| This thesis presents a new method to construct solutions for some nonlinear equations in Quantum Group Theory and provides an extension for the duality of finite dimensional (co)algebras.;The following self-inverse YB operators associated to (co)algebra structures are presented: ϕ = M ⊗ u + u ⊗ M -- I2 : A ⊗ A → A ⊗ A for (A, M, u), respectively y = Delta ⊗ epsilon + epsilon ⊗ Delta -- I2 : C ⊗ C → C ⊗ C for (C, Delta, epsilon).;For s ∈ kx we call the (co)algebras (A, M, u) and (A, sM, 1su ), respectively (C, Delta, epsilon) and ( C, sDelta, 1s3 ), equivalent. Equivalent (co)algebras give the same YB operator. Non-equivalent (co)algebras give different YB operators. We study the cases in which algebras and coalgebras give the same YB operator.;The following internal characterizations are given: For R, a self-inverse YB operator on V (dimkV = n), the following conditions are equivalent: (I)(a) rankk (R + I2) ≤ n; (b) ∃ x0 ≠ 0 such that R(x 0 ⊗ x) = x ⊗ x0 ∀x ∈ V. (II) R is associated to a certain algebra structure on V.;Let R be a self-inverse YB operator on V (dimkV = n and char k ≠ 3). The following conditions are equivalent: (I)(a) rankk (R + I2) ≤ n. (b) ∃W ⊂ V, W subspace of codimension one such that: Im(R -- T) ⊂ V ⊗ W (II) R is associated to a certain coalgebra structure on V.;We later generalize these constructions. We define: 4Aa,b,g:A⊗ A→A⊗A by 4Aa,b,g&parl0; a⊗b&parr0;=aab⊗1+b1⊗ab-g a⊗b for any a,b∈A&parl0;a,b,g∈ k and dimk(A) ≥ 2). Then 4Aa,b,g is an Yang-Baxter operator if and only if one of the following holds: (i) alpha = gamma ≠ 0, beta ≠ 0. (ii) beta = gamma ≠ 0, alpha ≠ 0. (iii) alpha = beta = 0, gamma ≠ 0. We then transfer the theory to coalgebras; so, y (c ⊗ d) = alphaepsilon( d)Delta(c) + betaepsilon(c)Delta( d) -- gammac ⊗ d is a Yang-Baxter operator. A classification of these operators is performed.;Let (L, [,]) a Lie algebra over k and z ∈ Z(L) = {z ∈ L : [z, x] = 0 ∀ ∈ L}. We define: 4La : L ⊗ L → L ⊗ L (x ⊗ y [x,y] ⊗ z + alpha y ⊗ x). If dimk( L) ≥ 3, then: 4La is a YB operator ⟺ alpha ≠ 0. A dual construction is perform for Lie coalgebras.;Next, the constructions of self-inverse YB operators enable us to extend the duality of finite dimensional algebras and coalgebras.;We present two examples of solutions related to the braid Hopf equation.;We give unifying versions of some basic concepts from the theory of algebras and the theory of coalgebras: the notion of ideal and a fundamental isomorphism theorem. |
