Realizations Of Yangian Algebras And Their Applications

Yangian algebra was introduced by Drinfeld in his fundamental paper [12]. In [12]，Drinfeld showed that Yangian algebra is an noncommutative and noncocommutative Hopfalgebra and the Yangian algebra Y(0) of simple Lie algebra0is a deformation of U(0[乜])in Hopf algebra class. Moreover, Yangian algebra was proved to be important for findingthe rational solutions of Yang-Baxter equation.For a simple Lie algebra0，its Yangian algebra Y(g) is generated by two partsgenerators x and J(x), where x G0. However, in general for the J(x) part, its actionon the representation of Yangian algebra is quite complicated. In2009, Bai，Ge,Jingintroduced the principal realization of Y(0【况).And they showed that using the principalbasis of the action of the J-part generators on the tensor representation of the naturerepresentation and it dual representation can be simplified greatly.The first main content of this dissertation is to generalize the results to the caseY(s[tv). In Chapter3, we give a beautiful result for the general Ycase by using theprincipal realization of YMoreover, we study the quantum number J2by usingthe principal basis of Y(s^), and give the action of J2on the maximal entangled statesexplicitly.For B，C，D type Lie algebras, Olshanski gave a kind of new algebras in his papers[55—57], called twisted Yangian algebras. For every B，C，D type Lie algebra0#，itstwisted Yangian algebra Y(qn) includes the universal enveloping algebra U(qn) of Bnas a subalgebra. And there exists an evaluation hommorphism ev: U(qn)—Y(qn)identical on the subalgebra U(qn)-Moreover, twisted Yangian algebra Y(qn) is a coidealOf f/(0【7V).The second main aim of this dissertation is to give the principal realization of twistedYangian algebras. In Chapter4, we will give the principal realization of twisted Yangianalgebras by using the main idea of Olshanski. Moreover, we will give a new interpretationof the principal realization from the discrete Fourier transform.In1987, Drinfeld gave a new realization for Yangian algebras and quantum aiffnealgebras in [16]. And he classified the finite dimensional irreducible representations ofYagian algebras by using the new realization of Yangian algebras. He also pointed thatYangian algebras can be realized by the R-matrix in the famous RTT realization. Usingthe RTT realization, we can express the Hopf algebra structure of Yangian easily. In [16]，Drinfeld claimed that the two realizations of Yangian algebra are isomorphic, and gavethe isomorphic map for the Y(s【tv) case. But he did not prove it, the complete proof was given by Brundan and Klechev in their paper [21]. For the B，C，D type, it is still an openproblem.The third main content of this dissertation is about the isomorphism of two realiza?tions of Yangian algebras. In Chapter5，we will show that the two realizations of YangianY(SO3) are isomorphic by using the Gauss decomposition. Moreover, we will discuss thegeneral case brielfy in Chapter5.