scholarly journals CARTAN SUBALGEBRAS IN DIMENSION DROP ALGEBRAS

2019 ◽  
pp. 1-31
Author(s):  
Selçuk Barlak ◽  
Sven Raum
Keyword(s):  
Conjugacy Classes ◽  
Natural Numbers ◽  
Cartan Subalgebras ◽  
Classes Of Matrices ◽  
Congruence Classes

We completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this parametrisation, we can provide examples of subhomogeneous $\text{C}^{\ast }$ -algebras with exactly $n$ Cartan subalgebras up to conjugacy. Moreover, we show that in many dimension drop algebras two Cartan subalgebras are conjugate if and only if their spectra are homeomorphic.

scholarly journals Congruence classes of matrices in GL2(Fq)

1993 ◽  
Vol 118 (1-3) ◽  
pp. 243-249 ◽  
Author(s):  
Priscilla S. Bremser
Keyword(s):  
Classes Of Matrices ◽  
Congruence Classes

Closures of conjugacy classes of matrices are normal

1979 ◽  
Vol 53 (3) ◽  
pp. 227-247 ◽  
Author(s):  
Hanspeter Kraft ◽  
Claudio Procesi
Keyword(s):  
Conjugacy Classes ◽  
Classes Of Matrices

Conjugacy Classes of Maximal Tori in Simple Real Algebraic Groups and Applications

1994 ◽  
Vol 46 (4) ◽  
pp. 699-717 ◽  
Author(s):  
Dragomir Ž. Doković ◽  
Nguyêñ Quôć Thăńg
Keyword(s):  
Lie Algebras ◽  
Lie Group ◽  
Short Proof ◽  
Algebraic Groups ◽  
Conjugacy Classes ◽  
Exponential Map ◽  
Simple Complex ◽  
Maximal Tori ◽  
Cartan Subalgebras

AbstractLet G be an almost simple complex algebraic group defined over R, and let G(R) be the group of real points of G. We enumerate the G(R)-conjugacy classes of maximal R-tori of G. Each of these conjugacy classes is also a single G(R)˚-conjugacy class, where G(R)˚ is the identity component of G(R), viewed as a real Lie group. As a consequence we also obtain a new and short proof of the Kostant-Sugiura's theorem on conjugacy classes of Cartan subalgebras in simple real Lie algebras.A connected real Lie group P is said to be weakly exponential (w.e.) if the image of its exponential map is dense in P. This concept was introduced in [HM] where also the question of identifying all w.e. almost simple real Lie groups was raised. By using a theorem of A. Borel and our classification of maximal R-tori we answer the above question when P is of the form G(R)˚.

scholarly journals Applying combinatorial results to products of conjugacy classes

2020 ◽  
Vol 23 (5) ◽  
pp. 917-923
Author(s):  
Rachel Deborah Camina
Keyword(s):  
Normal Subgroup ◽  
Conjugacy Class ◽  
Conjugacy Classes ◽  
Natural Numbers

AbstractLet {K=x^{G}} be the conjugacy class of an element x of a group G, and suppose K is finite. We study the increasing sequence of natural numbers {\{\lvert K^{n}\rvert\}_{n\geq 1}} and consider restrictions on this sequence and the algebraic consequences. In particular, we prove that if {\lvert K^{2}\rvert<\frac{3}{2}\lvert K\rvert} or if {\lvert K^{4}\rvert<2\lvert K\rvert}, then {K^{n}} is a coset of the normal subgroup {[x,G]} for all {n\geq 2} or 4, respectively. We then use these results to contribute to conjectures about the solubility of {\langle K\rangle} when {K^{n}} satisfies certain conditions.

The Argument from (Natural) Numbers

2018 ◽  
Author(s):  
Tyron Goldschmidt
Keyword(s):  
Natural Numbers ◽  
Existence Of God ◽  
Divine Attribute ◽  
Rule Out

This chapter considers Plantinga’s argument from numbers for the existence of God. Plantinga sees divine psychologism as having advantages over both human psychologism and Platonism. The chapter begins with Plantinga’s description of the argument, including the relation of numbers to any divine attribute. It then argues that human psychologism can be ruled out completely. However, what rules it out might rule out divine psychologism too. It also argues that the main problem with Platonism might also be a problem with divine psychologism. However, it will, at the least, be less of a problem. In any case, there are alternative, possibly viable views about the nature of numbers that have not been touched by Plantinga’s argument. In addition, the chapter touches on the argument from properties, and its relation to the argument from numbers.

Sign in / Sign up

Export Citation Format

Share Document