Introduction to linear Lie groups

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Presentation transcript:

Introduction to linear Lie groups PHY 745 Group Theory 11-11:50 AM MWF Olin 102 Plan for Lecture 31: Introduction to linear Lie groups Definitions Properties Examples Ref. J. F. Cornwell, Group Theory in Physics, Vol I and II, Academic Press (1984) 4/7/2017 PHY 745 Spring 2017 -- Lecture 31

4/7/2017 PHY 745 Spring 2017 -- Lecture 31

Norweigian mathematician Sophus Lie 1842-1899 Norweigian mathematician https://www.britannica.com/biography/Sophus-Lie 4/7/2017 PHY 745 Spring 2017 -- Lecture 31

Definition of a linear Lie group A linear Lie group is a group Each element of the group T forms a member of the group T’’ when “multiplied” by another member of the group T”=T·T’ One of the elements of the group is the identity E For each element of the group T, there is a group member a group member T-1 such that T·T-1=E. Associative property: T·(T’·T’’)= (T·T’)·T’’ Elements of group form a “topological space” Elements also constitute an “analytic manifold” Non countable number elements lying in a region “near” its identity 4/7/2017 PHY 745 Spring 2017 -- Lecture 31

Definition: Linear Lie group of dimension n A group G is a linear Lie group of dimension n if it satisfied the following four conditions: G must have at least one faithful finite-dimensional represention G which defines the notion of distance. 4/7/2017 PHY 745 Spring 2017 -- Lecture 31

Definition: Linear Lie group of dimension n -- continued Consider the distance between group elements T with respect to the identity E -- d(T,E). It is possible to define a sphere Md that constains all elements T’ 4/7/2017 PHY 745 Spring 2017 -- Lecture 31

Definition: Linear Lie group of dimension n -- continued There must exist There is a requirement that the corresponding representation is analytic 4/7/2017 PHY 745 Spring 2017 -- Lecture 31

4/7/2017 PHY 745 Spring 2017 -- Lecture 31

4/7/2017 PHY 745 Spring 2017 -- Lecture 31

4/7/2017 PHY 745 Spring 2017 -- Lecture 31

General transformation between rotated coordinates – Euler angles inertial body y x http://en.wikipedia.org/wiki/Euler_angles 4/7/2017 PHY 745 Spring 2017 -- Lecture 31

4/7/2017 PHY 745 Spring 2017 -- Lecture 31

Homomorphic mapping of SU(2) onto SO(3) 4/7/2017 PHY 745 Spring 2017 -- Lecture 31

Recall: 4/7/2017 PHY 745 Spring 2017 -- Lecture 31

4/7/2017 PHY 745 Spring 2017 -- Lecture 31

4/7/2017 PHY 745 Spring 2017 -- Lecture 31