
Lecture Notes 
Date 
Topic 
#1 
Lecture 1  Lie groups, Lie algebras, and their universal enveloping algebras (UEAs) have closely related representation theory. UEAs admit a remarkable deformation  quantum groups.
 Wednesday, Jan. 22,
 Outline of the course 
#2 
Lecture 2  Partition functions of lattice models may be described via endomorphisms of tensor products of vector spaces.
 Friday, Jan. 24,
 Lattice model partition functions 
#3 
Lecture 3  To determine partition function, analyze one row partititon function (aka "transfer matrix")
 Monday, Jan. 27,
 Evaluating partition functions 
#4 
Lecture 4  YangBaxter equations imply that transfer matrices commute
 Wednesday, Jan. 29,
 Quantum YangBaxter Equations 
#5 
Lecture 5  Quasitriangular Hopf algebra modules are a source of quantum YangBaxter Equations (QYBEs)
 Friday, Jan. 31,
 Conditions for QYBEs 
#6 
Lecture 6  Using the sixvertex model to calculate the number of alternating sign matrices
 Monday, Feb. 3,
 Evaluating partition functions 
#7 
Lecture 7  Relations among symmetric functions are proved as identities of partition functions
 Wednesday, Feb. 5,
 Symmetric functions as partition functions 
#8 
Lecture 8  Bialgebras are compatible algebras and coalgebras
 Friday, Feb. 7,
 Bialgebras via commutative diagrams 
#9 
Lecture 9  Bialgebra modules are a monoidal category; Hopf algebra antipode maps are antimorphisms, unique.
 Monday, Feb. 10,
 Properties of Hopf algebras 
#10 
Lecture 10  We provide a Hopf algebra structure on a deformation of the universal envoloping algebra of the Borel subgroup of SL(2)
 Wednesday, Feb. 12,
 Dualities, Deformation of Borel 
#11 
Lecture 11  The quantum deformation of the Borel subalgebra is selfdual.
 Friday, Feb. 14,
 Dual Hopf algebras 
#12 
Lecture 12  Relaxing isomorphism condition in cocommutative Hopf algebras leads to quantum groups.
 Monday, Feb. 17,
 Quasitriangular Hopf algebras 
#13 
Lecture 13  Quasitriangular hopf algebras give abstract QYBEs
 Wednesday, Feb. 19,
 Searching for parentheses 
#14 
Lecture 14  Properties of U_q(sl_2) as a Hopf algebra, start of representation theory
 Friday, Feb. 21,
 U_q(sl_2) 
#15 
Lecture 15  U(sl_2) representation theory and Hopfalgebra modules
 Monday, Feb. 24,
 Sketch of U(sl_2) reps 
#16 
 Wednesday, Feb. 26,
 PBW theorem, automorphism 
#17 
Lecture 17  Finite U_q(sl_2) modules are sums of weight spaces
 Friday, Feb. 28,
 Simple modules of U_q(sl_2), Part I 
#18 
Lecture 18  The modules L(n,+) and L(n,), n nonnegative, are the finite dimensional simple modules of U_q(sl_2)
 Monday, Mar. 2,
 Simple modules of U_q(sl_2), Part II 
#19 
Lecture 19  Finite dimensional modules are semisimple
 Wednesday, Mar. 4,
 Simple modules of U_q(sl_2), Part III 
#20 
Lecture 20  Simple modules at roots of unity have bounded dimension
 Friday, Mar. 6,
 Simple modules of U_q(sl_2), Part IV 