The q-deformed algebras su(n)q and their applications

Abstract

Since its beginning in the early 1980's, the subject of q-deformed algebras has expanded rapidly. Many of the techniques and structures of Lie algebras carry over to the q-deformations of Lie algebras. This thesis develops the Racah-Wigner algebra for q-deformations of Lie algebras and looks at some applications. Knowing the Racah-Wigner algebra, it was clear the recursive techniques developed by Butler and others for calculating coupling coefficients and 6j-symbols could be extended to q-deformed algebras. This allows a more general approach to finding coefficients for any q-deformation of a Lie algebra than the approaches previously known. This was the subject of a paper published in 1992 (Lienert and Butler, 1992a). The R-matrices are special elements appearing in q-deformed algebras and form the basis of many of the applications of q-deformed algebras. The symmetries and relations of R-matrices are tied in with those of the coupling coefficients and 6j-symbols. One of the relations satisfied by the coupling coefficients and R-matrices, the pentagonal equation, can be used as a recursion relation. The R-matrices can thus be calculated by a building up method similar to that used for the coupling coefficients. This new method allows R-matrices to be calculated far more efficiently. This work has been published (Lienert and Butler, 1992b). One of the applications of q-deformed algebras is to finding polynomials to describe knots. Based on the Racah-Wigner algebra of q-deformed algebras the diagrammatic approach of Guadagnini (1992) has been extended. Skein relations were known for knot polynomials based on q-deformed algebras but these are insufficient in most cases for calculating the polynomial of a knot. The diagrammatic techniques based on the Racah-Wigner algebra allowed new relations to be found and hence the calculation of two classes of knot polynomials. One class is equivalent to those already known, the other class it was hoped would be more powerful, allowing more knots to be distinguished. However little difference was found. This work has been prepared for publication. q-deformed algebras are the algebraic structure satisfied by solutions to the Yang-Baxter equation. The solutions to the Yang-Baxter equation are known as R-matrices, mentioned above. The Yang-Baxter equation is found in various guises in several areas of physics. One such appearance is as the factorization equation of 1+1 dimensional quantum field theory. Yang in 1967 gave this as the necessary equation to find solutions to a problem in many-body scattering in 1+1 dimensions (Yang, 1967). A similar equation arose in exactly solvable models of 2 dimensional statistical mechanics. Baxter was the first to find such a model and give the equation such a model satisfies (Baxter, 1972, 1982). Both the quantum field theory and statistical mechanics models are exactly solvable or integrable. Classical integrable models, such as those found in soliton theory, have long been known. A key equation in these is the classical Yang-Baxter equation which is related to the Jacobi identity of Lie algebras. The structure and representations of Lie algebras have been of much assistance in the study of integrable models. Faddeev, Sklyanin and others recognized the common equation in the work of Yang, Baxter and others and applied the techniques of the classical inverse method from integrable models to form the quantum inverse scattering method (Sklyanin et al, 1979; Kulish and Sklyanin, 1980). Sklyanin uncovered an algebraic structure satisfied by Baxter's known solution to the Yang-Baxter equation (Sklyanin, 1982). The structure was a one-parameter deformation of the universal enveloping algebra of the Lie algebra su(2). In studying further this structure and generalizing to other algebras, new solutions of the Yang-Baxter equation without spectral parameter have been found. The q-deformations of other algebras were defined by Drinfel'd (1985). They are 'quasi-triangular Hopf algebras'. The 'quasi-triangular' refers to their having a special element, the universal R-matrix, which satisfies the quantum Yang-Baxter equation. For ordinary Lie algebras, the R-matrix is simply the identity so that the q-deformed algebras are indeed a generalization of Lie algebras. The q-deformed algebras have a similar representation theory to Lie algebras because of their common structure (Rosso, 1988; Lusztig, 1988). The Racah-Wigner algebra also has similarities, but there is additional structure due to the deformation parameter q. The Racah-Wigner algebra has been studied in great detail for su(2)q by Nomura and others (Nomura, 1989; Hou et al, 1990a; Kirillov and Reshetikhin, 1988; Vaksman, 1989). The coupling coefficients and 6j-symbols are well known. For other groups, Reshetikhin (1987) covers some of the Racah-Wigner algebra. Very few coupling coefficients are known for other groups. Ma (1990a, 1990b) gives tables of some su(3)q coefficients. The Racah-Wigner algebra and in particular the coupling coefficients and 6jsymbols are important in the applications of both Lie algebras and q-deformed algebras. The algebra su(2)q, like its q = 1 equivalent, has found application in the study of atomic and molecular structure and spectra beginning from the work of Biedenharn (1989) and Macfarlane (1989) on the q-equivalent of harmonic oscillators. Since then many systems have been studied (Bonatsos et al, 1990; Chang and Yan, 1991; Iwao, 1990a, 1990b; Raychev et al, 1990; Celeghini et al, 1992). A better understanding of the structure of q-deformed algebra and knowledge of coupling coefficients will assist progress in these and other applications. The operator form of the universal R-matrix has long been known for many algebras (Burroughs, 1990; Reshetikhin, 1987). R-matrix can be expressed in matrix form by the way it acts on representations of su(n)q. This explicit matrix form has only been calculated for a few groups. It is known for all representations of su(2)q (Nomura, 1989; Kirillov and Reshetikhin, 1988), for a few of su(3)q (Ma, 1990a, 1990b) and the fundamental representation of other Lie groups (Reshetikhin, 1987). R-matrices find applications, both because they are solutions to the Yang-Baxter equation, and also because the Yang-Baxter equation without spectral parameter is related to the braid group relation. Knowing the explicit form of R-matrices can lead to new exactly solvable models. New matrix representations of braid groups can also be found. R-matrices have further properties that allow knot polynomials to be obtained. While R-matrices known from exactly solvable models give some knot polynomials, those found from the study of q-deformed algebras give a new hierarchy of knot polynomials. The Racah-Wigner algebra of q-deformed algebras needs to be well understood in order to apply q-deformed algebras in the areas mentioned above. R-matrices need to be explicitly calculated to find solutions of the Yang-Baxter equation or to find new matrix representations for braids. We address these problems in this thesis. In addition, we study one application in detail, that of finding knot invariants. In Chapter 2 we define the q-deformed algebras. The concept of Hopf algebras is introduced. The properties of the special element, the universal R-matrix, are given. The representation theory of q-deformed algebras is reviewed. The Racah-Wigner algebra for su(n)q is developed in Chapter 3. Vector coupling coefficients, recoupling coefficients and R-matrices are introduced. Equations relating the coefficients are proved. A recursive method is developed for calculating each type of matrix. Those for the vector coupling coefficients and recoupling coefficients are similar to those developed for the non-deformed case by Butler and Wybourne (Butler and Wybourne, 1976; Butler, 1976). A new recursive method for calculating R-matrices is described using the pentagonal relation as a recursion relation. This work has been published in two papers (Lienert and Butler, 1992a, 1992b). In Chapters 4 and 5 we give explicit examples of the recursive methods. The vector coupling coefficients, recoupling coefficients and R-matrices of su(2)q are calculated in Chapter 4. While all of these were previously known, the method used here is easily generalizable. This is illustrated in Chapter 5 where the primitive coupling coefficient and a large class of R-matrices are calculated for su(3)q. Chapter 6 is a review of knot theory and knot invariants. The braid group is defined and the process of obtaining knot polynomials from matrix representations of braid groups is outlined. The process of obtaining knot invariants from the q-deformed algebras su(n)q is described in Chapter 7. Explicit calculations of invariants based on the {1} and {2} representations are presented and their properties discussed. Having calculated the {2} polynomials for over 200 knots, we are able to draw some conclusions about the {2} polynomial. The {2}su(n)q polynomial is better than the {1}su(n)q polynomial at detecting the handedness or lack of handedness of knots with no exceptions being found. However, while it distinguished all knots having the same {1}su(n)q polynomial, it had as many pairs.