Abstract.We evaluate the integral appearing in the solution of the KnizhnikZamolodchikov equation associated with the vector representation of the simple Lie algebra g of type B n , C n or D n . This integral, which can be considered as a generalization of the Beta integral, is expressed in terms of an alternating product of the Gamma function.
IntroductionLet g be a simple Lie algebra of rank n, α 1 , . . . , α n its simple roots, and e i , f i , h i (i = 1, . . . , n) its Chevalley generators. Let V 1 and V 2 be its representations with the highest weight vectors v 1 , v 2 of highest weights λ 1 , λ 2 . Then the KnizhnikZamolodchikov equation is given aswhere φ takes values in the space V 1 ⊗ V 2 , ∈ g ⊗ g is the Casimir operator, and κ ∈ C\{0}. The solution of (1) which takes values in the space of singular vectors (or vacuum subspace)2000 Mathematics Subject Classification: Primary 33C60; Secondary 81T40.
Key words θm-curve, braid presentation, braid index.
MSC (2000) 57M15, 05C10In this paper we consider the braid index and the number of Seifert circles of a θm-curve in R 3 as a generalization of the concepts on an oriented link and establish a relation between them and the degree of Yokota's polynomial invariant.
In this paper we obtain a family of relations among the multiple zeta values by calculating the quantum [Formula: see text]-invariant of a framed oriented link, where Γ1,0 is the 7-dimensional irreducible representation of the exceptional simple Lie algebra [Formula: see text] over [Formula: see text].
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