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The Kontsevich integral and re-normalized link invariants arising from Lie superalgebras  [PDF]
Nathan Geer
Mathematics , 2009,
Abstract: We show that the coefficients of the re-normalized link invariants of the paper "Multivariable link invariants arising from Lie superalgebras of type I" are Vassiliev invariants which give rise to a canonical family of weight systems.
On the defining relations of the affine Lie superalgebras and their quantized universal enveloping superalgebras  [PDF]
Hiroyuki Yamane
Mathematics , 1996,
Abstract: In this paper, we give defining relations of the affine Lie superalgebras an and defining relations of a super-version of the Drinfeld[D]-Jimbo[J] affine quantized universal enveloping algebras. As a result, we can exactly define the affine quantized universal enveloping superalgebras with generators and relations. Moreover we give a Drinfeld's realizations of $U_h({\hat {sl}}(m|n)^{(1)})$.
Some remarks on quantized Lie superalgebras of classical type  [PDF]
Nathan Geer
Mathematics , 2005,
Abstract: In this paper we use the Etingof-Kazhdan quantization of Lie bi-superalgebras to investigate some interesting questions related to Drinfeld-Jimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the D-J type superalgebra associated to a Lie superalgebra of type A-G, with the distinguished Cartan matrix, is isomorphic to the E-K quantization of the Lie superalgebra. The first main result in the present paper is to extend this to arbitrary Cartan matrices. This paper also contains two other main results: 1) a theorem stating that all highest weight modules of a Lie superalgebra of type A-G can be deformed to modules over the corresponding D-J type superalgebra and 2) a super version of the Drinfeld-Kohno Theorem.
Integral Bases for the Universal Enveloping Superalgebras of Map Superalgebras II  [PDF]
Irfan Bagci,Samuel Chamberlin
Mathematics , 2013,
Abstract: In this paper we define integral forms for the universal enveloping superalgebras of the map superalgebras of Cartan type complex Lie superalgebras, and exhibit explicit integral bases for these integral forms.
The Kontsevich integral  [PDF]
Sergei Chmutov,Sergei Duzhin
Mathematics , 2005,
Abstract: This is an overview article on the Kontsevich integral written for the Encyclopedia of Mathematical Physics, to be published by Elsevier.
Etingof-Kazhdan quantization of Lie superbialgebras  [PDF]
Nathan Geer
Mathematics , 2004,
Abstract: For every semi-simple Lie algebra one can construct the Drinfeld-Jimbo algebra U. This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of U, Drinfeld used the KZ-equations to construct a quasi-Hopf algebra A. He proved that particular categories of modules over the algebras U and A are tensor equivalent. Analogous constructions of the algebras U and A exist in the case of Lie superalgebra of type A-G. However, Drinfeld's proof of the above equivalence of categories does not generalize to Lie superalgebras. In this paper, we will discuss an alternate proof for Lie superalgebras of type A-G. Our proof utilizes the Etingof-Kazhdan quantization of Lie (super)bialgebras. It should be mentioned that the above equivalence is very useful. For example, it has been used in knot theory to relate quantum group invariants and the Kontsevich integral.
Wheels, Wheeling, and the Kontsevich Integral of the Unknot  [PDF]
Dror Bar-Natan,Stavros Garoufalidis,Lev Rozansky,Dylan P. Thurston
Mathematics , 1997,
Abstract: We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne) for the relation between the two natural products on the space of Chinese characters. The two formulas use the related notions of "Wheels" and "Wheeling". We prove these formulas "on the level of Lie algebras" using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras.
Root Graded Lie Superalgebras  [PDF]
Malihe Yousofzadeh
Mathematics , 2015,
Abstract: We define root graded Lie superalgebras and study their connection with centerless cores of extended affine Lie superalgebras; our definition generalizes the known notions of root graded Lie superalgebras.
Extended Affine Lie Superalgebras  [PDF]
Malihe Yousofzadeh
Mathematics , 2013,
Abstract: We introduce the notion of extended affine Lie superalgebras and investigate the properties of their root systems. Extended affine Lie algebras, invariant affine reflection algebras, finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras are examples of extended affine Lie superalgebras.
Functional Integration and the Kontsevich Integral  [PDF]
Louis H. Kauffman
Mathematics , 1998,
Abstract: This paper is an exposition of heuristics related to Witten's functional integral, relating it to Vassiliev invariants and to the Kontsevich integrals that can be used to produce Vassiliev invariants of knots and links.In particular, we give a simplified version of the appearance of the Kontsevich integrals in the perturbation expansion of the functional integral.
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