Preface

The theory of classical matrix Lie algebras can be viewed from at least two

related but different perspectives. On the one hand, the special linear, orthogonal

and symplectic Lie algebras form four infinite series, An, Bn, Cn, Dn, which to-

gether with five exceptional Lie algebras, E6, £7, E$, F4, £?2, comprise a complete

list of the simple Lie algebras over the field of complex numbers. The structure of

these Lie algebras is uniformly described in terms of certain finite sets of vectors

in a Euclidean space called the root systems. The symmetries of the root systems

play a key role in the representation theory of all simple Lie algebras providing

the dimension and character formulas for the representations. On the other hand,

the matrix realizations of the classical Lie algebras allow some specific tools to be

used for their study which are not always available for the exceptional Lie algebras.

The theory of Yangians and twisted Yangians which we develop in this book is one

of such tools bringing in new symmetries and shedding new light on this classical

subject.

The Yangians and twisted Yangians are associative algebras whose defining re-

lations are written in a specific matrix form. We describe the structure of these

algebras and classify their finite-dimensional irreducible representations. The re-

sults exhibit many analogies with the representation theory of the classical Lie

algebras themselves, including the triangular decompositions of the (twisted) Yan-

gians and the parametrization of the representations by their highest weights. In

the simplest cases explicit constructions of the irreducible representations are also

given. Then we apply the Yangian symmetries to the classical Lie algebras. The ap-

plications include constructions of several families of Casimir elements, derivations

of the characteristic identities and Capelli identities, and explicit constructions of

all finite-dimensional irreducible representations of the classical Lie algebras via

weight bases of Gelfand-Tsetlin type.

Let us discuss the relationship between the classical Lie algebras and the

(twisted) Yangians in more detail. The term Yangian was introduced by V. G. Drin-

feld (in honor of C. N. Yang) in his fundamental paper (1985). In that paper,

Drinfeld also defined the quantized Kac-Moody algebras, which together with the

work of M. Jimbo (1985), who introduced these algebras independently, marked the

beginning of the era of quantum groups. The Yangians form a remarkable family

of quantum groups related to rational solutions of the classical Yang-Baxter equa-

tion. For each simple finite-dimensional Lie algebra a over the field C of complex

numbers, the corresponding Yangian is defined as a canonical deformation of the

universal enveloping algebra U(a[z]) for the polynomial current Lie algebra a[z\.

Importantly, the deformation is considered in the class of Hopf algebras, which

guarantees its uniqueness under some natural homogeneity conditions. Another

presentation of the Yangian for a was given later by Drinfeld (1988).

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