A \(q\)-analogue of \(U(\mathfrak{gl}(N+1))\), Hecke algebra, and the Yang-Baxter equation

DOI10.1007/BF00400222zbMath0602.17005MaRDI QIDQ1081676

Michio Jimbo

Publication date: 1986

Published in: Letters in Mathematical Physics (Search for Journal in Brave)

zbMATH Keywords

Yang-Baxter equationuniversal enveloping algebrar-matrixquantum R-matrix

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Quantum groups and related algebraic methods applied to problems in quantum theory (81R50) Yang-Baxter equations (16T25)

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I: Foundations, On the noncommutative Riemannian geometry of GLq(n), A two-parameter quantum \((2+1)\)-superspace and its deformed derivation algebra as Hopf superalgebra, New \(q\)-deformed oscillator algebra and thermodynamic model, Super duality and Kazhdan-Lusztig polynomials, Factorization of the \(\mathcal R\)-matrix for the quantum algebra \({U_q(s\ell_3)}\), Canonical and nonlinear vector coherent states of generalized models with spin-orbit interaction, Selection and identity rules for subductions of type A quantum Iwahori-Hecke algebras, Quantum symmetry algebras of spin systems related to Temperley–Lieb R-matrices, Multiplicative Slices, Relativistic Toda and Shifted Quantum Affine Algebras, The Inhomogeneous Invariance Quantum Group of Q-Deformed Boson Algebra with Continuous Parameters, Murphy elements from the double-row transfer matrix, New reflection matrices for theU_q(gl(m|n)) case, Reduction formula of form factors for the integrable spin-sXXZ chains and application to correlation functions, Quantum groups with invariant integrals, q-deformed fermion in many-particle systems and its application to BCS theory, Hecke algebras, 𝑈_{𝑞}𝑠𝑙_{𝑛}, and the Donald-Flanigan conjecture for 𝑆_{𝑛}, Quantum double of \(\text U_q((\mathfrak{sl}_2)^{\leqslant 0})\), Colored solutions of the Yang–Baxter equation from representations of Uqgl(2), Quantum $$\mathfrak{gl}_n,$$ q-Schur Algebras and Their Infinite/Infinitesimal Counterparts, Twisting 2-cocycles for the construction of new non-standard quantum groups, Inhomogeneous quantum groups for particle algebras, Generators and relations for Schur algebras, A generalization of Coxeter groups, root systems, and Matsumoto's theorem., INTRODUCTION TO QUANTUM INTEGRABILITY, Schur-Weyl reciprocity between quantum groups and Hecke algebras of type \(G(p,p,n)\), Stereographic variables for multidimensional quantum group covariant q-oscillators, Drinfeld constructions of the quantum affine superalgebra Uq(gl(m/n̂)), Probabilistic fuzzy sets and a related operator algebra, A super Frobenius formula for the characters of Iwahori–Hecke algebras, GENERAL THERMOSTATISTICAL PROPERTIES OF A Q-DEFORMED FERMI GAS TRAPPED IN A POWER-LAW POTENTIAL, BMW algebra, quantized coordinate algebra and type 𝐶 Schur–Weyl duality, The symmetric q-oscillator algebra: q-coherent states, q-Bargmann–Fock realization and continuous q-Hermite polynomials with 0 < q < 1, A FAMILY OF FACE MODELS RELATED TO Z_N-SYMMETRIC VERTEX MODEL OF BELAVIN, PARASTATISTICS ALGEBRA, YOUNG TABLEAUX AND THE SUPER PLACTIC MONOID, A geometric Schur-Weyl duality for quotients of affine Hecke algebras., Approach of spherical harmonics to the representation of the deformed su(1,1) algebra, Monomial Bases for Littleq-Schur Algebras at Even Roots of Unity, Equivalence of a tangle category and a category of infinite dimensional 𝑈_{𝑞}(𝔰𝔩₂)-modules, PBW bases of q-Schur algebras, DEFORMED HARMONIC OSCILLATOR AND NONLINEAR COHERENT STATES: NONCOMMUTATIVE QUANTUM SPACE APPROACH, Duality Relations for the Periodic ASEP Conditioned on a Low Current, A q-analog of the quantum central limit theorem for SUq(2), q complex, Raising and lowering operators for U q (gl(n)), Oscillator versus prefundamental representations, A family of quantum projective spaces and related $q$-hypergeometric orthogonal polynomials, The 𝑞-Schur² algebra, SOME NEW PROPERTIES CONCERNING THE q-DEFORMED CALCULUS AND THE q-DEFORMATION OF THE DIFFUSION EQUATION, The Jaynes-Cummings Model with a q Analogue of a Coherent State, Dynamics of aQ-analogue of the Quantum Harmonic Oscillator, Quantum algebra as the dynamical symmetry of the deformed Jaynes-Cummings model, A quantum homogeneous space of nilpotent matrices., Squeezing and quantum groups, Exactly solvable potentials and quantum algebras, No generalized statistics from dynamics in curved spacetime, Bethe Ansatz and thermodynamic limit of affine quantum group invariant extensions of the t–J model, 𝑞-Krawtchouk polynomials as spherical functions on the Hecke algebra of type 𝐵, Universal R-matrix and functional relations, q-Deformed Tamm–Dancoff oscillators: Coherent state and thermostatistics, Twisted Whittaker modules over \(U_q (\mathfrak{gl}_{n+1})\), Thermal properties of the \(q\)-deformed spin-one DKP oscillator, A solvable stochastic model for one-dimensional fracturing or catalysis processes, \(\imath\)Schur duality and Kazhdan-Lusztig basis expanded, The universal \(R\)-matrix and its associated quantum algebra as functionals of the classical \(r\)-matrix. The \(\mathrm{sl}_2\) case, Explicit central elements of \(U_q(\mathfrak{gl}(N+1))\), Invariant theory of \(\imath\,\)quantum groups of type AIII, Quantum symmetric pairs, On Poincaré-Birkhoff-Witt basis of the quantum general linear superalgebra, q-analog qudit Dicke states, Minor identities for Sklyanin determinants, Whittaker categories of quasi-reductive lie superalgebras and quantum symmetric pairs, Shift‐invariance for vertex models and polymers, Limits of traces of Temperley-Lieb algebras, Two relations for the antisymmetrizer in the Hecke algebra, Whittaker categories, properly stratified categories and Fock space categorification for Lie superalgebras, Khoroshkin-Tolstoy approach to quantum superalgebras, Weight-finite modules over the quantum affine and double quantum affine algebras of type \(\mathfrak{a}_1\), Algebraic Bethe ansatz for the open XXZ spin chain with non-diagonal boundary terms via \(U_{\mathfrak{q}}\mathfrak{sl}_2\) symmetry, Spin chains as modules over the affine Temperley-Lieb algebra, IntroSurvey of representation theory, Monodromy of the Casimir connection of a symmetrisable Kac-Moody algebra, The \(R\)-matrix presentation for the rational form of a quantized enveloping algebra, Unnamed Item, Stability of 𝚤canonical bases of irreducible finite type of real rank one, Monomial bases for $q$-Schur algebras, Pseudo-symmetric pairs for Kac-Moody algebras, Nonsymmetric Macdonald polynomials via integrable vertex models, Braid rigidity for path algebras, On diagonal solutions of the reflection equation, SOLITONIC ASPECTS OF q-FIELD THEORIES, Complete spectrum of quantum integrable lattice models associated to $\boldsymbol {\mathcal{U}_{q} (\widehat{gl_{n}})}$ by separation of variables, Tensor Product Representations of the Lie Superalgebra 𝔭(n) and Their Centralizers, A Q-operator for open spin chains I. Baxter’s TQ relation, A modified Brauer algebra as centralizer algebra of the unitary group, THE QUEER -SCHUR SUPERALGEBRA, THERMODYNAMIC PROPERTIES OF A QUANTUM GROUP BOSON GAS GL_p,q(2), COIDEAL SUBALGEBRAS IN QUANTUM AFFINE ALGEBRAS, Multiparameter quantum Schur duality of type B, Noncommutative phase and unitarization of GLp,q(2), A central extension of Uq sl(2|2)(1) and R-matrices with a new parameter, Integrable mixing of An−1 type vertex models, Heisenberg algebras and rational double affine Hecke algebras, Quantised affine algebras and parameter-dependent R-matrices, Effect of the Wigner–Dunkl algebra on the Dirac equation and Dirac harmonic oscillator, Theq-Onsager algebra and multivariableq-special functions, Kazhdan-Lusztig theory of super type D and quantum symmetric pairs, Schur-Weyl duality for quantum toroidal superalgebras, Generalisations of Hecke algebras from loop braid groups, Askey-Wilson braid algebra and centralizer of \(U_q (\mathfrak{sl}_2)\), Hom-Lie-Virasoro symmetries in Bloch electron systems and quantum plane in tight binding models, Universal K-matrices for quantum Kac-Moody algebras, BLM realization for 𝒰ℤ(𝔤𝔩 ̂n), Hecke Algebra Characters and Immanant Conjectures, Maximally extended $\boldsymbol {\mathfrak{sl}(2\vert 2)}$ , q-deformed $\boldsymbol{\mathfrak{d}{(2, 1;\epsilon)}}$ and 3D kappa-Poincaré, Unnamed Item, Cellular bases of generalised q-Schur algebras, Gröbner–Shirshov pair of irreducible modules over quantized enveloping algebra of type An, Affine flag varieties and quantum symmetric pairs, Temperley–Lieb, Brauer and Racah algebras and other centralizers of 𝔰𝔲(2), Reflection matrices withU_q[osp⁽²⁾(2|2m) symmetry], Quantum symmetries of q-difference equations, Finite-dimensional representations of quantum affine algebras at roots of unity, The quantum group SLq⋆(2) and quantum algebra Uq(sl2⋆) based on a new associative multiplication on 2 × 2 matrices, Presenting integral q-Schur algebras, On the elliptic 𝔤𝔩2 solid-on-solid model: Functional relations and determinants, Presenting cyclotomic q-Schur algebras, A COMPARISON OF INFINITESIMAL AND LITTLE q-SCHUR ALGEBRAS, Tensor products of $U^{\prime }_q\bigl (\widehat{sl}(2)\bigr )$Uq′sl̂(2)-modules and the big q2-Jacobi function transform, The center of 𝒰_q (𝔫_ω), RECTANGULAR YANG–BAXTER ALGEBRAS AND ALTERNATING A-TYPE INTEGRABLE VERTEX MODELS, Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality, On the representation theory of the infinite Temperley–Lieb algebra, Quantum groups and free differential algebras in field theory, On an affine quantum KPRV determinant at \(q=1\), The Yangian symmetry for three-spin-1/2 and four-spin-1/2 Heisenberg XXX models, Quantum superalgebra \(sl_q(2/1)\) on the Poincaré half-plane, The homological determinant of quantum groups of type 𝐴, On the quantum inverse scattering problem., Double centralizer properties, dominant dimension, and tilting modules, On the construction of trigonometric solutions of the Yang-Baxter equation, A TWO-PARAMETER DEFORMED SUSY ALGEBRA FOR SU_p/q(n)-COVARIANT (p,q)-DEFORMED FERMIONIC OSCILLATORS, Unnamed Item, New link polynomial obtained from octet representation of quantum sl (3) enveloping algebra, Analogues de la forme de Killing et du théorème d'Harish-Chandra pour les groupes quantiques, A new aspect of representations of \(U_q(s\hat l_2)\) -- root of unity case, Loop symmetry of integrable vertex models at roots of unity, Raising/Lowering Maps and Modules for the Quantum Affine Algebra, Dynamics with unitary phase operator: implications for Wigner's problem, Thermodynamics of a two-parameter deformed quantum group boson gas, THE q-BOSON REALIZATIONS OF THE QUANTUM GROUP U_q(C_n), A pairing theorem between a braided bialgebra and its dual bialgebra, Representations of twisted \(q\)-Yangians, Multiparameter deformations of the universal enveloping algebras of the simple Lie algebras A l for all l≥2 and the Yang–Baxter equation, A super Frobenius formula for the characters of cyclotomic Hecke algebras, Canonical basis for typeA,₃, Unnamed Item, Hecke algebraic properties of dynamical R -matrices. Application to related quantum matrix algebras, Howe duality for the quantum groups Uqu(m,n), Uqu(M), Monomial Bases for Little q-Schur Algebras s(2,r), Toroidal and level 0 Uq′(sln+1∧) actions on Uq(gln+1∧) modules, Presenting generalized 𝑞-Schur algebras, Unnamed Item, An interacting ensemble of particles in the context of quantum algebra, Quantum groups, Verma modules andq-oscillators: general linear case, On the Eigenfunctions for the Multi-species <i>q</i>-Boson System, Presenting q-Schur algebras as quotients of the quantized enveloping algebra of sln, Representations and classification of the compact quantum groups Uq(2) for complex deformation parameters, The openXXZand associated models atqroot of unity, Bose–Einstein condensation in a gas of Fibonacci oscillators, R matrices of three-state Hamiltonians solvable by coordinate Bethe ansatz, Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem, On boundary superalgebras, Yang–Baxter algebras, convolution algebras, and Grassmannians, Thermostatistics of a q -deformed relativistic ideal Fermi gas, Left-covariant first order differential calculus on quantum Hopf supersymmetry algebra, The center of q-Schur algebra U(2,r), CONFORMAL FIELD THEORY, QUANTUM GROUP AND BERRY PHASE, A REALIZATION OF THE ENVELOPING SUPERALGEBRA, The Hom–Yang–Baxter equation and Hom–Lie algebras, THE -SCHUR ALGEBRAS AND -SCHUR DUALITIES OF FINITE TYPE, Reduced qKZ equation and genuine qKZ equation, NONCOMMUTATIVE SYMMETRIC FUNCTIONS V: A DEGENERATE VERSION OF U_q(gl_N)

Cites Work

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• Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction