Outline Algebraic Bethe Ansatz for deformed Gaudin model Nuno Cirilo António Centro de Análise Funcional e Aplicações Departamento de Matemática, Instituto Superior Técnico Quantum Integrable Systems and Geometry September 2012, Olhão, Portugal N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Outline Outline Introduction 1 Quantum Integrable Systems Gaudin Models Quantum Inverse Scattering Method Deformed Gaudin Model 2 Classical r-matrix Sklyanin Bracket and Gaudin Algebra Algebraic Bethe Ansatz Conclusions 3 Summary Outlook N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Outline Outline Introduction 1 Quantum Integrable Systems Gaudin Models Quantum Inverse Scattering Method Deformed Gaudin Model 2 Classical r-matrix Sklyanin Bracket and Gaudin Algebra Algebraic Bethe Ansatz Conclusions 3 Summary Outlook N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Outline Outline Introduction 1 Quantum Integrable Systems Gaudin Models Quantum Inverse Scattering Method Deformed Gaudin Model 2 Classical r-matrix Sklyanin Bracket and Gaudin Algebra Algebraic Bethe Ansatz Conclusions 3 Summary Outlook N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Outline Introduction 1 Quantum Integrable Systems Gaudin Models Quantum Inverse Scattering Method Deformed Gaudin Model 2 Classical r-matrix Sklyanin Bracket and Gaudin Algebra Algebraic Bethe Ansatz Conclusions 3 Summary Outlook N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Quantum Integrable Systems In the framework of the quantum inverse scattering method (QISM) integrable systems can be classified by underlying dynamical symmetry algebras. More sophisticated solvable models correspond to Yangians, quantum affine algebras, elliptic quantum groups, etc. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Quantum Integrable Systems In the framework of the quantum inverse scattering method (QISM) integrable systems can be classified by underlying dynamical symmetry algebras. More sophisticated solvable models correspond to Yangians, quantum affine algebras, elliptic quantum groups, etc. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Quantum Integrable Systems In the framework of the quantum inverse scattering method (QISM) integrable systems can be classified by underlying dynamical symmetry algebras. More sophisticated solvable models correspond to Yangians, quantum affine algebras, elliptic quantum groups, etc. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Quantum Integrable Systems In the framework of the quantum inverse scattering method (QISM) integrable systems can be classified by underlying dynamical symmetry algebras. More sophisticated solvable models correspond to Yangians, quantum affine algebras, elliptic quantum groups, etc. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Quantum Integrable Systems In the framework of the quantum inverse scattering method (QISM) integrable systems can be classified by underlying dynamical symmetry algebras. More sophisticated solvable models correspond to Yangians, quantum affine algebras, elliptic quantum groups, etc. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Spin systems Model Quantum R ( λ, η ) -matrix Algebra XXX rational Yangian Y ( sl ( 2 )) quantum affine algebra U q ( � XXZ trigonometric sl ( 2 )) XYZ elliptic elliptic quantum group E τ,η ( sl ( 2 )) N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Outline Introduction 1 Quantum Integrable Systems Gaudin Models Quantum Inverse Scattering Method Deformed Gaudin Model 2 Classical r-matrix Sklyanin Bracket and Gaudin Algebra Algebraic Bethe Ansatz Conclusions 3 Summary Outlook N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Gaudin Models In this sense, one could say that the Gaudin models are the simplest quantum solvable systems being related to classical r-matrices. Gaudin models can be seen as a semi-classical limit of the quantum spin systems R ( λ ; η ) = I + η r ( λ ) + O ( η 2 ) . Gaudin Hamiltonians are related to classical r -matrix � H ( a ) = r ab ( z a − z b ) . b � = a Richardson Hamiltonian and Knizhnik-Zamolodchikov equations. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Gaudin Models In this sense, one could say that the Gaudin models are the simplest quantum solvable systems being related to classical r-matrices. Gaudin models can be seen as a semi-classical limit of the quantum spin systems R ( λ ; η ) = I + η r ( λ ) + O ( η 2 ) . Gaudin Hamiltonians are related to classical r -matrix � H ( a ) = r ab ( z a − z b ) . b � = a Richardson Hamiltonian and Knizhnik-Zamolodchikov equations. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Gaudin Models In this sense, one could say that the Gaudin models are the simplest quantum solvable systems being related to classical r-matrices. Gaudin models can be seen as a semi-classical limit of the quantum spin systems R ( λ ; η ) = I + η r ( λ ) + O ( η 2 ) . Gaudin Hamiltonians are related to classical r -matrix � H ( a ) = r ab ( z a − z b ) . b � = a Richardson Hamiltonian and Knizhnik-Zamolodchikov equations. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Gaudin Models In this sense, one could say that the Gaudin models are the simplest quantum solvable systems being related to classical r-matrices. Gaudin models can be seen as a semi-classical limit of the quantum spin systems R ( λ ; η ) = I + η r ( λ ) + O ( η 2 ) . Gaudin Hamiltonians are related to classical r -matrix � H ( a ) = r ab ( z a − z b ) . b � = a Richardson Hamiltonian and Knizhnik-Zamolodchikov equations. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Gaudin Models In this sense, one could say that the Gaudin models are the simplest quantum solvable systems being related to classical r-matrices. Gaudin models can be seen as a semi-classical limit of the quantum spin systems R ( λ ; η ) = I + η r ( λ ) + O ( η 2 ) . Gaudin Hamiltonians are related to classical r -matrix � H ( a ) = r ab ( z a − z b ) . b � = a Richardson Hamiltonian and Knizhnik-Zamolodchikov equations. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Gaudin Models In this sense, one could say that the Gaudin models are the simplest quantum solvable systems being related to classical r-matrices. Gaudin models can be seen as a semi-classical limit of the quantum spin systems R ( λ ; η ) = I + η r ( λ ) + O ( η 2 ) . Gaudin Hamiltonians are related to classical r -matrix � H ( a ) = r ab ( z a − z b ) . b � = a Richardson Hamiltonian and Knizhnik-Zamolodchikov equations. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Gaudin Models In this sense, one could say that the Gaudin models are the simplest quantum solvable systems being related to classical r-matrices. Gaudin models can be seen as a semi-classical limit of the quantum spin systems R ( λ ; η ) = I + η r ( λ ) + O ( η 2 ) . Gaudin Hamiltonians are related to classical r -matrix � H ( a ) = r ab ( z a − z b ) . b � = a Richardson Hamiltonian and Knizhnik-Zamolodchikov equations. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
Introduction QIS Deformed Gaudin Model GM Conclusions QISM Gaudin Models In this sense, one could say that the Gaudin models are the simplest quantum solvable systems being related to classical r-matrices. Gaudin models can be seen as a semi-classical limit of the quantum spin systems R ( λ ; η ) = I + η r ( λ ) + O ( η 2 ) . Gaudin Hamiltonians are related to classical r -matrix � H ( a ) = r ab ( z a − z b ) . b � = a Richardson Hamiltonian and Knizhnik-Zamolodchikov equations. N. Cirilo António, N. Manojlovi´ c and A. Stolin ABA for deformed Gaudin model
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