IOWA STATE UNIVERSITY

Mathematics 680A2: Hopf Algebras

Instructor: Jonathan Smith, 496 Carver, 4-8172 (voice mail)

e-mail: jdhsmithATiastateDOTedu (substitute punctuation)

Office Hours: Mon. 10 - 10.50 am, 3.10 - 4 pm; Wed. 10 - 10.50 am; Fri. 10 - 10.50 am (subject to change).

Grading: based on five graded homework assignments (last due 12/7).

Click here for information about special accommodations.

Textbook: S. Majid, A Quantum Groups Primer, Cambridge, 2002. ISBN 0-521-01041-2.

Study Plan: reserve 1 - 2 hours for homework between each pair of classes. Successful performance in the class depends critically on completion of the homework assignments.

Communication devices must remain switched off during the class periods.

Additional Mathematics course policies

Syllabus: Fundamentals of the theory and application of Hopf algebras, corresponding to material selected from Chapters 1 - 23 of the text, along with various supplementary topics.

Assignments

12/7 for 12/9: Read Majid, first four pages of Chapter 21 (but skip Proposition 21.4).

Click here for the Final Graded Homework in Portable Document Format (due Wednesday, 12/7).

12/2 for 12/5: Read Majid, Example 10.3. [Take $\delta_a$ on line 2 of page 61.]

Click here for Graded Homework #4 in Portable Document Format (due Friday, 11/18).

11/16 for 11/18: Read Majid, Propositions 9.2 and 10.1.

11/14 for 11/16: Read Majid, Chapter 9 (but skip Proposition 9.2 for now).

Note the corrections:

  • The associators on the right-hand side of the first hexagon in Figure 9.2 should not have been inverted.
  • The bottom-left crossing on the right-hand side of the first equation in Figure 9.3(a) needs to be switched.

11/11 for 11/14: Read Majid, first 3 pages of Chapter 22 and of Chapter 23.

11/2 for 11/4: Read Majid, Lemmas 5.2, 5.3.

11/2 for 11/4: Read Majid, Definition 5.1 and Example 5.4.

Click here for Graded Homework #3 in Portable Document Format (due Friday, 10/28).

10/26 for 10/28: Read Majid, Example 3.7.

10/19 for 10/21: Read Majid, Definition 3.4 and Proposition 3.5.

10/17 for 10/19: Read Majid, Chapter 3 through page 19, skip Proposition 3.2 and Definition 3.4.

10/14 for 10/17: Read Majid, from Proposition 2.7 to end of Chapter 2.

10/12 for 10/14: Read Majid, (ii) and (iv) on page 14.

10/5 for 10/12: Read Majid, page 13 below end of proof of Proposition 2.5.

Click here for Graded Homework #2 in Portable Document Format (due Wednesday, 10/5).

9/28 for 9/30: Read Majid, last paragraph on page 22.

9/26 for 9/28: Read Majid, Examples 2.2, 2.3. Note the corrections  Sb = - b  and  Sc = - c  for Example 2.2.

9/26 for 9/28: Read Majid, Example 1.5.

9/23 for 9/26: Read Majid, Lemma 7.1 and middle of page 6.

9/16 for 9/19: Read top two-thirds of Majid, page 12.

Click here for Graded Homework #1 in Portable Document Format (due Monday, 9/19).

9/16 for 9/19: Read Majid, page 7.

9/12 for 9/14: Read Majid, Corollary 1.9.

9/7 for 9/9: Read Majid: Definition 1.3, Examples 1.6 and 2.1.
  1. Confirm that Example 1.6 still works for an infinite group G.
  2. Explain why Example 2.1 does not work for an infinite group G.
9/2 for 9/7:
  1. If x is primitive and 1 is setlike, verify coassociativity at x.
  2. Write out a proof of Majid, Lemma 8.3.
8/31 for 9/2: Read Majid, Definition 1.1 through Definition 1.2. Then, consider a vector space with both an algebra and a coalgebra structure. Show that these structures cooperate to form a bialgebra if and only if the multiplication and unit are coalgebra homomorphisms.

8/29 for 8/31: Read the displayed equations in Example 1.6.

8/26 for 8/29: Read the displayed equations in Example 1.7.

8/24 for 8/26: Let K be a field. Let X and Y be sets of indeterminates, with disjoint union X+Y.
  1. Show that K[X+Y] is the tensor product of K[X] and K[Y].
  2. Explain why taking the union of X and Y, rather than the disjoint union, doesn't work to give you the tensor product.
8/22 for 8/24: Read Majid, page 2. Then consider the field C of complex numbers, the real span of 1 and i, and the skewfield H of quaternions, the real span of 1, i, j and k with ij=k, etc.
  1. Use the embedding of C in H, along with the product in H, to make H into a complex vector space.
  2. Explain why the embedding of C in H, along with the product in H, does not make H into a C-algebra.