8300: Quantum Groups Lecture Notes - Fall 2020

Notes will be posted here in the afternoons following the lecture.
In addition, here is a set of accumulating lecture notes, typeset by Henry Twiss.

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 -- Yang-Baxter equations imply that transfer matrices commute
Wednesday,
Jan. 29, 		Quantum Yang-Baxter Equations
#5
    Lecture 5 -- Quasitriangular Hopf algebra modules are a source of quantum Yang-Baxter Equations (QYBEs)
Friday,
Jan. 31, 		Conditions for QYBEs
#6
    Lecture 6 -- Using the six-vertex 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 self-dual.
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 Hopf-algebra 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 non-negative, 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

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