & ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
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6e820595996ad1d762ae21d9f0d011d9ff4c76d4 | subsection | 17 | 98 | Quantum group | Then we haveR^{+}_{0, 1 2 \cdots L}(\lambda ; \xi _1, \ldots , \xi _L) \,
\Delta ^{(L)}(x) = \sigma _c \circ \Delta ^{(L)}(x) \,
R^{+}_{0, 1 2 \cdots L}(\lambda ; \xi _1, \ldots , \xi _L)
\quad {\rm for \, \, all} \, \, x \in U_q(sl_2)Here parameters \lambda , \xi _1, \ldots , \xi _L are independent of
element x of U_q... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
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b410c58f520b954be14fd2b2cf1dc56868a27412 | subsection | 18 | 98 | Derivation in terms of the Temperley-Lieb algebra | Let us define U_j^{\pm } for j=0, 1, \ldots , L-1, byU^{\pm }_j = \left(
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & q^{\mp } & -1 & 0 \\
0 & -1 & q^{\pm } & 0 \\
0 & 0 & 0 & 0
\end{array}
\right)_{[j, j+1]} \, .They satisfy the defining relations of the Temperley-Lieb algebra:
U^{\pm }_{j} U^{\pm }_{j + 1} U^{\pm }_j & =... | {
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... | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
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6dbf45395299adde6892222600ac1dce878fa517 | subsection | 19 | 98 | Derivation in terms of the Temperley-Lieb algebra | From lemmas REF and REF
we have lemma REF , which is equivalent to
proposition REF .We now show that in the limit of taking u to - \infty ,
{\hat{R}}^{+}(u) is equivalent to
the spin-1/2 matrix representation of the universal R-matrix {\cal R}
of U_q(sl(2)).
An explicit expression of {\cal R} is given by{\cal R} = q^{... | {
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"raw": "T. Deguchi, K. Fabricius and B. M. McCoy, The sl_2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity, J. Stat. Phys. 102 (2001) 701–736.",
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5220dfd9c68e2a8841da5893fde67493e4aaa4d2 | subsection | 20 | 98 | Gauge transformations | Let us introduce operators {\Phi }_j
with arbitrary parameters \phi _j for j=0, 1, \ldots , L as follows:{\Phi }_j = \left(
\begin{array}{cc}
1 & 0 \\
0 & e^{\phi _j}
\end{array}
\right)_{[j]} =
I^{\otimes (j)} \otimes \left(
\begin{array}{cc}
1 & 0 \\
0 & e^{\phi _j}
\end{array}
\right) \otimes I^{\otimes (L-j)} .In t... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
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3a32e03595c61cd6c49ecedb8b71dd61e7e61059 | subsection | 21 | 98 | Gauge transformations | Then, the asymmetric monodromy matrices are transformed into the
symmetric one as follows.R_{0, 1 2 \cdots L}^{\pm }
= \left( \chi _{0 1 2 \cdots L} \right)^{\pm 1} \,
R_{0, 1 2 \cdots L} \left( \chi _{0 1 2 \cdots L} \right)^{\mp 1} \, .We note that the asymmetric R-matrices
{\check{R}}^{\pm }_{j, j+1}(u) are derived ... | {
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quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
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566647fdfb750907f2cd602befe05d600e23c24f | subsection | 22 | 98 | Affine quantum group symmetry | The affine quantum algebra U_q(\widehat{sl_2})
is an associative algebra over {\bf C} generated by
X_i^{\pm }, K_i^{\pm } for i=0,1 with the following relations:K_i K_i^{-1} & = & K^{-1}_i K_i = 1 \, , \quad K_i X_i^{\pm } K_i^{-1} = q^{\pm 2} X_i^{\pm } \, , \quad K_i X_j^{\pm } K_i^{-1} = q^{\mp 2} X_j^{\pm }
\quad (... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1049,
"openalex_id": "",
"raw": "M. Jimbo, A q-Difference Analogue of U({\\ cal g}) and the Yang-Baxter Equation, Lett. Math. Phys. 10 (1985) 63–69.",
"source_ref_id": "5240b04859cdc612969fa224a83d7392df2c84b8",
"start... | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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ad12fdecae10b917ac1395fd1878813ec4786eac | subsection | 23 | 98 | Affine quantum group symmetry | For a given finite-dimensional representation (\pi _V, V) of U_q(sl_2)
we have a finite-dimensional representation (\pi _{V(a)}, V(a))
of U_q(\widehat{sl_2}) through homomorphism \varphi _a, i.e.
\pi _{V(a)}(x)= \pi _V(\varphi _a(x)) for x \in U_q(\widehat{sl_2}).
We call (\pi _{V(a)}, V(a)) or V(a)
the evaluation repr... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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cdc4e1679ba01158bba290fd40af90ae858b9c1c | subsection | 24 | 98 | Affine quantum group symmetry | Through an explicit calculation we show that
U^{-} and \varphi _a \otimes \varphi _b
\left( \Delta (X_0^{\pm }) \right)
commute if a=b:{[}
U^{-}, \, \varphi _a \otimes \varphi _a
\left( \Delta (X_0^{\pm }) \right) {]} = 0 \, .We derive (REF ) through (REF ).In the spin-1/2 representation of U_q(sl_2), we thus have
the ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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a1216b734ad87d8a450e8bb8577e23e2879e4c75 | subsection | 25 | 98 | Affine quantum group symmetry | Combining (REF ) and (REF )
We have the following relations:R^{+}_{12}(\lambda _1, \lambda _2)
\, \, \varphi _{0}^{\otimes 2} \left( \Delta (X_1^{\pm }) \right)
& = & \varphi _{0}^{\otimes 2}
\left( \tau \circ \Delta (X_1^{\pm }) \right) \, \,
R^{+}_{12}(\lambda _1, \lambda _2) \, ,
\\
R^{+}_{12}(\lambda _1, \lambda _2... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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300ab721e109ae4757a05a0fd0cd67eeb6adf395 | subsection | 26 | 98 | Affine quantum group symmetry | We have\varphi _{2\lambda _1} \otimes \varphi _{2 \lambda _2}
\left( \Delta (X_0^{\pm }) \right)
= (\chi _{12})^2 \, \,
\varphi _{0} \otimes \varphi _{0}
\left( \Delta (X_0^{\pm }) \right) \, \, (\chi _{12})^{-2}Thus, relations (REF ) are now expressed as follows.R^{+}_{12}(\lambda _1, \lambda _2) \, \varphi _{2 \lambd... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
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33c209bba37ab9e1a9408647fddb61ec32507dfa | subsection | 27 | 98 | Symmetry relations of | Let us generalize relations (REF ).
Making an extensive use of relations
(REF ),
we can show commutation relations
for R_p^{\sigma } for all permutations \sigma .
In fact, we can prove (REF ) also by the method
for showing proposition REF .
In Appendix A we shall show in proposition REF
how we generalize the symmetry ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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6b69805df1a09e8294ecc97a8b09efec88e7150e | subsection | 28 | 98 | Projection operators | Let us recall that {\check{R}}_{12}^{+}(u) has been defined by{\check{R}}_{12}^{+}(u) = \Pi _{12} R_{12}^{+}(u) \,.We define operator P_{12}^{+} byP_{1 2}^{+} = {\check{R}}_{1 2}^{+}(\eta )Explicitly we haveP_{12}^{+}
= \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & {\frac{q}{[2]}} & {\frac{1}{[2]}} & 0 \\
0 & {\frac{... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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d4e93c712ebeb6334c5600ba9c9a5e3951390aac | subsection | 29 | 98 | Projection operators | Similarly, we define projection operators
acting on V_{j \, j+1 \cdots \, j+\ell -1} recursively byP^{(\ell )}_{j \, j+1 \cdots j+\ell -1} =
P^{(\ell -1)}_{j \, j+1 \cdots j+\ell -2}
{\check{R}}_{j+\ell -2, \, j+\ell -1}^{+}((\ell -1)\eta )
P^{(\ell -1)}_{j \, j+1 \cdots j+\ell -2}Hereafter we shall abbreviate P^{(\ell... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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4a46c235591a1baa14e7ddc8b1b14c428967b922 | subsection | 30 | 98 | The case of tensor product of spin- | We first consider the case of tensor product of
spin-s representations.
We set L=N_{s} \ell .
We introduce a set of parameters,
\xi _1^{(\ell )}, \xi _2^{(\ell )}, ..., \xi _L^{(\ell )}, as follows:\xi _{(k-1)\ell +j}^{(\ell )} = {\zeta }_k - (j-1) \eta +
(\ell -1) \eta /2 \qquad \mbox{for} \quad j=1, \ldots \ell , \, ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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660e38cd7de61eba9afb01fb9411ff28b64e26f6 | subsection | 31 | 98 | The case of tensor product of spin- | We define the monodromy matrix
T^{(\ell +)}_0(u; \zeta _1, \ldots , \zeta _{N_{s}})
acting on the tensor product of spin-s representations,
V^{(2s)}(\zeta _1) \otimes V^{(2s)}(\zeta _{N_{s}}) byT^{(\ell +)}_0(\lambda _0; \zeta _1, \ldots , \zeta _{N_{s}}) =
\prod _{k=0}^{N_{s}-1} P^{(\ell )}_{\ell k+1} \, \cdot \,
R^{+... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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0389b7e0d4c935f462854740663bcb8620c13728 | subsection | 32 | 98 | The case of mixed spins | Let us consider the tensor product of representations
with different spins, s_1, s_2, \ldots , s_r. | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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c0ff9585cdee1c23d6e7eedf95f3104a1a554cba | subsection | 33 | 98 | The case of mixed spins | Here we introduce
\ell _j by \ell _j=2 s_j for j=1, 2, \ldots , r, and
we assume that \ell _1+ \ell _2 + \cdots + \ell _r = L. Å@
Let us introduce a set of parameters,
\xi _1^{({\mbox{$\ell $}})}, \xi _2^{({\mbox{$\ell $}})}, ...,
\xi _L^{({\mbox{$\ell $}})}, as follows:\xi _{\ell _1 + \cdots + \ell _{k-1} +j}^{({\mbox... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
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99f9af610131f44002ffa8783b326918553b89b5 | subsection | 34 | 98 | Higher-spin | We now define the basis vectors of the (\ell +1)-dimensional
irreducible representation of U_q(sl_2),
|| \ell , n \rangle for n=0, 1, \ldots , \ell as follows.
We define ||\ell , 0 \rangle by||\ell , 0 \rangle = |1 \rangle _1 \otimes |1 \rangle _2 \otimes \cdots |1 \rangle _\ellHere |\alpha \rangle _j for \alpha =1, 2
... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
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94e47cef5ac59bf18ef8c9cab4295a2e6839175f | subsection | 35 | 98 | Higher-spin | Let us define the q-factorial, [n]_q!, by[n]_q ! = [n]_q [n-1]_q \cdots [1]_q \, .For integers m and n satisfying m \ge n
we define the q-binomial coefficients as follows\left[
\begin{array}{c}
m \\
n
\end{array}
\right]_q
= {\frac{[m]_q !}{[m-n]_q ! [n]_q!}}Then we have the following expression of the conjugate vector... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
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a5200b75c272315449fcd31bbf616bf3bd4c4cc2 | subsection | 36 | 98 | Higher-spin | Choosing the normalization factors N(\ell ,n),
we can derive the following symmetric expression of the
L-operator:L^{(\ell )}(\lambda ) =
{\frac{1}{ 2 \sinh (u+\ell \eta /2)}}
\left(
\begin{array}{cc}
z K^{1/2} - z^{-1} K^{-1/2} & 2 \sinh \eta X^{-} \\
2 \sinh \eta X^{+} & z K^{-1/2} - z^{-1} K^{1/2}
\end{array}
\right... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
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"nlin.SI",
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fdc1fd268c55666d64df8ba2831cec420dfa0fdd | subsection | 37 | 98 | Algebraic Bethe-ansatz method for higher-spin cases | We now discuss the eigenvalues of the transfer matrix
of an integrable higher-spin XXZ spin chain
constructed by the fusion method. | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
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3a33f885f9c5e2cf508858639850c3a478c135e0 | subsection | 38 | 98 | Algebraic Bethe-ansatz method for higher-spin cases | We consider the case of mixed spins, where we define
the transfer matrix on
the tensor product of spin-s_j representations for
j=1, 2, \ldots , r.We define A, B, C, and D operators of the algebraic Bethe ansatz
for higher-spin cases
by the following matrix elements of the monodromy matrix:\left(
\begin{array}{cc}
A^{({... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
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5e41b7b3d8a219925fbf59aa86913da31cb4a841 | subsection | 39 | 98 | Algebraic Bethe-ansatz method for higher-spin cases | Noting\prod _{k=1}^{r} P^{(\ell _k)}_{\ell (k-1)+1} | 0 \rangle = | 0 \rangle \, ,it is easy to show the following relations:A^{({\mbox{$\ell $}})}(\lambda ) |0 \rangle & = &
a^{({\mbox{$\ell $}})}(\lambda ; \lbrace \zeta _k \rbrace ) | 0 \rangle \, , \\
D^{({\mbox{$\ell $}})}(\lambda ) |0 \rangle & = &
d^{({\mbox{$\el... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
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80479546f28cb4d48ad8bd90012e463502067865 | subsection | 40 | 98 | The | In order to formulate the derivation of
the pseudo-diagonalized forms of B and C operators for the XXZ case,
we briefly formulate some symbols and review some useful formulas
shown in Ref. in SS5.1.
First, we introduce the F-basis .Definition 15 (Partial F and total F)
We define partial F byF_{1, \, 2 \cdots n} & = & e... | {
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quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
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0058d80b94f90c6c6bdc67eb1e7c7d45b926f3ca | subsection | 41 | 98 | Basic properties of the | Let us introduce some important properties of the R-matrix of the
XXZ spin-chain.The R-matrix is invariant under the charge conjugation.
For the symmetric R-matrix,
we define the charge conjugation operator {\cal C} by{\cal C}_{1 2 \cdots n} = \sigma ^x_1 \cdots \sigma _n^{x}For a given operator A \in End(V(\lambda _1)... | {
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"Tetsuo Deguchi",
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d8d32ec37e184f44e49c0106b2fe81c0dc1d0b6c | subsection | 42 | 98 | Basic properties of the | We shall use it when we pseudo-diagonalize the B operators.Definition 21 For X_{1 \cdots n}(\lambda _1, \cdots , \lambda _n)
\in End(V(\lambda _1) \otimes \cdots \otimes V(\lambda _n))
we define X^{\dagger }_{1 \cdots n} byX^{\dagger }_{1 \cdots n}(\lambda _1, \ldots , \lambda _n)
= X^{t_1 \cdots t_n}_{1 \cdots n}(-\la... | {
"cite_spans": []
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quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
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2622997889eb188584ceb92c6acb328e3ea1cb59 | subsection | 43 | 98 | Basic properties of the | It is easy to show R_{12}^{\dagger } = R_{21}.Lemma 22 Under the \dagger operation the monodromy matrix is
given by the following:R^{\dagger }_{0, 1 \cdots n} = R_{0, 1 \cdots n}^{-1}
= R_{1 \cdots n, \, 0}
\\We define operators A^{\dagger }_{1 \cdots n},
B^{\dagger }_{1 \cdots n}, C^{\dagger }_{1 \cdots n} and
D^{\da... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
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"Tetsuo Deguchi",
"Chihiro Matsui"
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d49dc48ca0a6a2b8273a064452f2863bfe5d3b95 | subsection | 44 | 98 | The diagonalized forms of operators | Let us give the diagonalized forms of the A and D operators
.The following criterion for the F-basis to be non-singular
should be useful.Proposition 24 The determinants of the partial and total F matrices are given by{\rm det} F_{0, 1 \cdots n}
= \prod _{j=1}^{n} b(\lambda _{0}-\xi _{j}) \, , \quad {\rm det} F_{1\cdots... | {
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quantum-group symmetry | [
"Tetsuo Deguchi",
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"cond-mat.stat-mech",
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855974739f3d6d16dce3e6e28d046f7fa3736d8a | subsection | 45 | 98 | Pseudo-diagonalization of the | Let us recall that the matrix elements of the monodromy matrix
R^{+}_{0, 1 \cdots L} are related to the symmetric ones as follows.& & R^{+}_{0, 1 2 \cdots L}(u; \xi _1, \ldots , \xi _L) =
\left(
\begin{array}{cc}
A^{+}_{12 \cdots L}(u; \xi _1, \ldots , \xi _L) &
B^{+}_{12 \cdots L}(u; \xi _1, \ldots , \xi _L) \\
C^{+}_... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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c9b6ea77cb2223d3356567837b5d50ddf1aa51fc | subsection | 46 | 98 | Pseudo-diagonalization of the | We remark that more generally, we have commutation relations
(REF ) for the affine quantum group
U_q({\widehat{sl_2}}).In this subsection we abbreviate the superscript + for the asymmetric
monodromy matrix, for simplicity. In fact, the essential parts of
formulas such as the fundamental commutation relations
are invari... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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1ab8f563ac7d952da2f9d774871add10f459b859 | subsection | 47 | 98 | Pseudo-diagonalization of the | We define {\widehat{\delta }}_{1 \cdots n}
and {\widehat{\delta }}_{0, 1 \cdots n} by{\widehat{\delta }}_{1 \cdots n} & = &
\prod _{1\le j <k \le n} {\widehat{\delta }}_{j k}(\lambda _j, \lambda _k)
\\
{\widehat{\delta }}_{0, 1 \cdots n} & = & {\widehat{\delta }}_{0 1 \cdots n}
{\widehat{\delta }}^{-1}_{1 \cdots n}
= \... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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535166055fd7df425aebbe29d38b1b5992dacf3f | subsection | 48 | 98 | Pseudo-diagonalization of the | We have\widetilde{\Delta }_{1 \cdots n}(X^{-}) & = &
F_{2 \cdots n} F_{1, 2 \cdots n} {\Delta }^{(n-1)}(X^{-})
F^{t_1 \cdots t_n}_{n \cdots 2, 1} F_{2 \cdots n}^{-1}
\widehat{\delta }_{1, 2 \cdots n}Putting F_{1, 2 \cdots n} = e_1^{11} + e_1^{22} R_{1, 2 \cdots n} and
F^{t_1 \cdots t_n}_{n \cdots 2, 1}= e_1^{22} + R_{2... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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... |
8725ddf1df2e352de77216b477cd518367234930 | subsection | 49 | 98 | Pseudo-diagonalization of the | The case of n=1 is trivial.
Let us assume the case of n-1.
In eq. (REF ),
the first term gives the following:
X_1^{-} \widehat{\delta }_{1, 2\cdots n}
= X_1^{-} \widehat{\delta }_1^{1 \cdots n}.
Assuming (REF ) for \widehat{\Delta }_{2 \cdots n} and putting it
into the second term of (REF ),
we havee_1^{11} \widetilde{... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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34416cb3985b90dcd0dd508d2f7e68789d348e0a | subsection | 50 | 98 | Pseudo-diagonalization of the | Similarly, we havee_1^{22} \widetilde{D}_{2 \cdots n}(\xi _1)
\widetilde{\Delta }_{2 \cdots n}(X^{-}) e_1^{22}
\widehat{\delta }_{1, 2 \cdots n}
= e_1^{22} \sum _{i=2}^{n} X_i^{-} \widehat{\delta }_{i}^{1 \cdots n}Thus, we have the case of n as follows.{\widetilde{\Delta }}_{1 \cdots n}(X^{-})
& = & X_1^{-} {\widehat{\... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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d3de5b948022152ace5a1d530b93ec948ba49970 | subsection | 51 | 98 | Pseudo-diagonalization of the | Putting 1 - q b_{01} = c_{0i}^{-} in (REF )
we show{\widetilde{B}}_{1 \cdots n}(\lambda ) =
\sum _{i=1}^{n} c_{0i}^{-}
X_i^{-} {\widetilde{D}}_{1 \cdots i-1 \, i+1 \cdots n}
{\widehat{\delta }}_{i}^{1 \cdots n} \, .After some calculation, we have (REF ).Similarly, making use of lemmas REF ,
REF and REF ,
we can show th... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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fbf2b70b62c28f781d32964321ba77cd1efb19e1 | subsection | 52 | 98 | Pseudo-diagonalized forms of
the symmetric | Let us show the pseudo-diagonalized forms of
the B and C operators
of the symmetric monodromy matrix R_{0, 1 \cdots n}.
Here we recall that expressions
(REF ) and (REF ) are for
\widetilde{B}^{+}_{1 2 \cdots n}(\lambda ) and
\widetilde{C}^{+}_{1 2 \cdots n}(\lambda ),
respectively. They are matrix elements of
the asymm... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1374,
"openalex_id": "",
"raw": "N. Kitanine, J.M. Maillet and V. Terras, Form factors of the XXZ Heisenberg spin-1/2 finite chain, Nucl. Phys. B 554 [FS] (1999) 647–678",
"source_ref_id": "efe99452be2366a35304b0c64d11b12e7d... | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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aa49614b60721e98285c32c5ee6e52176f62f0f1 | subsection | 53 | 98 | Formula for higher-spin scalar products | Let us consider the case of tensor product of spin-s representations.
We recall that \ell =2s and L=\ell N_{s}.
We introduce parameters \xi _j^{(\ell ; \epsilon )} for j=1, 2, \ldots , L,
as follows:\xi _{(k-1) \ell + j}^{(\ell ; \epsilon )}
= \zeta _k - (j-1) \eta + \ell \eta /2 + \epsilon r_j \qquad j=1, \ldots \ell ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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4b5200b01e5bc0d895693b8ee623e4cceecc1c6b | subsection | 54 | 98 | Formula for higher-spin scalar products | We define the scalar product
S^{(\ell )}_{n}(\lbrace \mu _j \rbrace , \lbrace \lambda _{\alpha } \rbrace ; \lbrace \zeta _k \rbrace )
by the following:S^{(\ell )}_{n}(\lbrace \mu _j \rbrace , \lbrace \lambda _{\alpha } \rbrace ; \lbrace \zeta _k \rbrace )
=
\langle 0 | \, C^{(\ell )}(\mu _1) \cdots C^{(\ell )}(\mu _n)
... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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0.00... |
2a233e8e82209994477d00e2752ed620247aacaa | subsection | 55 | 98 | Formula for higher-spin scalar products | (REF )
of lemma REF as follows.& & \langle 0 | \, C^{(\ell )}_{1 \cdots N_{s}}(\mu _1; \lbrace \zeta _j \rbrace )
\cdots C^{(\ell )}_{1 \cdots N_{s}}(\mu _n; \lbrace \zeta _j \rbrace )
B^{(\ell )}_{1 \cdots N_{s}}(\lambda _1; \lbrace \zeta _j \rbrace )
\cdots B^{(\ell )}_{1 \cdots N_{s}}(\lambda _n; \lbrace \zeta _j \r... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
] | 2,008 | en | Physics | [
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5516ee56d8e03ade3e65e3b4ad022b46df087c4b | subsection | 56 | 98 | Formula for higher-spin scalar products | Moreover, we have
\langle 0| P_{1 \cdots L}^{(\ell ) \, {\bar{\chi }}} = \langle 0| and
P_{1 \cdots L}^{(\ell ) \, {\bar{\chi }}} | 0 \rangle = | 0 \rangle . | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
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cf3f3340ab6c716cc8e112fb6b4a729ad970e9cf | subsection | 57 | 98 | Formula for higher-spin scalar products | We thus have& & \langle 0 | \, C^{(\ell )}_{1 \cdots N_{s}}(\mu _1; \lbrace \zeta _j \rbrace )
\cdots C^{(\ell )}_{1 \cdots N_{s}}(\mu _n; \lbrace \zeta _j \rbrace )
B^{(\ell )}_{1 \cdots N_{s}}(\lambda _1; \lbrace \zeta _j \rbrace )
\cdots B^{(\ell )}_{1 \cdots N_{s}}(\lambda _n; \lbrace \zeta _j \rbrace )
\, | 0 \ran... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
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9b29bd41c8e1d63da961670f097228ba8250ccc9 | subsection | 58 | 98 | Formula for higher-spin scalar products | Here we remark that
the operator F_{L \cdots 2 1} appears in the
pseudo-diagonalization process of the B and C operators,
as shown in Section 5, and also that
the determinant of F_{L \cdots 2 1} vanishes at \epsilon =0,
when parameters \xi _j are given by eq. (REF ).
In fact, if we put some inhomogeneous parameters \xi... | {
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"end": 535,
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"raw": "K. Fabricius and B. M. McCoy, Evaluation Parameters and Bethe Roots for the Six-Vertex Model at Roots of Unity, in Progress in Mathematical Physics Vol. 23 (MathPhys Odyssey 2001), edited by M. Kash... | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
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21910a4edd32ca87f45d3a9af2239e8b7324179a | subsection | 59 | 98 | Formula for higher-spin scalar products | We set the inhomogeneous parameters as follows:\xi _{\ell _1 + \cdots \ell _{k-1} + j}^{({\mbox{$\ell $}}; \epsilon )}
= \zeta _k - (j-1) \eta +
(\ell -1) \eta /2 + \epsilon r_j \qquad j=1, \ldots \ell ; \,
k= 1, \ldots , r \, .Let us define P^{({\mbox{$\ell $}})}_{1 2 \cdots L}
and P^{({\mbox{$\ell $}}) \, {\bar{\chi ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
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4c9e1f06db22a347066894c0f5c1ca853fcaff8e | subsection | 60 | 98 | Formula for higher-spin scalar products | The scalar product of the mixed-spin XXZ spin chain is
reduced into that of the spin-1/2 XXZ spin chain as follows:S^{({\mbox{$\ell $}})}_{n}(\lbrace \mu _j \rbrace , \lbrace \lambda _{\alpha } \rbrace ; \lbrace \zeta _k \rbrace )
=
\lim _{\epsilon \rightarrow 0} \Bigg [ S^{(1)}_{n}(\lbrace \mu _j \rbrace ,
\lbrace \la... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.01.002 10.1016/j.nuclphysb.2011.05.013 | 0807.1847 | Form factors of integrable higher-spin XXZ chains and the affine
quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
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faa9e585e5707bb2a6bf193ee4068fc83614afa4 | subsection | 61 | 98 | Determinant expressions of the scalar products | Let us review the result of the spin-1/2 case .
Suppose that \lambda _{\alpha } for \alpha =1, 2, \ldots , n,
are solutions of the Bethe ansatz equations with
in homogeneous parameters \xi _j for j=1, 2, \ldots , L,
the scalar product is defined byS_n(\lbrace \mu _j \rbrace , \, \lbrace \lambda _{\alpha } \rbrace ; \lb... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 47,
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"raw": "N. Kitanine, J.M. Maillet and V. Terras, Form factors of the XXZ Heisenberg spin-1/2 finite chain, Nucl. Phys. B 554 [FS] (1999) 647–678",
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quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
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0a6f090b691a7d057af764047f2229d7ac20eaca | subsection | 62 | 98 | Determinant expressions of the scalar products | Then, the exact expression of the scalar product has been shown
through the pseudo-diagonalized forms of the B and C operators as follows
:& & S_n(\lbrace \mu _j \rbrace , \, \lbrace \lambda _{\alpha } \rbrace ; \lbrace \xi _k \rbrace )
= \langle 0 | \prod _{j=1}^{n} \widetilde{C}(\mu _j)
\prod _{\alpha =1}^{n} \wideti... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1197,
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"raw": "N. Kitanine, J.M. Maillet and V. Terras, Form factors of the XXZ Heisenberg spin-1/2 finite chain, Nucl. Phys. B 554 [FS] (1999) 647–678",
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quantum-group symmetry | [
"Tetsuo Deguchi",
"Chihiro Matsui"
] | [
"cond-mat.stat-mech",
"hep-th",
"math-ph",
"math.MP",
"math.QA",
"nlin.SI",
"quant-ph"
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