Math 662--Section 600 -- Fall 2016

Lie Algebras and Their Representations

TR 2:20-3:35pm
BLOC 111

/~rowell/M662-Fall2016.html

Homework: (turn in all problems).

HW1 (due Tuesday Sept. 20): 1) Prove that if B(x,y) is a symmetric, non-degenerate, bilinear form on finite dimensional vector space V then the corresponding orthogonal Lie algebra o(B,V) is isomorphic to o(S,V) where S is the standard dot product S(x,y)=x_1y_1+...+x_ny_n. 2) Prove that R^3 with the cross product is a (real) Lie algebra. Is there a relationship with the real Lie algebra su(2)? 3) Consider the real algebra H spanned by 1,i,j,k where i,j,k are the quaternion units, as a Lie algebra with the commutator as bracket. Is it related to one (or both) of the real Lie algebras in 2) above? Hint consider H/R, where R=R*1 is the subalgebra spanned by 1. 4) Check that the radical rad(K) of the Killing form of g satisfies the conditions of Cartan's criterion for solvability.
HW2: (due Thursday Oct. 13): 1) Let g be a real Lie algebra, and g_C its complexification: as a vector space it is g tensored with C over R. What is the bracket [x,y] on g_C? 2) Determine the adjoint representation and Killing form for o(S,V) with V=C^3. Is the Killing form non-degenerate? Is the adjoint representation faithful? 3) Compute the Casimir for the 3-dimensional representation of o(S,V) with dim(V)=3 DIRECTLY, by finding a basis and dual basis.
HW3: (due Nov. 22): 1) Prove that the roots systems for so(5) and sp(4) are isomorphic. 2) Let (R,E) be a roots system, with Weyl group W. Show that W is a normal subgroup of the group of automorphisms of (R,E) (that is, the group of linear automorphisms of E, preserving R as a set, and preserving the Cartan integers). 3) Fill in the details of the proofs of the results in 10.2 and 10.3 of Humphreys book.

Course Description:

This course is an introduction to the theory of complex Lie algebras and their representations.

Course Information:

	Instructor:	Dr. Eric Rowell
	Office Hours:	TBA

	Office:	510B Blocker
	Office Phone:	sorry, no phone.
	E-mail:	rowell@math.tamu.edu

	Textbooks:	Humphreys, Introduction to Lie algebras and representation theory (free download from TAMU library) Fulton and Harris, Representation theory (recommended, free download from TAMU library)

	Course Syllabus:	The goal of the course is to understand two fundamental results about Lie algebras: 1) the classification of semis-simple, finite dimensional complex Lie algebras and 2) highest weight theory.

	Exams:	No exams

	Homework:	Written homework will be due approximately 6 times.

	Prerequisites:	M653 or equivalent and some familiarity with linear algebra.

Grading:

Your final grade will be based upon the graded HW assignments

Course Policies:

	Missed Work:	Making up missed work will be arranged according to University policies only. See Student Rule 7 for a description of university approved excuses: http://student-rules.tamu.edu/rule07

	Academic Dishonesty:	It is not permissible to hand in the work of others for a grade. You are allowed to discuss homework with others, but your write-ups are expected to be done on your own and in your own words. Copying the work of others may be prosecuted to the full extent possible under University policies. For additional information please visit: http://aggiehonor.tamu.edu “An Aggie does not lie, cheat, or steal, or tolerate those who do.”

	Disability Assistance	The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Disability Services, currently located in the Disability Services building at the Student Services at White Creek complex on west campus or call 979-845-1637. For additional information, visit http://disability.tamu.edu.

Page maintained by Eric Rowell, Dept. of Mathematics, Texas A&M University.

Math 662--Section 600 -- Fall 2016

Lie Algebras and Their Representations

TR 2:20-3:35pmBLOC 111

Homework: (turn in all problems).

Course Description:

Course Information:

Grading:

Course Policies:

TR 2:20-3:35pm
BLOC 111