Lie Groups, Number Theory, and Vertex Algebras

In this research, we present some basic facts about Lie algebra and Lie groups. We shall require only elementary facts about the general definition and knowledge of a few of the more basic groups, such as Euclidean groups. Then we introduce the Heisenberg group which is the most well-known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as representation theory, partial differential equations and number theory... It also offers the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis.

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Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of SLn(ℤp)

Forum Mathematicum ◽

10.1515/forum-2019-0260 ◽

2020 ◽

Vol 32 (2) ◽

pp. 319-338 ◽

Cited By ~ 2

Author(s):

Jishnu Ray

Keyword(s):

Number Theory ◽

Lie Groups ◽

Number Fields ◽

Principal Congruence ◽

Class Numbers ◽

Group Algebras ◽

Generators And Relations ◽

Congruence Kernel ◽

Iwahori Subgroup ◽

Iwasawa Algebra

AbstractIwasawa algebras of compact p-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of p-adic Lie groups. We previously determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over {\mathbb{Z}_{p}} which were uniform pro-p groups in the sense of Dixon, du Sautoy, Mann and Segal. In this paper, for prime {p>n+1}, we determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-p Iwahori subgroup of {\mathrm{GL}_{n}(\mathbb{Z}_{p})} which is not, in general, a uniform pro-p group.

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The Sub- Laplacian on The Heisenberg Group and Itʼs Spectral Properties: مؤثر لابلاس الجزئي على زمرة هايزنبرغ وخصائصه الطيفية

Arab Journal of Sciences & Research Publishing - المجلة العربية للعلوم و نشر الأبحاث ◽

10.26389/ajsrp.s181019 ◽

2020 ◽

Vol 6 (1) ◽

Author(s):

Soha Ali Salamah

Keyword(s):

Number Theory ◽

Heisenberg Group ◽

Lie Groups ◽

Spectral Properties ◽

Complete Analysis ◽

Nilpotent Lie Groups ◽

One Dimensional ◽

Underlying Space ◽

Adjoint Extension ◽

The One

In this paper we talk about the spectral theory of the sub-Laplacian on the Heisenberg group. Then we give a complete analysis of the spectrum of the unique self- adjoint extension of this sub-Laplacian on the one-dimensional Heisenberg group. The Heisenberg group is the most known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as representation theory, partial differential equations and number theory... It also offers the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis. The results in this paper are valid for the sub-Laplacian on the n-dimensional Heisenberg group, in which the underlying space is, but we have chosen to present the results for the one-dimensional Heisenberg group ℍ for the sake of simplicity and transparency.

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