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Math 747 Introduction to Lie algebras. Spring 2023/2024.
• Math 747 Syllabus Syllabus

• Math 846: Association Schemes. Spring 2022/2023.
• Math 846 Syllabus Syllabus
• Lecture 1 W 1/25 Definition of association scheme and examples
• Lecture 2 F 1/27 More examples of association schemes
• Lecture 3 M 1/30 Algebraic characterization of association schemes
• Lecture 4 W 2/1 The primitive idempotents of a commutative association schemes
• Lecture 5 F 2/3 The first and second orthogonality relation
• Lecture 6 M 2/6 The Krein condition
• Lecture 7 W 2/8 The dual Bose-Mesner algebra and the subconstituent algebra
• Lecture 8 F 2/10 The triple product relations
• Lecture 9 M 2/13 The primary T-module
• Lecture 10 W 2/15 Duality and fusion
• Lecture 11 F 2/17 Primitivity
• Lecture 12 M 2/20 Primitivity, cont.
• Lecture 13 W 2/22 Subschemes
• Lecture 14 F 2/24 Quotient schemes
• Lecture 15 M 2/27 P-polynomial association schemes
• Lecture 16 W 3/1 Q-polynomial association schemes
• Lecture 17 F 3/3 Abelian group schemes are self-dual
• Lecture 18 M 3/6 The Hamming association schemes
• Lecture 19 W 3/8 The subconstituent algebra of a hypercube
• Lecture 20 F 3/10 The Johnson association scheme
• Lecture 21 M 3/20 Spherical representations
• Lecture 22 W 3/22 The balanced set condition
• Lecture 23 F 3/24 Balanced set characterization of the Q-polynomial property
• Lecture 24 M 3/27 The linear programming problem
• Lecture 25 W 3/29 Subsets of an association scheme
• Lecture 26 F 3/31 MacWilliams inequality
• Lecture 27 M 4/3 Codes in a P-polynomial scheme
• Lecture 28 W 4/5 Designs in a Q-polynomial scheme
• Lecture 29 F 4/7 The weight enumerator
• Lecture 30 4/10 Bounds on the degree and strength
• Lecture 31 4/12 Bounds on the degree and strength; case of equality
• Lecture 32 4/14 Bounds on the degree and strength; case of equality
• Lecture 33 4/17 Relative t-designs
• Lecture 34 4/19 Linear programming in the hypercube
• Lecture 35 4/21 Some open problems
• Appendix 4/21 Lectures 1 through 35
• Lecture 36 4/24 Ashwin Maran talk on quantum isomorphism of graphs
• Lecture 37A 4/26 Quantum isomorphism of graphs, notes by Austen Fan, Ashwin Maran, Benjamin Young
• Lecture 37B 4/26 Yanli Liu lecture on strongly-regular graphs
• Lecture 38A 4/28 Yiwen Bai lecture on classical t-designs, I
• Lecture 38B 4/28 Yuxiao Fu lecture on classical t-designs, II
• Lecture 39B 5/1 Ruocheng Yang lecture on the parameters of a code and the Hamming code
• Lecture 40A 5/3 Yangrong Zhao lecture on the weight enumerator of codes and dual codes
• Lecture 40B 5/3 Keru Zhou lecture on the binary Golay code
• Lecture 41 5/5 Hongyu Zhu lecture on the ternary Golay code

• Math 846: Algebraic Graph Theory. Spring 2021/2022.
• Math 846 Syllabus Syllabus
• Abbreviations List of Abbreviations
• Lecture 1 W 1/26 Graphs and their spectra; regular graphs; walks and paths
• Lecture 2 F 1/28 Bipartite graphs; the adjacency algebra and dual adjacency algebra; the subconstituent algebra T
• Lecture 3 M 1/31 Irreducible T-modules; T-module isomorphisms; some parameters attached to an irreducible T-module
• Lecture 4 W 2/2 The hypercube and its spectrum
• Lecture 5 F 2/4 The concept of a tridiagonal pair; using a hypercube to construct tridiagonal pairs that are bipartite and dual-bipartite
• Lecture 6 M 2/7 Hermitean and Euclidean spaces; inner product matrices; positive semidefinite matrices; bipartite and reducible matrices
• Lecture 7 W 2/9 The Perron-Frobenius theorem and applications to graphs
• Lecture 8 F 2/11 An eigenvalue bound; distance-regular graphs; the intersection numbers; examples of distance-regular graphs
• Lecture 9 M 2/14 Automorphisms of graphs; distance-transitivity implies distance-regularity; some polynomials attached to a distance-regular graph
• Lecture 10 W 2/16 The intersection matrix; the Norman Biggs multiplicity formula
• Lecture 11 F 2/18 More intersection numbers; some orthogonality relations for polynomials
• Lecture 12 M 2/21 The intersection numbers are determined by the spectrum; the geometry of the eigenspaces; the cosine sequence
• Lecture 13 W 2/23 The representations of a distance-regular graph; the Krawtchouk polynomials
• Lecture 14 F 2/25 The Krein parameters; the dual distance matrices
• Lecture 15 M 2/28 The Krein condition and an application.
• Lecture 16 W 3/2 A geometric interpretation of the Krein parameters; Norton algebras
• Lecture 17 F 3/4 The Q-polynomial property and the dual adjacency matrix; some polynomials attached to the dual adjacency matrix; Askey-Wilson duality
• Lecture 18 M 3/7 A geometric interpretation of the Q-polynomial property
• Lecture 19 W 3/9 The tridiagonal relations
• Lecture 20 F 3/11 Proof of the tridiagonal relations
• Lecture 21 M 3/21 Proof of the tridiagonal relations, cont.
• Lecture 22 W 3/23 The eigenvalues and dual eigenvalues in closed form; the bipartite/dual-bipartite case
• Lecture 23 F 3/25 The bipartite/dual-bipartite case; formulas for the intersection numbers
• Lecture 24 M 3/28 The bipartite/dual bipartite case; the tridiagonal relations in Z3-symmetric form
• Lecture 25 W 3/30 The bipartite/dual bipartite case; type II matrices and spin models
• Lecture 26 F 4/1 The bipartite/dual-bipartite case; the Nomura classification
• Lecture 27 M 4/4 The bipartite/dual-bipartite case; Hadamard matrices and graphs
• Lecture 28 W 4/6 The bipartite/dual-bipartite case; the double cover of the Higman-Sims graph
• Lecture 29 F 4/8 Tridiagonal pairs from a Q-polynomial distance-regular graph; nonisomorphic irreducible T-modules are orthogonal; the primary T-module
• Lecture 30 M 4/11 The primary T-module; reduction rules and bases
• Lecture 31 W 4/13 The concept of a Leonard pair; a Leonard pair on the primary T-module; the Askey-Wilson relations
• Lecture 32 F 4/15 The Askey-Wilson relations attached to the primary T-module
• Lecture 33 M 4/18 Pascasio characterization of the Q-polynomial property
• Lecture 34 W 4/20 Distance-regular graphs with classical parameters
• Lecture 35 F 4/22 Balanced set characterization of the Q-polynomial property
• Lecture 36 M 4/25 The triple intersection numbers
• Lecture 37 W 4/27 Some relations involving the raising, lowering, and flat maps that are implied by the Q-polynomial property; the split decomposition
• Lecture 38 F 4/29 More on the split decomposition; bounds on the endpont and dual endpoint of an irreducible T-module
• Lecture 39 M 5/2 Tridiagonal pairs and tridiagonal systems, the split decomposition of a tridiagonal system
• Lecture 40 W 5/4 The raising and lowering maps with respect to the split decomposition, the tetrahedron diagram, four mutually opposite flags
• Lecture 41 F 5/6 A q-geometric tridiagonal pair gives an irreducible module for the q-tetrahedron algebra
• Lecture 42 Appendix How the q-tetrahedron algebra is related to Uqsl2hat
• Summary slides Tridiagonal pairs and Uqsl2hat
• Amin Idelhaj Lecture Ramanujan graphs
• Charles Wang Lecture The diameter of bipartite distance-regular graphs
• Yanli Liu Lecture 2-Homogeneous bipartite DRGs Section 3
• Kenneth Ma Lecture 2-Homogeneous bipartite DRGs Section 4
• Karthik Ravishankar Lecture 2-Homogeneous bipartite DRGs Section 5
• Daniel Szabo Lecture 2-Homogeneous bipartite DRGs Section 6
• Jiaming Xu Lecture 2-Homogeneous bipartite DRGs Section 7
• Benjamin Young Lecture 2-Homogeneous bipartite DRGs Section 8
• Yufei Zhan Lecture 2-Homogeneous bipartite DRGs Section 9

• Math 542: Modern Algebra, Spring 2016
• Math 542 Syllabus Syllabus
• Math 542 Homework Homework (To be updated as we go along)
• Lecture 1, W 1/20 Rings of fractions
• Lecture 2, F 1/22 The field of fractions
• Lecture 3, M 1/25 The Chinese remainder theorem; example
• Lecture 4, W 1/27 The Chinese remainder theorem
• Lecture 5, F 1/29 Euclidean domains
• Lecture 6, M 2/1 Euclidean domains and universal side divisors
• Lecture 7, W 2/3 Principal ideal domains
• Lecture 8, F 2/5 Unique factorization domains
• Lecture 9, M 2/8 A PID is a UFD
• Lecture 10, W 2/10 Primes in the Gaussian integers
• Lecture 11, F 2/12 Expressing an integer as the sum of two squares
• Lecture 12, M 2/15 Polynomial rings
• Lecture 13, W 2/17 The Gauss lemma
• Lecture 14, F 2/19 Irreducibility criteria
• Lecture 15, M 2/22 The Eisenstein irreducibility condition
• Lecture 16, W 2/24 For a finite field the group of units is cyclic
• Lecture 17, F 2/26 The group of units for Z mod N
• Lecture 18, M 2/29 Modules for rings
• Lecture 19, W 3/2 Homomorphisms of R-modules
• Lecture 20, F 3/4 Quotients of R-modules
• Lecture 21, M 3/7 Direct products of R-modules
• Lecture 22, W 3/9 Free R-modules
• Lecture 23, F 3/11 Vector spaces
• Lecture 24, M 3/14 Vector spaces and linear transformations
• Midterm W 3/16 Midterm exam questions
• Midterm Solutions W 3/16 Midterm exam solutions
• Lecture 25, F 3/18 Transition matrices
• Lecture 26, M 3/28 The tensor product of vector spaces
• Lecture 27, W 3/30 The dual of a vector space
• Lecture 28, F 4/1 The transpose of a linear transformation
• Lecture 29, M 4/4 Determinants
• Lecture 30, W 4/6 Properties of the determinant
• Lecture 31, F 4/8 The inverse of a matrix
• Lecture 32, M 4/11 Modules over a PID
• Lecture 33, W 4/13 Torsion modules
• Lecture 34, F 4/15 Rank and torsion
• Lecture 35, M 4/18 Submodules of free modules over a PID
• Lecture 36, W 4/20 Submodules of free modules over a PID, cont.
• Lecture 37, F 4/22 Finitely generated modules over a PID
• Lecture 38, M 4/25 Elementary divisors and invariant factors
• Lecture 39, W 4/27 The rational canonical form
• Lecture 40, F 4/29 The Smith normal form
• Lecture 41, M 5/2 The Jordan canonical form
• Lecture 42, W 5/4 Examples involving the Jordan canonical form
• Lecture 43, F 5/6 Finding the eigenvalues of some tridiagonal matrices using hidden symmetry
• Final Exam, S 5/8 Final exam questions
• Final Exam Solutions, S 5/8 Final exam solutions
Math 320: Linear Algebra and Differential Equations, Spring 2014.
• Math 320 Syllabus Syllabus
• Lecture 1, W 1/22 Section 1.1
• Lecture 2, F 1/24 Section 1.2
• Lecture 3, M 1/27 Section 1.3
• Lecture 4, W 1/29 Section 1.4
• Lecture 5, F 1/31 Section 1.5
• Lecture 6, M 2/3 Section 2.1
• Lecture 7, W 2/5 Section 2.2
• Lecture 8, F 2/7 Section 2.4
• Lecture 9, M 2/10 Section 3.1
• Lecture 10, W 2/12 Section 3.2
• Lecture 11, F 2/14 Section 3.3
• Lecture 12, M 2/17 Section 3.4
• Lecture 13, W 2/19 Section 3.5
• Lecture 14, F 2/21 Sections 3.5,3.6
• Lecture 15, M 2/24 Section 3.6
• Lecture 16, W 2/26 Section 3.6 (loose ends)
• Lecture 17, M 3/3 Section 4.1
• Lecture 18, W 3/5 Section 4.2
• Lecture 19, F 3/7 Section 4.3
• Lecture 20, M 3/10 Section 4.4
• Lecture 21, W 3/12 Section 4.5
• Lecture 22, F 3/14 Section 4.6
• Lecture 23, M 3/24 Section 4.7
• Lecture 24, W 3/26 Section 5.1
• Lecture 25, F 3/28 Section 5.2
• Lecture 26, M 3/31 Section 5.3
• Lecture 27, W 4/2 Section 5.5
• Lecture 28, F 4/4 Section 6.1
• Lecture 29, M 4/7 Section 6.2
• Lecture 30, W 4/9 Section 6.3
• Lecture 31, F 4/11 Section 6.3 (loose ends)
• Lecture 32, W 4/16 Section 7.1
• Lecture 33, F 4/18 Section 7.2
• Lecture 34, M 4/21 Section 7.3
• Lecture 35, W 4/23 Section 7.5
• Lecture 36, F 4/25 Section 7.5 (continued)
• Lecture 37, M 4/28 Section 8.1
• Lecture 38, W 4/30 Section 8.1 (continued)
• Lecture 39, F 5/2 Section 8.2
• Lecture 40, M 5/5 Section 8.2 (continued)
• Lecture 41, W 5/7 Section 8.3
• Lecture 42, F 5/9 Section 8.3 (continued)
Math 210: What follows is a list of Math 210 Homework solutions.
• Math 210 syllabus for Fall 2012
• Typos contains a list of typos in the text "Finite Mathematics" 5th Edition, by Maki and Thompson.
• Section 1.1 HW solutions
• Section 1.2 HW solutions
• Section 1.3 HW solutions
• Section 1.4 HW solutions
• Section 1.R HW solutions
• Section 2.1 HW solutions
• Section 2.2 HW solutions
• Section 2.3 HW solutions
• Section 2.4 HW solutions
• Section 2.R HW solutions
• Section 3.1 HW solutions
• Section 3.2 HW solutions
• Section 3.3 HW solutions
• Section 3.4 HW solutions
• Section 3.5 HW solutions
• Section 3.R HW solutions
• Section 4.1 HW solutions
• Section 4.2 HW solutions
• Section 4.R HW solutions
• Section 5.1 HW solutions
• Section 5.2 HW solutions
• Section 5.3 HW solutions
• Section 5.R HW solutions
• Section 6.1 HW solutions
• Section 6.2 HW solutions
• Section 6.R HW solutions
• Section 7.1 HW solutions
• Section 7.2 HW solutions
• Section 7.3 HW solutions
• Section 7.R HW solutions
• Section 8.1 HW solutions
• Section 8.2 HW solutions
• Section 8.3 HW solutions
• Section 8.R HW solutions
• Section 9.1 HW solutions
• Section 9.2 HW solutions
• Section 9.3 HW solutions
Some of my recent talks are given below.
• ndpt.pdf The nucleus of a Q-polynomial distance-regular graph (50 minutes)
terwilliCMS2024.pdf The S3-symmetric tridiagonal algebra (20 minutes)
• terwilMilw2024.pdf The Norton-balanced condition for Q-polynomial distance-regular graphs (20 minutes)
• tatt.pdf Generalizing the Q-polynomial property to graphs that are not distance-regular (50 minutes)
• spt.pdf Spin models and distance-regular graphs of q-Racah type (50 minutes)
• terwillBled2023.pdf The Z3-symmetric down-up algebra (20 minutes)
• talkMilw2023.pdf The q-Shuffle algebra, the alternating elements, and the positive part of quantum affine sl2 (45 minutes)
• slcTerwilliger.pdf A Q-polynomial structure associated with the projective geometry LN(q) (25 minutes)
• terAltTDDRG.pdf Tridiagonal pairs, alternating elements, and distance-regular graphs (50 minutes)
• noncom.pdf Tridiagonal pairs and the quantum affine sl2 algebra (50 minutes)
• cclp.pdf Compatibility and companions for Leonard pairs (50 minutes)
• ta.pdf The alternating central extension of the q-Onsager algebra (50 minutes)
• talkField2021.pdf Q-polynomial distance-regular graphs and the positive part of the quantum affine sl2 algebra (45 minutes)
• sl.pdf Leonard pairs, spin models, and distance-regular graphs (60 minutes)
• acext.pdf The alternating central extension for the positive part of the quantum affine sl2 algebra
• spinBled2019.pdf Leonard pairs, spin models, and distance-regular graphs (20 minutes)
• terwilli2019.pdf The alternating PBW basis for the positive part of the quantum affine sl2 algebra
• catalan2018.pdf Catalan words and q-shuffle algebras
• infdim.pdf An infinite-dimensional Square_q module obtained from the q-shuffle algebra for affine sl2
• lusztigAutTalk2018.pdf The Lusztig automorphisms of the q-Onsager algebra
• tdpairs2018.pdf Tridiagonal pairs and applications
• tbtdpair.pdf Totally bipartite tridiagonal pairs
• eltalk.pdf Tridiagonal pairs of q-Racah type, the Bockting operator, and L-operators for Uq(L(sl2))
• www2016.pdf Leonard triples of q-Racah type; Combinatorics seminar talk Fall 2016
• lrtriple60min.pdf Lowering-Raising triples and Uq(sl2); Colloquium talk at DePaul U. 2016
• skeinUAW2015.pdf Topological aspects of the Z3-symmetric Askey-Wilson relations
• lrtTalk2014.pdf Lowering-Raising triples and Uq(sl2)
• billiard.pdf Billiard Arrays and finite-dimensional irreducible Uqsl2-modules
• tdpInAlgGrTh.pdf Tridiagonal pairs in Algebraic Graph Theory
• lpAndQtet.pdf Leonard pairs and the q-tetrahedron algebra
• univAWalgebra.pdf The universal Askey-Wilson algebra
• classTDpair.pdf A classification of sharp tridiagonal pairs
• rahmanSl3.pdf The Rahman polynomials and the Lie algebra sl3
The pdf file for the following paper is given below. P. Terwilliger. The incidence algebra of a uniform poset. Coding theory and design theory Part I. 193--212 IMA Vol. Math. Appl. 20 Springer New York, 1990.
• uniformposet.pdf
The pdf file for the following paper is given below. P. Terwilliger. Leonard pairs and dual polynomial sequences. The paper was submitted to LAA on May 15, 1987, but never published.
• leonardpair.pdf
The pdf file for the following paper is given below. Counting 4-vertex configurations in P and Q polynomial association schemes. Algebras, Groups, and Geometries 2 (1985) 541--554.
• counting4vertex.pdf
The following contains the course notes from Math 846 Algebraic Graph Theory, Spring term 2009.
• Part 1 846 notes
• Part 2 846 notes
• Part 3 846 notes
• Part 4 846 notes
• Part 5 846 notes
• Part 6 846 notes
The following contains the notes from six lectures on distance-regular graphs given at De La Salle University in Manila, from May 24 to June 5, 2010.
• Lecture 1 Manila notes
• Lecture 2 Manila notes
• Lecture 3 Manila notes
• Lecture 3 appendix Manila notes
• Lecture 4 Manila notes
• Lecture 5 Manila notes
• Lecture 6 Manila notes
The following contains the course notes from Math 805 Special Functions, Fall term 2010. The course was about the Askey scheme of orthogonal polynomials. We focussed on the terminating branch and its relationship to Leonard pairs.
• Part 1 Intro/Hermite polynomials
• Part 2 General orthogonal polynomials
• Part 3 The Kratchouk polynomials
• Part 4 The Krawtchouk polynomials cont.
• Part 5 Leonard systems and their classification
• Part 6 The terminating branch of the Askey scheme via Leonard systems
• Part 7 Leonard systems of q-Racah type
The following contains the course notes from Math 846 The Double Affine Hecke Algebra, Fall term 2011. The course was about the DAHA of rank 1. We focussed on the ring theoretic aspects as well as the connection to the Askey-Wilson polynomials.
• Part 1 Intro/linear basis for DAHA
• Part 2 DAHA and the universal Askey-Wilson algebra
• Part 3 The centralizer of the generator t0
• Part 4 The center of DAHA
• Part 5 Modules for DAHA
• Part 6 Connection to the Askey-Wilson polynomials.
• Part 7 The action of DAHA on the space of Laurent polynomials in one variable
The following contains the course notes from Math 475: Combinatorics, for Fall term 2012. The textbook was: Richard Brualdi. Introductory Combinatorics, 5th Ed.
• Math 475 Syllabus Syllabus
• Chapters 1,2 Permutations and Combinations
• Chapter 3 The Pigeonhole Principle
• Chapter 4 Generating Permutations and Combinations
• Chapter 5 The Binomial Coefficients
• Chapter 6 The Inclusion-Exclusion Principle and Applications
• Chapter 7 Recurrence Relations and Generating Functions
• Chapter 8 Special Counting Sequences
• Chapter 14 Polya Counting
The following contains the course notes from Math 846: Tridiagonal Pairs and Related Topics, for Fall term 2013.
• Lecture 1, W 9/4 Introduction
• Lecture 2, F 9/6 TD pairs and graphs
• Lecture 3, M 9/9 The dual adjacency matrix
• Lecture 4, W 9/11 The path graph
• Lecture 5, F 9/13 Distance-regularity with respect to a vertex
• Lecture 6, M 9/16 The hypercubes
• Lecture 7, W 9/18 The hypercubes and sl2
• Lecture 8, F 9/20 The hypercubes and the tetrahedron algebra
• Lecture 9, M 9/23 The equitable basis for sl2
• Lecture 10, W 9/25 Evaluation modules for the tetrahedron algebra
• Lecture 11, F 9/27 The Dirac basis for sl2
• Lecture 12, M 9/30 The Drinfeld polynomial
• Lecture 13, W 10/2 The Krein parameters of a DRG
• Lecture 14, F 10/4 The Krein parameters of a DRG, cont.
• Lecture 15, M 10/7 TD pairs and Leonard pairs; preliminaries
• Lecture 16, W 10/9 Totally bipartite TD pairs
• Lecture 17, F 10/11 Totally bipartite TD pairs, cont.
• Lecture 18, M 10/14 Totally bipartite TD pairs, cont.
• Lecture 19, W 10/16 Totally bipartite TD pairs and Uq(sl2)
• Lecture 20, F 10/18 The equitable presentation of Uq(sl2)
• Lecture 21, M 10/21 The split decomposition of a TD system
• Lecture 22, W 10/23 The shape vector and the tetrahedron diagram
• Lecture 23, F 10/25 The tridiagonal relations
• Lecture 24, M 10/28 Double lowering maps
• Lecture 25, W 10/30 Double lowering maps, cont.
• Lecture 26, F 11/1 Double lowering maps and the q-exponential function
• Lecture 27, M 11/4 Double lowering maps; the operators K,B,M.
• Lecture 28, W 11/6 Double lowering maps; the classification
• Lecture 29, F 11/8 Double lowering maps and recurrent sequences
• Lecture 30, M 11/11 Leonard pairs of q-Racah type
• Lecture 31, W 11/13 The algebra Uq(sl2)
• Lecture 32, F 11/15 Rotators for Uq(sl2) modules
• Lecture 33, M 11/25 Rotators and Lusztig operators
• Lecture 34, W 11/27 The dual space of a Uq(sl2)-module
• Lecture 35, W 12/4 Uq(sl2)-modules from the equitable point of view
• Lecture 36, F 12/6 Uq(sl2)-modules from the equitable point of view, cont.
• Lecture 37, M 12/9 Twelve bases for an irreducible Uq(sl2)-module
• Lecture 38, W 12/11 Irreducible Uq(sl2)-modules; the bilinear form
• Lecture 39, F 12/13 Irreducible Uq(sl2)-modules; transition matrices
The following contains the course notes from Math 846: Quantum groups and Hopf algebras, for Fall term 2015. We used the textbook by Kassel.
• Math 846 Syllabus Syllabus
• Lecture 1, W 9/2 Introduction
• Lecture 2, F 9/4 First examples
• Lecture 3, W 9/9 M(2) and GL(2)
• Lecture 4, F 9/11 SL(2) and gradings
• Lecture 5, M 9/14 Filtrations
• Lecture 6, W 9/16 Ore extensions
• Lecture 7, F 9/18 Noetherian rings
• Lecture 8, M 9/21 Tensor products
• Lecture 9, W 9/23 Tensor products and Hom
• Lecture 10, F 9/25 Free A-modules
• Lecture 11, M 9/28 Dual vector spaces
• Lecture 12, W 9/30 The transpose map
• Lecture 13, F 10/2 tensor products of algebras
• Lecture 14, M 10/5 tensor algebras and free algebras
• Lecture 15, W 10/7 Coalgebras
• Lecture 16, F 10/9 Algebras and coalgebras
• Lecture 17, M 10/12 Coalgebra morphisms
• Lecture 18, W 10/14 Bialgebras
• Lecture 19, F 10/16 Primitive elements
• Lecture 20, M 10/19 The antipode S
• Lecture 21, W 10/21 Hopf algebras
• Lecture 22, F 10/23 Some Hopf algebra morphisms
• Lecture 23, M 10/26 The Hopf algebras GL(2), SL(2)
• Lecture 24, W 10/28 Hopf algebra modules
• Lecture 25, F 10/30 Morphisms for Hopf algebra modules
• Lecture 26, M 11/2 Comodules
• Lecture 27, W 11/4 Examples of comodules
• Lecture 28, F 11/6 H-comodule algebras
• Lecture 29, M 11/9 The quantum plane
• Lecture 30, W 11/11 The q-determinant
• Lecture 31, F 11/13 The Hopf algebras GLq(2), SLq(2)
• Lecture 32, M 11/16 Lie algebras
• Lecture 33, W 11/18 H-module algebras
• Lecture 34, F 11/20 Bialgebra dualities
• Lecture 35, M 11/30 A duality between U(sl2) and M(2)
• Lecture 36, M 11/23 The algebra Uq(sl2) (talk by Mao Li)
• Lecture 37, M 11/23 The Yang-Baxter equation (talk by Zhaochen Wang)
• Lecture 38, W 11/25 Modules for Uq(sl2) (talk by Zheng Lu)
• Lecture 39, W 12/2 The Hopf algebra Uq(sl2) (talk by Jason Steinberg)
• Lecture 40, F 12/4 Uq(sl2) and the quantum plane (talk by Dongxi Ye)
• Lecture 41, M 12/7 A duality between Uq(sl2) and Mq(2)
• Lecture 42, W 12/9 A duality between Uq(sl2) and SLq(2)
• Lecture 43, F 12/11 A bilinear form for Uq(sl2)-modules
• Lecture 44, M 12/14 Quantum groups in statistical mechanics (talk by Hans Chaumont)
• Lecture 45, M 12/14 The equitable presentation of Uq(sl2)
The following contains the course notes from Math 846: Hopf algebras in Combinatorics, Fall 2016.
• Math 846 Syllabus Syllabus
• Lecture 1, W 9/7/16 Introduction
• Lecture 2, F 9/9/16 coalgebras
• Lecture 3, M 9/12/16 bialgebras
• Lecture 4, W 9/14/16 bialgebra examples
• Lecture 5, F 9/16/16 Gradings
• Lecture 6, M 9/19/16 The antipode
• Lecture 7, W 9/21/16 Hopf algebra examples
• Lecture 8, F 9/23/16 The Takeuchi formula
• Lecture 9, M 9/26/16 Subcoalgebras
• Lecture 10, W 9/28/16 Primitive elements
• Lecture 11, F 9/30/16 Lie algebras and Hopf algebras
• Lecture 12, M 10/3/16 Commutative Hopf algebras
• Lecture 13, W 10/5/16 Duality
• Lecture 14, F 10/7/16 The shuffle algebra
• Lecture 15, M 10/10/16 The shuffle algebra, cont.
• Lecture 16, W 10/12/16 Symmetric functions
• Lecture 17, F 10/14/16 The Hopf algebra structure
• Lecture 18, M 10/17/16 The dominance order and applications
• Lecture 19, W 10/19/16 Schur functions
• Lecture 20, F 10/21/16 Schur functions and column strict tableaux
• Lecture 21, M 10/24/16 Skew Schur functions
• Lecture 22, W 10/26/16 A generating function
• Lecture 23, F 10/28/16 The power symmetric functions revisited
• Lecture 24, M 10/31/16 The Hall inner product
• Lecture 25, W 11/2/16 An alternate view of Schur funcions
• Lecture 26, F 11/4/16 An alternate view of Schur funcions, cont.
• Lecture 27, M 11/7/16 Littlewood-Richardson coefficients
• Lecture 28, W 11/9/16 The Pieri rule
• Lecture 29, F 11/11/16 The skew Pieri rule
• Lecture 30, M 11/14/16 Positive self dual Hopf algebras
• Lecture 31, W 11/16/16 Primitive elements and symmetric algebras
• Lecture 32, F 11/18/16 Primitive elements in a PSH
• Lecture 33, M 11/21/16 Decomposing a PSH into Hopf subalgebras
• Lecture 34, W 11/23/16 PSH algebras and symmetric functions
• Lecture 35, M 11/28/16 PSH algebras and symmetric functions, cont.
• Lecture 36, W 11/30/16 Zhaochen Wang lecture
• Lecture 37, F 12/2/16 Shuai Shao lecture
• Lecture 38, M 12/5/16 David Wanger lecture
• Lecture 39, W 12/7/16 Michael Powers lecture
• Lecture 40, F 12/9/16 Hang Huang lecture
• Lecture 41, M 12/12/16 Rebecca Eastham lecture
• Lecture 42, W 12/14/16 Benjamin Branman lecture
The following contains the course notes from Math 846: Introduction to quantum groups, Spring 2019. We use the book by Janzten.
• Lecture 1, W 1/23/19 A basis for Uq(sl2)
• Lecture 2, F 1/25/19 The Casimir element C for Uq(sl2)
• Lecture 3, M 1/28/19 The center of Uq(sl2)
• Lecture 4, W 1/30/19 Highest weight modules for Uq(sl2)
• Lecture 5, F 2/1/19 The fd irred Uq(sl2)-modules
• Lecture 6, M 2/4/19 When is a fd Uq(sl2)-module semi simple?
• Lecture 7, W 2/6/19 Some infinite diml Uq(sl2)-modules
• Lecture 8, F 2/8/19 The quantum adjoint action for Uq(sl2)
• Lecture 9, M 2/11/19 The Harish-Chandra map for Uq(sl2)
• Lecture 10, W 2/13/19 The coproduct and counit for Uq(sl2)
• Lecture 11, F 2/15/19 The antipode for Uq(sl2)
• Lecture 12, M 2/18/19 Hopf algebras
• Lecture 13, W 2/20/19 Some canonical maps
• Lecture 14, F 2/22/19 The quantum trace
• Lecture 15, M 2/25/19 More canonical maps
• Lecture 16, W 2/27/19 The R matrix
• Lecture 17, F 3/1/19 The R matrix, continued
• Lecture 18, M 3/4/19 Link diagrams and the R matrix
• Lecture 19, W 3/6/19 L-operators for Uq(sl2)
• Lecture 20, F 3/8/19 Finite diml semisimple Lie algebras g
• Lecture 21, M 3/11/19 The quantum group Uq(g)
• Lecture 22, W 3/13/19 The square of the antipode S for Uq(g)
• Lecture 23, F 3/15/19 The square of the antipode S for Uq(g), cont.
• Lecture 24, M 3/25/19 The quantum adjoint action
• Lecture 25, W 3/27/19 A basis for tilde Uq
• Lecture 26, F 3/29/19 The map tilde Uq ->Uq
• Lecture 27, M 4/1/19 The triangular decomposition of Uq
• Lecture 28, W 4/3/19 Representations of Uq
• Lecture 29, F 4/5/19 The quantum trace for Uq
• Lecture 30, M 4/8/19 The Uq-modules M(lambda)
• Lecture 31, W 4/10/19 the Uq-modules L(lambda)
• Lecture 32, F 4/12/19 fd irred Uq-modules
• Lecture 33, M 4/15/19 classification of fd irred Uq-modules
• Lecture 34, W 4/17/19 The braid group action
• Lecture 35, F 4/19/19 The braid group action, cont.
• Lecture 36, M 4/22/19 Kevin Kristensen talk
• Lecture 37, W 4/24/19 Peter Nielsen talk
• Lecture 38, F 4/26/19 Dadong Peng talk
• Lecture 39, M 4/29/19 Peter Ruan talk
• Lecture 40 W 5/1/19 Ben Branman talk
• Lecture 41 F 5/3/19 Seth Davis talk
The following contains the course notes from Math 846: Crystal Bases in Algbraic Combinatorics, Fall 2019. We use the book by Bump and Schilling.
• Lecture 1, W 9/4/19 Root systems
• Lecture 2, F 9/6/19 The simple roots
• Lecture 3, M 9/9/19 The Weyl group
• Lecture 4, W 9/11/19 The simple root systems
• Lecture 5, F 9/13/19 Kashiwara crystals
• Lecture 6, M 9/16/19 Examples of crystals
• Lecture 7, W 9/18/19 More examples of crystals
• Lecture 8, F 9/20/19 The tensor product of crystals
• Lecture 9, M 9/23/19 Morphisms of crystals
• Lecture 10, W 9/25/19 The signature rule
• Lecture 11, F 9/27/19 The character of a crystal
• Lecture 12, M 9/30/19 Dynkin diagrams and Levi branching
• Lecture 13, W 10/2/19 Crystals from tableaux
• Lecture 14, F 10/4/19 Crystals from tableaux, cont.
• Lecture 14 Artwork, F 10/4/19 Some crystal artwork
• Lecture 15, M 10/7/19 Crystals and Schur functions
• Lecture 16, W 10/9/19 Crystals of type A2
• Lecture 17, F 10/11/19 Stembridge crystals
• Lecture 18, M 10/14/19 Virtual crystals
• Lecture 19, W 10/16/19 Virtual crystals, cont.
• Lecture 20, F 10/18/19 Examples of virtual crystals
• Lecture 21, M 10/21/19 Properties of virtual crystals
• Lecture 22, W 10/23/19 Properties of virtual crystals, cont.
• Lecture 23, F 10/25/19 Miniscule crystals
• Lecture 24, M 10/28/19 The root systems E6, E7, E8, F4
• Lecture 25, W 10/30/19 Characterizing miniscule crystals
• Lecture 26, F 11/1/19 Adjoint crystals
• Lecture 27, M 11/4/19 Normal crystals
• Lecture 28, W 11/6/19 An involution for normal crystals
• Lecture 29, F 11/8/19 The crystal Binfinity
• Lecture 30, M 11/11/19 Tableaux of type Br, Cr, Dr
• Lecture 31, W 11/13/19 Tableaux of type Br, Cr, Dr, cont.
• Lecture 32, F 11/15/19 Schensted insertion
• Lecture 33, M 11/18/19 The RSK correspondance
• Lecture 34, W 11/20/19 Extended RSK
• Lecture 35, F 11/22/19 The box/ball view
• Lecture 36, M 11/25/19 Dual RSK
• Lecture 37, W 11/27/19 Dual Schensted insertion
• Lecture 38, M 12/2/19 The plactic monoid (Edwin Baeza and Owen Goff)
• Lecture 39, W 12/4/19 Crystals and Schensted insertion (Jiaqi Hou)
• Lecture 39A, W 12/4/19 Crystals and Schensted insertion, cont. (Changhun Jo)
• Lecture 40, F 12/6/19 Crystals of skew tableaux (Yunxuan Li)
• Lecture 40A, F 12/6/19 Levi branching (Peter Ruan)
• Lecture 41, M 12/9/19 Bicrystals (Ian Seong and Feng Sheng)
• Lecture 42, W 12/11/19 Intro to See-Saw (Yu Sun)
• Lecture 42A, W 12/11/19 Littlewood-Richardson coefficients (Evan Sorensen)