# The Izergin-Korepin model

Abstract : Among the models of interacting classical statistical mechanics the yang—baxter (yb) integrable systems play a special role. The central model in the theory of yb integrable systems is the six vertex model. many powerful techniques were developed to study the six vertex model. the model under consideration is the izergin—korepin (ik) nineteen vertex model, which can be viewed as a generalization of the six vertex model. our aim is to understand the physics of the ik model using the extensions of the methods which were applied to the six vertex model. We review the algebraic bethe ansatz for the ik model based on the nineteen-vertex $r$-matrix and propose a new presentation for the eigenstate of the relevant transfer matrix. we also address the question of the calculation of the scalar products of the ik model. an important object in the theory of scalar products is the domain wall boundary partition function. for this partition function defined for the ik model we derive a recurrence relation and solve it in a special case. we move on to the representation theory of the underlying quantum group ($u_q(a_2^{(2)})$), for which we compute all higher dimensional irreducible representations which are relevant for the ik model (kirillov—reshetikhin (kr) modules). the latter is accomplished in the so-called drinfeld presentation of quantum groups. this presentation has technical advantages for computations of the $r$-matrices by means of the khoroshkin—tolstoy (kt) formula. we use this to compute the $r$-matrix in a tensor product of the fundamental representation and a generic higher dimensional kr module. on the other hand, the drinfeld presentation makes apparent the connection between the borel subalgebras of the quantum group $u_q(a_2^{(2)})$ and the $q$-deformed oscillator algebras (osc$_q$). the latter algebras are closely related to the representation theoretic definition of special transfer matrices: the $q$-operators; these operators are central in the theory of functional relations of integrable models. we use the osc$_q$ type algebras in the kt formula to compute some $l$-matrices which are used to build the $q$-operators. finally, we consider a special case of the ground state of the ik model when the deformation parameter $q$ is equal to a root of unity. in this case we compute explicitly the ground state eigenvalues of various transfer matrices including the $q$-operator. we use the latter result to compute the components of the ground state of the ik model for small systems.
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Cited literature [51 references]

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2015PA066357.pdf
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• HAL Id : tel-01266625, version 1

### Citation

Alexandr Garbali. The Izergin-Korepin model. Operator Algebras [math.OA]. Université Pierre et Marie Curie - Paris VI, 2015. English. ⟨NNT : 2015PA066357⟩. ⟨tel-01266625⟩

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