NOTES ON THE CONSTRUCTIONS OF MAXIMAL COMMUTATIVE SUBALGEBRA OFMn(k)

2001 ◽  
Vol 29 (10) ◽  
pp. 4333-4339 ◽  
Author(s):  
Youngkwon Song
Keyword(s):  
Commutative Subalgebra ◽  
Maximal Commutative Subalgebra

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 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.

On a Class of Automaton Algebras

2016 ◽  
Vol 60 (1) ◽  
pp. 31-38 ◽  
Author(s):  
Ferran Cedó ◽  
Jan Okniński
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Module ◽  
Commutative Subalgebra

AbstractWe show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

QUASI-INVARIANCE FORMULAS FOR COMPONENTS OF QUANTUM LÉVY PROCESSES

2004 ◽  
Vol 07 (01) ◽  
pp. 131-145 ◽  
Author(s):  
UWE FRANZ ◽  
NICOLAS PRIVAULT
Keyword(s):  
Lie Algebras ◽  
Lévy Processes ◽  
Unitary Operator ◽  
Levy Processes ◽  
Absolutely Continuous ◽  
Commutative Subalgebra ◽  
Independent Increments ◽  
Commutative Subalgebras ◽  
Current Representation ◽  
General Method

A general method for deriving Girsanov or quasi-invariance formulas for classical stochastic processes with independent increments obtained as components of Lévy processes on real Lie algebras is presented. Letting a unitary operator arising from the associated factorizable current representation act on an appropriate commutative subalgebra, a second commutative subalgebra is obtained. Under certain conditions the two commutative subalgebras lead to two classical processes such that the law of the second process is absolutely continuous w.r.t. to the first. Examples include the Girsanov formula for Brownian motion as well as quasi-invariance formulas for the Poisson process, the Gamma process,15,16 and the Meixner process.

Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Grzegorz Bajor ◽  
Leon van Wyk ◽  
Michał Ziembowski
Keyword(s):  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Arbitrary Graph ◽  
Unital Ring ◽  
Commutative Subalgebra ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Commutative Subalgebras

Abstract Considering prime Leavitt path algebras L K ⁢ ( E ) {L_{K}(E)} , with E being an arbitrary graph with at least two vertices, and K being any field, we construct a class of maximal commutative subalgebras of L K ⁢ ( E ) {L_{K}(E)} such that, for every algebra A from this class, A has zero intersection with the commutative core ℳ K ⁢ ( E ) {\mathcal{M}_{K}(E)} of L K ⁢ ( E ) {L_{K}(E)} defined and studied in [C. Gil Canto and A. Nasr-Isfahani, The commutative core of a Leavitt path algebra, J. Algebra 511 2018, 227–248]. We also give a new proof of the maximality, as a commutative subalgebra, of the commutative core ℳ R ⁢ ( E ) {\mathcal{M}_{R}(E)} of an arbitrary Leavitt path algebra L R ⁢ ( E ) {L_{R}(E)} , where E is an arbitrary graph and R is a commutative unital ring.

Computing embedding numbers and branches of orders via extensions of the Bruhat-Tits tree

2019 ◽  
Vol 15 (10) ◽  
pp. 2067-2088
Author(s):  
Luis Arenas-Carmona ◽  
Claudio Bravo
Keyword(s):  
Local Field ◽  
Matrix Algebra ◽  
Explicit Representation ◽  
Past Work ◽  
Field Extensions ◽  
The Past ◽  
Commutative Subalgebra ◽  
Maximal Orders

Let [Formula: see text] be a local field and let [Formula: see text] be the two-by-two matrix algebra over [Formula: see text]. In our previous work, we developed a theory that allows the computation of the set of maximal orders in [Formula: see text] containing a given suborder. This set is given as a subgraph of the Bruhat–Tits (BT)-tree that is called the branch of the order. Branches have been used to study the global selectivity problem and also to compute local embedding numbers. They can usually be described in terms of two invariants. To compute these invariants explicitly, the strategy in our past work has been visualizing branches through the explicit representation of the BT-tree in terms of balls in [Formula: see text]. This is easier for orders spanning a split commutative subalgebra, i.e. an algebra isomorphic to [Formula: see text]. In fact, we have successfully used this idea in the past to compute embedding numbers for the split algebra. In the present work, we develop a theory of branches over field extensions that can be used to extend our previous computations to orders spanning a field. We use the same idea to compute branches for orders generated by arbitrary pairs of non-nilpotent pure quaternions, generalizing previous results due to the first author and Saavedra. We assume throughout that [Formula: see text].

scholarly journals Open quantum random walks, quantum Markov chains and recurrence

2019 ◽  
Vol 31 (07) ◽  
pp. 1950020 ◽  
Author(s):  
Ameur Dhahri ◽  
Farrukh Mukhamedov
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Markov Processes ◽  
Transition Operator ◽  
Quantum Random Walks ◽  
Open Quantum ◽  
Commutative Subalgebra ◽  
Open Quantum Random Walks ◽  
Quantum Markov Chains

In the present paper, we construct QMCs (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution [Formula: see text] of OQRW. This sheds new light on some properties of the measure [Formula: see text]. As an example, we simply mention that the measure can be considered as a distribution of some functions of certain Markov processes. Furthermore, we study several properties of QMC and associated measures. A new notion of [Formula: see text]-recurrence of QMC is studied, and the relations between the concepts of recurrence introduced in this paper and the existing ones are established.

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