Abstract
For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang–Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang–Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper we present detailed considerations of the above scheme on the example of the algebra Uq(sl(2)) leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra.
Highlights
The “Yang-Baxter maps” [1] are invertible maps of a Cartesian product of two identical sets X,R : X X ÞÑ X X (1.1)satisfying the “functional” or “set-theoretic” Yang-Baxter equation [2], R12 ̋ R13 ̋ R23 “ R23 ̋ R13 ̋ R12 . (1.2)This equation states an equality of two different composition of the three maps R12, R13 and R23, acting on different factors in a product of three sets XXX (for instance, R12 coincides with the map (1.1) in the first and second factors and acts as an identity map in the third one; the maps R13 and R23 are defined ).The interest to the Yang-Baxter maps is motivated mostly by their connection to discrete integrable evolution equations
For any quasitriangular algebra A the map (1.5) is a solution the set-theoretic Yang-Baxter equation (1.2), where the algebra A serves as a set X. At this point it should be noted that a quasi-triangular algebra A is, not just an abstract structureless set (as it is implicitly assumed in the setting of Eq(1.2)), but has algebraic relations between its elements, which must be taken into account in order for the Eq(1.2) to hold
We show that the map (1.5) defines Hamiltonian evolution equations for a discrete integrable quantum system in 2D with an algebra of observables formed by a tensor power of the quantum algebra A
Summary
The “Yang-Baxter maps” [1] are invertible maps of a Cartesian product of two identical sets X ,. Let A be a quasitriangular Hopf algebra, there exists an invertible element RPAbA, belonging to the tensor product of two algebras A, called the universal R-matrix, which by construction satisfies the quantum Yang-Baxter equation. At this point it should be noted that a quasi-triangular algebra A is, not just an abstract structureless set (as it is implicitly assumed in the setting of Eq(1.2)), but has algebraic relations between its elements, which must be taken into account in order for the Eq(1.2) to hold This is a slight generalization of the meaning of Eq(1.2), but this is a well needed generalization, since the most important applications of the Yang-Baxter maps to dynamical systems naturally require to equip the set X with an additional structure for purposes of quantization or considerations of Hamiltonian systems. Note the discrete integrable equations arising here for the algebra Uqpslp2qq are related to the discrete Liouville equation, various variants of which were previously considered in [19,20,21,22]
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