scholarly journals Cocommutative elements form a maximal commutative subalgebra in quantum matrices

2018 ◽  
Vol 17 (09) ◽  
pp. 1850179
Author(s):  
Szabolcs Mészáros
Keyword(s):  
Trace Formula ◽  
Quantum Matrices ◽  
Root Of Unity ◽  
Commutative Subalgebra ◽  
Maximal Commutative Subalgebra ◽  
Coordinate Rings

In this paper, we prove that the subalgebras of cocommutative elements in the quantized coordinate rings of [Formula: see text], [Formula: see text] and [Formula: see text] are the centralizers of the trace [Formula: see text] in each algebra, for [Formula: see text] being not a root of unity. In particular, it is not only a commutative subalgebra as it was known before, but it is a maximal one.

scholarly journals MAXIMAL COMMUTATIVE SUBALGEBRAS INVARIANT FOR CP-MAPS: (COUNTER-)EXAMPLES

2008 ◽  
Vol 11 (04) ◽  
pp. 523-539 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
FRANCO FAGNOLA ◽  
MICHAEL SKEIDE
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann ◽  
Commutative Subalgebra ◽  
Commutative Subalgebras ◽  
Maximal Commutative Subalgebra

We solve, mainly by counterexamples, many natural questions regarding maximal commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a von Neumann algebra. In particular, we discuss the structure of the generators of norm continuous semigroups on [Formula: see text] leaving a maximal commutative subalgebra invariant and show that there exist Markov CP-semigroups on Md without invariant maximal commutative subalgebras for any d > 2.

scholarly journals PBW THEOREMS AND FROBENIUS STRUCTURES FOR QUANTUM MATRICES

2007 ◽  
Vol 49 (3) ◽  
pp. 479-488
Author(s):  
FABIO GAVARINI
Keyword(s):  
Function Algebra ◽  
Direct Application ◽  
Hopf Subalgebra ◽  
Frobenius Morphism ◽  
Quantum Matrices ◽  
Root Of Unity ◽  
Frobenius Structures ◽  
Quantum Function

AbstractLet $G \in \{{\it Mat}_n(\C), {GL}_n(\C), {SL}_n(\C)\}$, let $\Oqg$ be the quantum function algebra – over $\Z [q,q^{-1}]$ – associated to G, and let $\Oeg$ be the specialisation of the latter at a root of unity ϵ, whose order ℓ is odd. There is a quantum Frobenius morphism that embeds $\Og,$ the function algebra of G, in $\Oeg$ as a central Hopf subalgebra, so that $\Oeg$ is a module over $\Og$. When $G = {SL}_n(\C)$, it is known by [3], [4] that (the complexification of) such a module is free, with rank ℓdim(G). In this note we prove a PBW-like theorem for $\Oqg$, and we show that – when G is Matn or GLn – it yields explicit bases of $\Oeg $ over $ \Og$ over $\Og,$. As a direct application, we prove that $\Oegl$ and $\Oem$ are free Frobenius extensions over $\Ogl$ and $\Om$, thus extending some results of [5].

scholarly journals PRIME IDEALS INVARIANT UNDER WINDING AUTOMORPHISMS IN QUANTUM MATRICES

2002 ◽  
Vol 13 (05) ◽  
pp. 497-532 ◽  
Author(s):  
K. R. GOODEARL ◽  
T. H. LENAGAN
Keyword(s):  
Tensor Product ◽  
Prime Ideal ◽  
Matrix Algebra ◽  
Prime Ideals ◽  
Quantum Matrices ◽  
Root Of Unity

The main goal of the paper is to establish the existence of tensor product decompositions for those prime ideals P of the algebra [Formula: see text] of quantum n × n matrices which are invariant under winding automorphisms of A, in the generic case (q not a root of unity). More specifically, every such P is the kernel of a map of the form [Formula: see text] where A → A ⊗ A is the comultiplication, A+ and A- are suitable localized factor algebras of A, and P± is a prime ideal of A± invariant under winding automorphisms. Further, the algebras A±, which vary with P, can be chosen so that the correspondence (P+, P-) ↦ P is a bijection. The main theorem is applied, in a sequel to this paper, to completely determine the winding-invariant prime ideals in the generic quantum 3 × 3 matrix algebra.

scholarly journals Maximal commutative subalgebras of a Grassmann algebra

2019 ◽  
Vol 18 (07) ◽  
pp. 1950139
Author(s):  
Victor A. Bovdi ◽  
Ho-Hon Leung
Keyword(s):  
Grassmann Algebra ◽  
New Approach ◽  
Commutative Subalgebra ◽  
Intersecting Family ◽  
Grassmann Algebras ◽  
Commutative Subalgebras ◽  
Maximal Commutative Subalgebra

We provide a new approach to the investigation of maximal commutative subalgebras (with respect to inclusion) of Grassmann algebras. We show that finding a maximal commutative subalgebra in Grassmann algebras is equivalent to constructing an intersecting family of subsets of various odd sizes in [Formula: see text] which satisfies certain combinatorial conditions. Then we find new maximal commutative subalgebras in the Grassmann algebra of odd rank [Formula: see text] by constructing such combinatorial systems for odd [Formula: see text]. These constructions provide counterexamples to conjectures made by Domoskos and Zubor.

scholarly journals Cocycle Deformations of Coordinate Rings of Quantum Matrices

1997 ◽  
Vol 189 (1) ◽  
pp. 23-33 ◽  
Author(s):  
Mitsuhiro Takeuchi
Keyword(s):  
Quantum Matrices ◽  
Coordinate Rings

The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’Aini Aris ◽  
Azlina Jumadi
Keyword(s):  
Convex Polytopes ◽  
Polynomial Systems ◽  
Matrix Formulation ◽  
Bivariate Polynomials ◽  
Homogeneous Coordinates ◽  
The Matrix ◽  
Projective Vector ◽  
Coordinate Rings ◽  
Resultant Matrix ◽  
The Given

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

scholarly journals Ramification Theory for Extensions of Degree p. II

1972 ◽  
Vol 46 ◽  
pp. 97-109
Author(s):  
Susan Williamson
Keyword(s):  
Valuation Ring ◽  
Quotient Field ◽  
Root Of Unity ◽  
Ramification Index ◽  
Ramification Theory ◽  
Rank One ◽  
The Absolute

Let k denote the quotient field of a complete discrete rank one valuation ring R of unequal characteristic and let p denote the characteristic of R̅; assume that R contains a primitive pth root of unity, so that the absolute ramification index e of R is a multiple of p — 1, and each Gallois extension K ⊃ k of degree p may be obtained by the adjunction of a pth root.

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