Tenth UCB/UCSB Algebra Day Saturday, December 4, 2010 939 Evans Hall, UCB PROGRAM ******* 8:45- 9:20 Arrival, Refreshments (989 Evans) ********************************* 9:20- 9:30 Welcome and Introductions (939 Evans) 9:30-10:10 Ken Goodearl: "Why do Cauchon diagrams connect real, complex, and quantum matrices?" 10:20-11:00 Karel Casteels: "Determining primitivity of H-prime ideals in quantum matrices" 11:10-11:50 Garrett Johnson: "Quantum Affine Algebras as Smash Products" 12:00-12:40 Adrian Wadsworth: "Valuations on division algebras and their associated graded rings" 12:50- 1:50 Lunch: Food Court on Hearst Ave (weather permitting) **************************************************** 2:00- 2:40 Birge Huisgen-Zimmermann: "Degenerations of finite- dimensional representations" 2:50- 3:30 T. Y. Lam: "Rings of Square Stable Range One" 3:30- 4:00 Coffee Hour (989 Evans) *********** *********** 4:00- 4:40 Tom Howard: "Complexity as a homological invariant" 4:50- 5:30 Monica Vazirani: "Simple KLR modules" 6:30- 9:00 Dinner: Restaurant: TBA ******* *************** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Abstracts of Talks ================== Speaker: Adrian Wadsworth (UCSD) Title: Valuations on division algebras and their associated graded rings Abstract: In studying valuations on division algebras, the associated graded division rings have been a valuable tool. The graded ring has a simpler structure than the valued division ring, but preserves many significant properties. We illustrate this with calculations of SK_1 of division algebras over Henselian valued fields. ================================================================ Speaker: Monica Vazirani (UCD) Title: Simple KLR modules Abstract: Khovanov-Lauda-Rouquier (KLR) algebras have played a fundamental role in categorifying quantum groups. I will discuss the structure of their simple modules, in particular that they carry the structure of a crystal graph. This is joint work with Aaron Lauda. =============================================================== Speaker: Tom Howard (UCSB) Title: Complexity as a homological invariant Abstract: Let M be a finite dimensional module over a finite dimensional algebra A. The complexity of M measures the extent to which M fails to have finite projective dimension by recording the rate at which terms in a minimal projective resolution grow. Three questions arise. 1) How can we compute complexity? 2) Which values of complexity can occur? 3) What good is knowing complexities? Last year I discussed the first question for a large class of algebras. This year I will describe which values of complexity may occur for this class of algebras, and will promote the theory of complexity as a source of homological invariants. ============================================================== Speaker: Karel Casteels (UCSB) Title: Determining primitivity of H-prime ideals in quantum matrices Abstract: By results of Goodearl and Letzter, it is known the the prime spectrum of the algebra of quantum matrices may be partitioned into finitely many "H-strata", where H is an algebraic torus acting on the algebra. Moreover, each H-stratum is homeomorphic to a Laurent polynomial ring, and contains a unique prime ideal invariant under the action of H (an H-prime ideal). In this talk, I'll show how to use a combinatorial diagram (known in the literature as a "Cauchon diagram" or "le-diagram") to calculate the Krull dimension of a given H-stratum by associating a certain permutation to its corresponding Cauchon diagram. It is then a simple matter of counting the number of even cycles in the permutation. In particular, an H-prime is primitive precisely when there are no even cycles. This is joint work with Jason Bell and Stephane Launois. ============================================================= Speaker: T. Y. Lam (UCB) Title: Rings of square stable range one Abstract: Let S be a (von Neumann) regular ring. Then S is unit-regular iff S has stable range 1. Can we recognize the strongly regular rings also by a suitable stable range condition? And what if we replace the class of regular rings by the (broader) class of exchange rings? We'll solve these problems by introducing a new stable range on rings, called the "square stable range". Surprisingly, the idea of this new stable range hails from a far corner in linear algebra. This is joint work with D. Khurana and Z. Wang. ============================================================ Speaker: Garrett Johnson (UCSB) Title: Quantum affine algebras as smash products Abstract: De Concini, Kac, and Procesi defined a family of subalgebras of a quantized universal enveloping algebra U_q(g) associated to elements of the corresponding Weyl group of g. Some examples for the finite Weyl groups include the algebras of quantum matrices, quantum affine space, and quantum Euclidean space. I will describe a smash product construction that produces DKP algebras associated to affine Weyl groups from some of the finite-type DKP algebras listed above. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Algebra Day Participants ************************ >From UCB: G. Bergman, L. Williams, T. Y. Lam, J. Chen, Z. Wang. >From UCD: M. Vazirani. >From U. Pacific: A. Dugas. >From UCSB: K. Goodearl, B. Zimmermann-Huigen, W. Jacob, K. Casteels, C. Nowlin, G. Johnson, T. Howard, J. Galstad. >From UCSD: A. Wadsworth, M. Reyes. >From BYU: P. Nielsen. ============================================================= Algebra Day Information *********************** (1) In order to accommodate 8 speakers, all talks are 40 minutes (like the last Alg. Day in UCSB). And the breaks are 10 minutes, except for the mid-afternoon coffee break. We can keep to the announced schedule only if all talks begin and end on time. (2) The lecture room (939 Evans) will have an overhead projector. It is also equipped for powerpoint presenta- tions, but the organizers have not used the equipment before. You are welcome to try it, provided that you have a back-up plan (a.k.a. transparencies!) in case of equipment malfunction. (3) Arrival information: 7 guests have reservations at WFC Women's Faculty Club. To go there, take Gayley Dr. on the east boundary of the campus. If you're driving south on Gayley, there is a "little road" off of Gayley Drive right after you pass the Haas Business School, which takes you a few hundred feet down to the Optometry Building. WFC is just around the corner. If you're driving north on Gayley, go past the I-House (at the intersection with Bancroft). At the end of the block, make a (slightly illegal) U-turn. Right after the U-turn, you'll see the afore-mentioned "little road" on your right. (4) WFC offers a continental breakfast. The WFC guests can either have breakfast there, or come to 989 Evans Hall to have light refreshments. We hope you'll come to Evans Hall as early as possible, in order to mingle with other participants who are not staying at WFC. (5) Evans Hall is locked on Saturday, so coming into the building may be somewhat of a challenge. We'll try to have some doors "popped open", but don't know if this would work. If you can't get in, Dr. Zhou Wang has agreed to help out. His cell phone # is 510-926-7339. Lam doesn't have a cell phone, but his landlines are 510-642-4378 (office), and 510-845-7930 (home) in case you run into any difficulties.