Yangians and their applications

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Publisher Summary This chapter discusses the Yangians theory and their applications. The discovery of the Yangians is motivated by quantum inverse scattering theory. The Yangians form a remarkable family of quantum groups related to rational solutions of the classical Yang–Baxter equation. For each simple finite-dimensional Lie algebra α over the field of complex numbers, the corresponding Yangian Y (α) is defined as a canonical deformation of the universal enveloping algebra U (α[ x ]) for the polynomial current Lie algebra α[x]. The deformation is considered in the class of Hopf algebras which guarantees its uniqueness under natural “homogeneity” conditions. For any simple Lie algebra α, the Yangian Y (α) contains the universal enveloping algebra U (α) as a subalgebra. The Lie algebra α is regarded as fixed point subalgebra of an involution σ of the appropriate general linear Lie algebra. The defining relations of the Yangian is written in a form of a single ternary (or RTT) relation on the matrix of generators. It originates from quantum inverse scattering theory. The Yangians are primarily regarded as a vehicle for producing rational solutions of the Yang-Baxter equation which plays a key role in the theory of integrable models.