A work on Representation Theorem’s and
Algebraic Applications

Sahar Jaafar Mahmood Abumalah

doi:https://doi.org/10.14445/22315373/IJMTT-V34P503

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 34 | Number 1 | Year 2016 | Article Id. IJMTT-V34P503 | DOI : https://doi.org/10.14445/22315373/IJMTT-V34P503

A work on Representation Theorem’s and Algebraic Applications

Sahar Jaafar Mahmood Abumalah

Citation :

Sahar Jaafar Mahmood Abumalah, "A work on Representation Theorem’s and Algebraic Applications," International Journal of Mathematics Trends and Technology (IJMTT), vol. 34, no. 1, pp. 9-12, 2016. Crossref, https://doi.org/10.14445/22315373/IJMTT-V34P503

Abstract

In mathematics a representation theorem is a theorem which states that each abstract structure with certain properties is isomorphic to a concrete structure. There are several examples of theorems of representation in various fields of mathematics: . In algebra, the theorem of Cayley states that each group is isomorphic to a group transformed a whole. The representation theory studies the properties of abstract groups through their representation as transformations of vector spaces. Additionally, also in algebra, Stone representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets. A variant of this theorem channeled reticles requires each distributive lattice is isomorphic to a reticle sub-lattice power set of a set. . In category theory, the Yoneda lemma explains how arbitrary functors in the category of sets can be seen as hom functions. . In set theory, the Mostowski collapsing theorem states that every well-founded extensional structure is isomorphic to a transitive set with the relation of belonging (∈). In functional analysis, the Riesz representation theorem is currently a list of many theorems. One identifies the dual space C0 (X) with the set of regular measurements in X.in geometry, the Whitney embedding theorems embed any abstract manifold in some Euclidean space. The Nash embedding theorem embeds an abstract Riemannian manifold isometrically in a Euclidean space. But the aim of this work is the algebraic applications, specially the representation theory.

Keywords

But the aim of this work is the algebraic applications, specially the representation theory.

References

[1] J. L. Alperin, Local Representation Theory: Modular Representations as an Introduction to the Local Representation Theory of Finite Groups, , 1986
[2] Armand Borel, Essays in the History of Lie Groups and Algebraic Groups, AMS, 2001
[3] Armand Borel et W. Casselman, Automorphic Forms, Representations, and L-functions, AMS, 197
[4] Charles W. Curtis et Irving Reiner, Representation Theory of Finite Groups and Associative Algebras, John Wiley & Sons, 1962
[5] William Fulton et Joe Harris, Representation Theory : A First Course
[6] Stephen Gelbart, « An Elementary Introduction to the Langlands Program », Bull. Amer. Math. Soc., vol. 10, no 2, 1984, p. 177-219
[7] Roe Goodman et Nolan R. Wallach, Representations and Invariants of the Classical Groups, CUP, 1998
[8] James Gordon et Martin Liebeck, Representations and Characters of Groups, CUP, 1993
[9] James E. Humphreys , Introduction to Lie Algebras and Representation, Theory, Birkhäuser, 1972
[10] James E. Humphreys, Linear Algebraic Groups, Springer, coll. « GTM » (no 21), 1975
[11] Jens Carsten Jantzen , Representations of Algebraic Groups, AMS, 2003
[12] Anthony W. Knapp (de), Representation Theory of Semisimple Groups : An Overview Based on Examples,, 2001
[13] T. Y. Lam, « Representations of Finite Groups: A Hundred Years », Notices Amer. Math. Soc., vol. 45, no 3 et 4, 1998, 361-372 (Part I), 465-474 (Part II)
[14] Peter J. Olver, Classical Invariant Theory, CUP, 1999
[15] Jean-Pierre Serre, Linear Representations of Finite Groups
[16] R. W. Sharpe, Differential Geometry : Cartan's Generalization of Klein's Erlangen Program, Springer, coll. « GTM » (no 166), 1997
[17] Daniel Simson, Andrzej Skowronski et Ibrahim Assem, Elements of the Representation Theory of Associative Algebras, CUP, 2007
[18] S. Sternberg, Group Theory and Physics, CUP, 1994
[19] Introduction to representation theory Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina January 10, 2011
[20] Arthur Cayley, "On the theory of groups, as DEPENDING on the symbolic equation θn = 1", Philos. Mag., Vol. 7, No. 4, 1854, p. 40-47.
[21] William Burnside, Theory of Groups of Finite Order, Cambridge, 2004 (1st ed. 1911) , p. 22.
[22] Camille Jordan, Treaty substitutions and algebraic equations, Paris, Gauther-Villars, 1870, p. 60-61.