scholarly journals Modified quantum dimensions and re-normalized link invariants

2009 ◽  
Vol 145 (1) ◽  
pp. 196-212 ◽  
Author(s):  
Nathan Geer ◽  
Bertrand Patureau-Mirand ◽  
Vladimir Turaev
Keyword(s):  
Lie Algebras ◽  
Lie Superalgebras ◽  
Quantum Dilogarithm ◽  
Quantum Invariants ◽  
Simple Lie Algebras ◽  
Link Invariants ◽  
Root Of Unity ◽  
Usual Quantum ◽  
Quantum Dimensions ◽  
Trivial Link

AbstractIn this paper we give a re-normalization of the Reshetikhin–Turaev quantum invariants of links, using modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly, we give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable and generalize the multivariable Alexander polynomial. The second example is a hierarchy of link invariants arising from nilpotent representations of quantized$\mathfrak {sl}(2)$at a root of unity. These invariants contain Kashaev’s quantum dilogarithm invariants of knots.

Integral modular categories and integrality of quantum invariants at roots of unity of prime order

1998 ◽  
Vol 1998 (505) ◽  
pp. 209-235 ◽  
Author(s):  
G Masbaum ◽  
H Wenzl
Keyword(s):  
Lie Algebras ◽  
Prime Order ◽  
Roots Of Unity ◽  
Local Orientation ◽  
Homfly Polynomial ◽  
Algebraic Integers ◽  
Quantum Invariants ◽  
Root Of Unity ◽  
Orientation Reversal

Abstract It is shown how to deduce integrality properties of quantum 3-manifold invariants from the existence of integral subcategories of modular categories. The method is illustrated in the case of the invariants associated to classical Lie algebras constructed in [42], showing that the invariants are algebraic integers provided the root of unity has prime order. This generalizes a result of [31], [32] and [29] in the sl2-case. We also discuss some details in the construction of invariants of 3-manifolds, such as the S-matrix in the PSUk case, and a local orientation reversal principle for the colored Homfly polynomial.

CHERN-SIMONS THEORY AND KAUFFMAN POLYNOMIALS

1990 ◽  
Vol 05 (06) ◽  
pp. 1165-1195 ◽  
Author(s):  
YONG-SHI WU ◽  
KENGO YAMAGISHI
Keyword(s):  
Lie Algebras ◽  
Simons Theory ◽  
Wilson Loops ◽  
Expectation Values ◽  
Simple Lie Algebras ◽  
Link Invariants ◽  
Chern Simons ◽  
Knots And Links ◽  
Chern Simons Theory

We report on a study of the expectation values of Wilson loops in D=3 Chern-Simons theory. The general skein relations (of higher orders) are derived for these expectation values. We show that the skein relations for the Wilson loops carrying the fundamental representations of the simple Lie algebras SO(n) and Sp(n) are sufficient to determine invariants for all knots and links and that the resulting link invariants agree with Kauffman polynomials. The polynomial for an unknotted circle is identified to the formal characters of the fundamental representations of these Lie algebras.

scholarly journals MULTIVARIABLE LINK INVARIANTS ARISING FROM LIE SUPERALGEBRAS OF TYPE I

2010 ◽  
Vol 19 (01) ◽  
pp. 93-115 ◽  
Author(s):  
NATHAN GEER ◽  
BERTRAND PATUREAU-MIRAND
Keyword(s):  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Parameter Family ◽  
Type I ◽  
Link Invariant ◽  
Link Invariants ◽  
Quantum Dimensions

In this paper, we construct new links invariants from a type I basic Lie superalgebra 𝔤. The construction uses the existence of an unexpected replacement of the vanishing quantum dimension of typical module, by non-trivial "fake quantum dimensions". Using this, we get a multivariable link invariant associated to any one parameter family of irreducible 𝔤-modules.

scholarly journals CODES, S-STRUCTURES, AND EXCEPTIONAL LIE ALGEBRAS

2019 ◽  
Vol 62 (S1) ◽  
pp. S14-S27 ◽  
Author(s):  
ISABEL CUNHA ◽  
ALBERTO ELDUQUE
Keyword(s):  
Lie Algebras ◽  
Simple Lie Algebras ◽  
Nonassociative Algebras ◽  
Exceptional Lie Algebras

AbstractThe exceptional simple Lie algebras of types E7 and E8 are endowed with optimal $\mathsf{SL}_2^n$ -structures, and are thus described in terms of the corresponding coordinate algebras. These are nonassociative algebras which much resemble the so-called code algebras.

Models of isotropic simple lie algebras

1979 ◽  
Vol 7 (17) ◽  
pp. 1835-1875 ◽  
Author(s):  
B.N. Allison
Keyword(s):  
Lie Algebras ◽  
Simple Lie Algebras

Nilpotent Orbits in Simple Lie Algebras and their Transverse Poisson Structures

2008 ◽  
Author(s):  
P. A. Damianou ◽  
H. Sabourin ◽  
P. Vanhaecke ◽  
Rui Loja Fernandes ◽  
Roger Picken
Keyword(s):  
Lie Algebras ◽  
Poisson Structures ◽  
Nilpotent Orbits ◽  
Simple Lie Algebras
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