An official website of the United States government
Abstract We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantised enveloping algebras of Kac–Moody type. Our methods are based on star products on noncommutative $${\mathbb N}$$ -graded algebras. The resulting defining relations are expressed in terms of continuous q -Hermite polynomials and a new family of deformed Chebyshev polynomials.
more »
« less
Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations
To Nicolás Andruskiewitsch on his 60th birthday, with admiration We introduce bivariate versions of the continuous [Formula: see text]-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac–Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.
more »
« less
- Award ID(s):
- 1901830
- PAR ID:
- 10221076
- Date Published:
- Journal Name:
- Journal of Algebra and Its Applications
- Volume:
- 20
- Issue:
- 01
- ISSN:
- 0219-4988
- Page Range / eLocation ID:
- 2140016
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We study the dynamics of the unicritical polynomial family [Formula: see text]. The [Formula: see text]-values for which [Formula: see text] has a strictly preperiodic postcritical orbit are called Misiurewicz parameters, and they are the roots of Misiurewicz polynomials. The arithmetic properties of these special parameters have found applications in both arithmetic and complex dynamics. In this paper, we investigate some new such properties. In particular, when [Formula: see text] is a prime power and [Formula: see text] is a Misiurewicz parameter, we prove certain arithmetic relations between the points in the postcritical orbit of [Formula: see text]. We also consider the algebraic integers obtained by evaluating a Misiurewicz polynomial at a different Misiurewicz parameter, and we ask when these algebraic integers are algebraic units. This question naturally arises from some results recently proven by Buff, Epstein, and Koch and by the second author. We propose a conjectural answer to this question, which we prove in many cases.more » « less
-
The density matrix formalism is a fundamental tool in studying various problems in quantum information processing. In the space of density matrices, the most well-known measures are the Hilbert–Schmidt and Bures–Hall ensembles. In this work, the averages of quantum purity and von Neumann entropy for an ensemble that interpolates between these two major ensembles are explicitly calculated for finite-dimensional systems. The proposed interpolating ensemble is a specialization of the [Formula: see text]-deformed Cauchy–Laguerre two-matrix model and new results for this latter ensemble are given in full generality, including the recurrence relations satisfied by their associated bi-orthogonal polynomials when [Formula: see text] assumes positive integer values.more » « less
-
Magneto-intersubband resistance oscillations (MISOs) of highly mobile 2D electrons in symmetric GaAs quantum wells with two populated subbands are studied in magnetic fields [Formula: see text] tilted from the normal to the 2D electron layer at different temperatures [Formula: see text]. The in-plane component ([Formula: see text]) of the field [Formula: see text] induces magnetic entanglement between subbands, leading to beating in oscillating density of states (DOS) and to MISO suppression. Model of the MISO suppression is proposed. Within the model, a comparison of MISO amplitude in the entangled and disentangled ([Formula: see text]) 2D systems yields both difference frequency of DOS oscillations, [Formula: see text], and strength of the electron–electron interaction, described by parameter [Formula: see text], in the 2D system. These properties are analyzed using two methods, yielding consistent but not identical results for both [Formula: see text] and [Formula: see text]. The analysis reveals an additional angular dependent factor of MISO suppression. The factor is related to spin splitting of quantum levels in magnetic fields.more » « less
-
Let [Formula: see text] be an integer and [Formula: see text] be a finite field with [Formula: see text] elements. We prove several results on the distribution in short intervals of polynomials in [Formula: see text] that are not divisible by the [Formula: see text]th power of any non-constant polynomial. Our main result generalizes a recent theorem by Carmon and Entin on the distribution of squarefree polynomials to all [Formula: see text]. We also develop polynomial versions of the classical techniques used to study gaps between [Formula: see text]-free integers in [Formula: see text]. We apply these techniques to obtain analogs in [Formula: see text] of some classical theorems on the distribution of [Formula: see text]-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S0219498821400168
