scholarly journals ROOTS OF UNITY ASSOCIATED TO STRONGLY DETECTED BOUNDARY SLOPES

2009 ◽  
Vol 18 (12) ◽  
pp. 1623-1636
Author(s):  
SRIKANTH KUPPUM ◽  
XINGRU ZHANG
Keyword(s):  
Roots Of Unity ◽  
Root Of Unity ◽  
Boundary Slope ◽  
Boundary Slopes

We found a family of infinitely many hyperbolic knot manifolds each member of which has a strongly detected boundary slope with associated root of unity of order 4.

Polynomials With {0, +1, -1} Coefficients and a Root Close to a Given Point

1997 ◽  
Vol 49 (5) ◽  
pp. 887-915 ◽  
Author(s):  
Peter Borwein ◽  
Christopher Pinner
Keyword(s):  
Algebraic Number ◽  
Real Root ◽  
Small Degree ◽  
Pisot Number ◽  
Roots Of Unity ◽  
Best Approximations ◽  
Rate Of Approximation ◽  
Root Of Unity ◽  
Extremal Polynomials ◽  
Image Position

AbstractFor a fixed algebraic number α we discuss how closely α can be approximated by a root of a {0, +1, -1} polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, k, of the polynomial at α.In particular we obtain the following. Let BN denote the set of roots of all {0, +1, -1} polynomials of degree at most N and BN(α k) the roots of those polynomials that have a root of order at most k at α. For a Pisot number α in (1, 2] we show thatand for a root of unity α thatWe study in detail the case of α = 1, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When k = 0 or 1 we can describe the extremal polynomials explicitly.

scholarly journals QUANTUM GROUPS AT ROOTS OF UNITY AND MODULARITY

2006 ◽  
Vol 15 (10) ◽  
pp. 1245-1277 ◽  
Author(s):  
STEPHEN F. SAWIN
Keyword(s):  
Representation Theory ◽  
Quantum Groups ◽  
Roots Of Unity ◽  
Quantum Field ◽  
Root Of Unity ◽  
Basic Representation ◽  
Topological Quantum ◽  
Quantum Version ◽  
Simply Connected ◽  
Connected Lie Group

We develop the basic representation theory of all quantum groups at all roots of unity (that is, for q any root of unity, where q is defined as in [18]), including Harish–Chandra's theorem, which allows us to show that an appropriate quotient of a subcategory gives a semisimple ribbon category. This work generalizes previous work on the foundations of representation theory of quantum groups at roots of unity which applied only to quantizations of the simplest groups, or to certain fractional levels, or only to the projective form of the group. The second half of this paper applies the representation theory to give a sequence of results crucial to applications in topology. In particular, for each compact, simple, simply-connected Lie group we show that at each integer level the quotient category is in fact modular (thus leading to a Topological Quantum Field Theory), we determine when at fractional levels the corresponding category is modular, and we give a quantum version of the Racah formula for the decomposition of the tensor product.

IMMERSED AND VIRTUALLY EMBEDDED BOUNDARY SLOPES IN ARITHMETIC MANIFOLDS

2004 ◽  
Vol 13 (05) ◽  
pp. 587-596
Author(s):  
ANNEKE BART
Keyword(s):  
Finite Index ◽  
Hyperbolic Manifold ◽  
Totally Geodesic ◽  
Boundary Slope ◽  
Embedded Boundary ◽  
Boundary Slopes

Given a Bianchi Group [Formula: see text], and a Hyperbolic manifold M, where π1(M) is of finite index in Γd, we show that all boundary slopes are realized as the boundary slope of an immersed totally geodesic surface and hence are virtually embedded boundary slopes.

Representations of Quantum Heisenberg Algebras

1994 ◽  
Vol 46 (5) ◽  
pp. 920-929 ◽  
Author(s):  
Marc A. Fabbri ◽  
Frank Okoh
Keyword(s):  
Heisenberg Algebra ◽  
Roots Of Unity ◽  
Direct Sums ◽  
Von Neumann ◽  
One Dimensional ◽  
Infinite Dimensional ◽  
Root Of Unity ◽  
Heisenberg Algebras ◽  
Neumann Theorem ◽  
Von Neumann Theorem

AbstractA Lie algebra is called a Heisenberg algebra if its centre coincides with its derived algebra and is one-dimensional. When is infinite-dimensional, Kac, Kazhdan, Lepowsky, and Wilson have proved that -modules that satisfy certain conditions are direct sums of a canonical irreducible submodule. This is an algebraic analogue of the Stone-von Neumann theorem. In this paper, we extract quantum Heisenberg algebras, q(), from the quantum affine algebras whose vertex representations were constructed by Frenkel and Jing. We introduce the canonical irreducible q()-module Mq and a class Cq of q()-modules that are shown to have the Stone-von Neumann property. The only restriction we place on the complex number q is that it is not a square root of 1. If q1 and q2 are not roots of unity, or are both primitive m-th roots of unity, we construct an explicit isomorphism between q1() and q2(). If q1 is a primitive m-th root of unity, m odd, q2 a primitive 2m-th or a primitive 4m-th root of unity, we also construct an explicit isomorphism between q1() and q2().

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.

UNITARY BRAID REPRESENTATIONS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 985-1006
Author(s):  
HANS WENZL
Keyword(s):  
Quantum Groups ◽  
Braid Group ◽  
Deformation Parameter ◽  
Hecke Algebras ◽  
Tensor Products ◽  
Roots Of Unity ◽  
Von Neumann ◽  
Root Of Unity ◽  
Brauer Algebras ◽  
Original Motivation

The original motivation for studying unitarizable representations of the braid group was to construct interesting examples of subfactors of II1 von Neumann factors. This is possible for representations factoring through Hecke algebras or q-Brauer algebras only if the deformation parameter is a root of unity. Information gained from this work can be applied to studying tensor products of representations of quantum groups at roots of unity and invariants of 3-manifolds.

scholarly journals Power partitions and a generalized eta transformation property

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Don Zagier
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
General Type ◽  
Circle Method ◽  
Exact Formula ◽  
Transformation Property ◽  
Roots Of Unity ◽  
Root Of Unity ◽  
Famous Paper ◽  
The One

In their famous paper on partitions, Hardy and Ramanujan also raised the question of the behaviour of the number $p_s(n)$ of partitions of a positive integer~$n$ into $s$-th powers and gave some preliminary results. We give first an asymptotic formula to all orders, and then an exact formula, describing the behaviour of the corresponding generating function $P_s(q) = \prod_{n=1}^\infty \bigl(1-q^{n^s}\bigr)^{-1}$ near any root of unity, generalizing the modular transformation behaviour of the Dedekind eta-function in the case $s=1$. This is then combined with the Hardy-Ramanujan circle method to give a rather precise formula for $p_s(n)$ of the same general type of the one that they gave for~$s=1$. There are several new features, the most striking being that the contributions coming from various roots of unity behave very erratically rather than decreasing uniformly as in their situation. Thus in their famous calculation of $p(200)$ the contributions from arcs of the circle near roots of unity of order 1, 2, 3, 4 and 5 have 13, 5, 2, 1 and 1 digits, respectively, but in the corresponding calculation for $p_2(100000)$ these contributions have 60, 27, 4, 33, and 16 digits, respectively, of wildly varying sizes

ON THE QUANTUM COMPLEXITY OF EVALUATING THE TUTTE POLYNOMIAL

2010 ◽  
Vol 19 (06) ◽  
pp. 727-737
Author(s):  
HAMED AHMADI ◽  
PAWEL WOCJAN
Keyword(s):  
Braid Group ◽  
Quantum Computer ◽  
Tutte Polynomial ◽  
Roots Of Unity ◽  
Homfly Polynomial ◽  
Root Of Unity ◽  
Quantum Complexity ◽  
Homfly Polynomials ◽  
Alternating Links ◽  
Alternating Link

We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q, 1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at the point (e2πi/5, e-2πi/5) is DQC1-complete and at points [Formula: see text] for some integer k is in BQP. To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids. To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible Jones–Wenzl representations of the Hecke algebra have finite order. We show that for each braid b we can efficiently construct a braid [Formula: see text] such that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of [Formula: see text] are alternating links.

scholarly journals Generalised cluster algebras and $q$-characters at roots of unity

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Anne-Sophie Gleitz
Keyword(s):  
Cluster Algebra ◽  
Integral Form ◽  
The Other ◽  
Cluster Algebras ◽  
Roots Of Unity ◽  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Root Of Unity ◽  
International Audience

International audience Shapiro and Chekhov (2011) have introduced the notion of <i>generalised cluster algebra</i>; we focus on an example in type $C_n$. On the other hand, Chari and Pressley (1997), as well as Frenkel and Mukhin (2002), have studied the <i>restricted integral form</i> $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ of a quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ where $q=ε$ is a root of unity. Our main result states that the Grothendieck ring of a tensor subcategory $C_{ε^\mathbb{z}}$ of representations of $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ is a generalised cluster algebra of type $C_{l−1}$, where $l$ is the order of $ε^2$. We also state a conjecture for $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$, and sketch a proof for $l=2$. Shapiro et Chekhov (2011) ont introduit la notion d'<i>algèbre amassée généralisée</i>; nous étudions un exemple en type $C_n$. Par ailleurs, Chari et Pressley (1997), ainsi que Frenkel et Mukhin (2002), ont étudié la <i>forme entière restreinte</i> $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ d'une algèbre affine quantique $U_q(\widehat{\mathfrak{g}})$ où $q=ε$ est une racine de l'unité. Notre résultat principal affirme que l'anneau de Grothendieck d'une sous-catégorie tensorielle $C_{ε^\mathbb{z}}$ de représentations de $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ est une algèbre amassée généralisée de type $C_{l−1}$, où $l$ est l'ordre de $ε^2$. Nous conjecturons une propriété similaire pour $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$ et donnons un aperçu de la preuve pour $l=2$.

scholarly journals Lie polynomials in an algebra defined by a linearly twisted commutation relation

2021 ◽  
pp. 2250175
Author(s):  
Rafael Reno S. Cantuba
Keyword(s):  
Commutation Relation ◽  
Roots Of Unity ◽  
Slope Parameter ◽  
Elementary Approach ◽  
Root Of Unity ◽  
Deformation Parameters ◽  
Linear Polynomial ◽  
Characterization Problem ◽  
Heisenberg Algebras ◽  
Defining Relation

We present an elementary approach to characterizing Lie polynomials on the generators [Formula: see text] of an algebra with a defining relation in the form of a twisted commutation relation [Formula: see text]. Here, the twisting map [Formula: see text] is a linear polynomial with a slope parameter, which is not a root of unity. The class of algebras defined as such encompasses [Formula: see text]-deformed Heisenberg algebras, rotation algebras, and some types of [Formula: see text]-oscillator algebras, the deformation parameters of which, are not roots of unity. Thus, we have a general solution for the Lie polynomial characterization problem for these algebras.

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