scholarly journals Entanglement in fermionic chains and bispectrality

2021 ◽  
pp. 2140001
Author(s):  
Nicolas Crampé ◽  
Rafael I. Nepomechie ◽  
Luc Vinet
Keyword(s):  
Signal Processing ◽  
Lie Algebras ◽  
Tridiagonal Matrix ◽  
Root Of Unity ◽  
Deformed Algebra ◽  
Bispectral Problem

Entanglement in finite and semi-infinite free Fermionic chains is studied. A parallel is drawn with the analysis of time and band limiting in signal processing. It is shown that a tridiagonal matrix commuting with the entanglement Hamiltonian can be found using the algebraic Heun operator construct in instances when there is an underlying bispectral problem. Cases corresponding to the Lie algebras [Formula: see text] and [Formula: see text] as well as to the q-deformed algebra [Formula: see text] at [Formula: see text] a root of unity are presented.

A New Class of Representations of EALA Coordinated by Quantum Tori in Two Variables

2002 ◽  
Vol 45 (4) ◽  
pp. 672-685 ◽  
Author(s):  
S. Eswara Rao ◽  
Punita Batra
Keyword(s):  
Lie Algebras ◽  
Primitive Root ◽  
Irreducible Module ◽  
Affine Lie Algebras ◽  
Quantum Torus ◽  
New Class ◽  
Root Of Unity ◽  
Quantum Tori ◽  
Finite Dimensional

AbstractWe study the representations of extended affine Lie algebras where q is N-th primitive root of unity (ℂq is the quantum torus in two variables). We first prove that ⊕ for a suitable number of copies is a quotient of . Thus any finite dimensional irreducible module for ⊕ lifts to a representation of . Conversely, we prove that any finite dimensional irreducible module for comes from above. We then construct modules for the extended affine Lie algebras which is integrable and has finite dimensional weight spaces.

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.

Parafermion theory from the q-deformed algebra with q a root of unity

2020 ◽  
Vol 17 (03) ◽  
pp. 2050045
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Partition Function ◽  
Specific Heat ◽  
Set Theory ◽  
Fock Space ◽  
Infinite Dimensional ◽  
Root Of Unity ◽  
Deformed Algebra ◽  
Bosonic Case ◽  
Fermionic Case ◽  
Bosonic Limit

In this paper, we find the [Formula: see text]-deformed algebra with the finite- and infinite-dimensional Fock space and both the fermionic limit and the bosonic limit. Using the cardinality of set theory, we propose the Hamiltonian interpolating bosonic case and fermionic case, which enables us to construct the proper partition function and internal energy. As examples, we discuss the specific heat of free [Formula: see text] parafermion gas model and [Formula: see text] parafermion star.

scholarly journals Groups of automorphisms of local fields of period p and nilpotent class < p, II

2017 ◽  
Vol 28 (10) ◽  
pp. 1750066
Author(s):  
Victor Abrashkin
Keyword(s):  
Finite Field ◽  
Galois Group ◽  
Lie Algebras ◽  
Explicit Form ◽  
Field Extension ◽  
Local Fields ◽  
Root Of Unity ◽  
Groups Of Automorphisms ◽  
Nilpotent Class ◽  
Primitive Formula

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be the maximal quotient of period [Formula: see text] and nilpotent class [Formula: see text] of the Galois group of a maximal [Formula: see text]-extension of [Formula: see text]. We describe the ramification filtration [Formula: see text] and relate it to an explicit form of the Demushkin relation for [Formula: see text]. The results are given in terms of Lie algebras attached to the appropriate [Formula: see text]-groups by the classical equivalence of the categories of [Formula: see text]-groups and Lie algebras of nilpotent class [Formula: see text].

POSSIBLE IMPLICATIONS OF INTEGRABLE SYSTEMS FOR STRING THEORY

1991 ◽  
Vol 06 (06) ◽  
pp. 977-988 ◽  
Author(s):  
A. GERASIMOV ◽  
D. LEBEDEV ◽  
A. MOROZOV
Keyword(s):  
String Theory ◽  
Lie Algebras ◽  
Spectral Parameter ◽  
Integrable Systems ◽  
Integrable Models ◽  
Lax Representation ◽  
Two Dimensional ◽  
Deformed Algebra ◽  
Area Preserving

A kind of program for the unification of conformal and two-dimensional integrable models is described. Integrable systems, defined by the condition of vanishing curvature (Lax representation), can be derived from universal generalizations of Wess-Zumino-Witten action, including one more integration (over spectral parameter λ). Such actions, in their turn, should be derivable from some membrane-like action, related to the Kirillov-Kostant form for some quantum two-loop Lie algebras (like complexification of Fairlie’s deformed algebra of two-dimensional area-preserving reparametrizations).

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.

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