An introduction to quantized Lie groups and algebras

We give a selfcontained introduction to the theory of quantum groups according to Drinfeld highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After having defined Poisson-Lie groups we study their relation to Lie-bi algebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of $sl_2$ is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal $R$-matrix for the quantum $sl_2$ algebra. In the last section we deduce all finite dimensional irreducible representations for $q$ a root of unity. We also give their tensor product decomposition (fusion rules) which is relevant to conformal field theory.Comment: 38 page

Effect of blood's velocity on blood resistivity

Blood resistivity is an important quantity whose value influences the results of various methods used in the study of heart and circulation. In this paper, the relationship between blood resistivity and velocity of blood flow was evaluated and analyzed based upon a probe using six-ring electrodes and a circulatory model. The experimental results indicated that the change in blood resistivity was only ±1.1% when the velocity of blood flow changed from 2.83 to 40 cm/s and it rose to 23% when the velocity was lower than 2.83 cm/s

Finite W-algebras

Finite versions of W-algebras are introduced by considering (symplectic) reductions of finite dimensional simple Lie algebras. In particular a finite analogue of $W^{(2)}_3$ is introduced and studied in detail. Its unitary and non-unitary, reducible and irreducible highest weight representations are constructed.Comment: 1

Exact Solution of the Quantum Calogero-Gaudin System and of its q-Deformation

A complete set of commuting observables for the Calogero-Gaudin system is diagonalized, and the explicit form of the corresponding eigenvalues and eigenfunctions is derived. We use a purely algebraic procedure exploiting the co-algebra invariance of the model; with the proper technical modifications this procedure can be applied to the $q-$deformed version of the model, which is then also exactly solved.Comment: 20 pages Late

The spin 1/2 Calogero-Gaudin System and its q-Deformation

The spin 1/2 Calogero-Gaudin system and its q-deformation are exactly solved: a complete set of commuting observables is diagonalized, and the corresponding eigenvectors and eigenvalues are explicitly calculated. The method of solution is purely algebraic and relies on the co-algebra simmetry of the model.Comment: 15 page

On the Hopf algebras generated by the Yang-Baxter R-matrices

We reformulate the method recently proposed for constructing quasitriangular Hopf algebras of the quantum-double type from the R-matrices obeying the Yang-Baxter equations. Underlying algebraic structures of the method are elucidated and an illustration of its facilities is given. The latter produces an example of a new quasitriangular Hopf algebra. The corresponding universal R-matrix is presented as a formal power series.Comment: 10 page