Yangians and their applications

On the R-Matrix Realization of Yangians and their Representations

“…The universal enveloping algebra U a[x] has a grading defined on the generators by declaring the degree of F ij x s to be equal to s. Then ψ is obviously a homomorphism of graded algebras. Hence, we may assume that C is homogeneous of degree, say, d. Write 17) summed over the indices i a , j a , r a such that…”

“…17. Let L(λ(u)) and L(µ(u)) be two irreducible highest weight modules over X(a) with λ(u) = (λ −n (u), .…”

“…In the case of A type, i.e., a = sl N , the RT T -presentation of the corresponding Yangian turns out to be particularly useful in the applications of the R-matrix techniques to the classical Lie algebras; see e.g. the review paper [17] and references therein. Moreover, this presentation is most convenient for the study of various subalgebras of the A type Yangian which play an important role in the applications to the quantum spin chain models; see e.g.…”

Yangian of the Strange Lie Superalgebra of Qn-1 Type, Drinfel'd Approach

“…V. Drinfel'd shows this object to be isomorphic to the Yangin of gl(n). It is the RFT approach that is usual for papers devoted to Yangians and defines the Yangian as the algebra generated by matrix elements of Yangians irreducible representations according to Drinfel'd (see [4] and [6,7,8] for Yangians of the Lie superalgebras). More precisely, the Yangian can be viewed as the Hopf algebra generated by matrix elements of a matrix T (u) (so-called transfer matrix) with the commutation defining relations:…”

Метод Сдвига Инвариантов И Модель Годена

“…Для классических алгебр Ли g это было сделано (дру-гими методами) Ольшанским и Назаровым (см. [14], [11]), а в случае g = sl r также Тарасовым [16].…”