Braided Algebra Research Articles

A quantum deformation of invariants of binary forms

The modern representation theory has its roots in the classical theory of invariants of binary forms of Caley, Silvester, Clebsch, Gordan, Capelli etc. We use the theory of the quantum group U q(sl(2, IR) in order to develop a quantum theory of invariants. Higher binary forms are introduced on the basis of braided algebras. We define quantised invariants and give basic examples. We show that the symbolic method of Clebsch and Gordan works also in the quantised case. We calculate the quantised discriminant of the quadratic form, the quantised catalecticant of the quartic form and further invariants without a classical analogon.

Q-oscillator representations of Hermitian braided matrices

Representations of the braided algebra of in a Hilbert space are constructed. The generators of this algebra are expressed in terms of two independent q-oscillators. It is shown that, apart from a factor of U(1), these generators reduce to the ordinary angular momentum algebra generators in the limit.

QUANTUM INTEGRABILITY OF NON-ULTRALOCAL MODELS THROUGH BAXTERIZATION OF QUANTIZED BRAIDED ALGEBRA

A scheme suitable for describing quantum non-ultralocal models, including supersymmetric ones, is proposed. Braided algebras are generalized to be used through Baxterization for constructing braided quantum Yang—Baxter equations. Supersymmetric and some known non-ultralocal models are derived in the framework of this approach. As further applications of the scheme, the construction of new quantum integrable non-ultralocal models, like mKdV and anyonic supersymmetric models including deformed anyonic superalgebra, is outlined.

EXACT BETHE ANSATZ SOLUTION OF NON-ULTRALOCAL QUANTUM mKdV MODEL

A lattice regularized Lax operator for the non-ultralocal modified Korteweg de Vries (mKdV) equation is proposed at the quantum level with the basic operators satisfying a q-deformed braided algebra. From the associated quantum R and Z-matrices the exact integrability of the model is proved through the braided quantum Yang-Baxter equation which is a suitably generalized equation for the non-ultralocal models. Using the algebraic Bethe ansatz the eigenvalue problem of the quantum mKdV model is exactly solved and its connection with the spin-1/2 XXZ chain is established, facilitating the investigation of the corresponding conformal properties.

Intersecting Braids and Intersecting Knot Theory

An extension of the Artin Braid Group is considered, with the introduction of new operatores that generate double and triple intersections. The extended Alexander theorem, relating intersecting closed braids and intersecting knots is proved for double and triple intersections, and a counter example is given for the case of quadruple intersections. Intersecting knot invariants are also constructed via Markov traces defined on the intersecting braid algebra representations, and the extended Turaev representation is discussed as an example.

Link invariants for intersecting loops

We generalize the braid algebra to the case of loops with intersections. We introduce the Reidemeister moves for four-and six-valent vertices to have a theory of rigid vertex equivalence. By considering representations of the extended braid algebra, we derive skein relations for link polynomials, which allow us to generalize any link polynomial to the intersecting case. We perturbatively show that the HOMFLY polynomials for intersecting links correspond to the vacuum expectation value of the Wilson line operator of the Chern-Simons theory. We make contact with quantum gravity by showing that these polynomials are simply related with some solutions of the complete set of constraints with cosmological constant Λ, for loops including triple self intersections. Previous derivations of this result were restricted to the four-valent case.