# Algebra and Geometry Seminar

### Info

Welcome to the Algebra and Geometry Seminar at Iowa State University, organized by Jonas Hartwig, Jason McCullough, and Tathagata Basak.

During Fall 2021, the seminar runs virtually (and/or in Carver 401 if circumstances allow) on Tuesdays at 2:00pm–2:50pm. To attend, you should be on the seminar mailing list, where invitation links to the talks will be posted. If you wish to be added or removed from the mailing list, please contact one of the organizers.

Topics include:

• associative algebras and commutative rings,
• representation theory and Lie theory,
• connections to combinatorics, geometry and physics.

### Talks

August 31, 2021
Meet and greet
September 7, 2021
Jason McCullough (ISU)
A $$\mathbb{Z}$$-graded K-algebra R is Koszul if K has a linear free resolution over R. It is known that quotients of a polynomials ring or exterior algebra or tensor algebra that are Koszul must have quadratic relations. It is sufficient to have a Groebner basis of quadratic relations – we call these algebras G-quadratic. Neither of these implications is reversible in general. There is an intermediate notion, called LG-quadratic, which describe algebras that are quotients of G-quadratic algebras by a regular sequence of linear forms. The goal of this talk will be to show that there are G-quadratic quotients of exterior algebras that are not LG-quadratic, and Koszul quotients that are not LG-quadratic. This is joint work with Zach Mere (former ISU ugrad, now Utah grad student).
September 14, 2021
Generic Gelfand-Tsetlin Representations of Nonstandard Quantized Orthogonal Algebras
Jordan Disch (ISU)
We construct infinite-dimensional analogs of classical representations of the nonstandard quantized enveloping algebra $$U_q(\mathfrak{so}_n)$$ by rationalizing the classical formulas from finite-dimensional representations. These also provide representations for the universal enveloping algebra $$U(\mathfrak{so}_n)$$ as $$q$$ approaches $$1$$. We use these new representations to embed $$U_q(\mathfrak{so}_n)$$ into a skew group algebra of shift operators.
September 21, 2021
The canonical module of a Koszul algebra
Matthew Mastroeni (ISU)
We study under what conditions the canonical module of a Cohen-Macaulay Koszul algebra has a linear free resolution over its Koszul algebra, motivated by a question of Stillman asking whether this holds for all superlevel Koszul algebras. Using recent work of D’Alì and Venturello, we show that Stillman’s question has a negative answer in general. However, we also highlight a number of cases where the canonical module does have a linear resolution and give some consequences for resolutions of modules over certain rings of geometric interest. This talk is based on joint preliminary work with Paolo Mantero.
September 28, 2021
General Lines in Projective Space and the Koszul Property
Josh Rice (ISU)
A graded k-algebra is said to be Koszul if the minimal R-free graded resolution of k is linear. In this talk we study the Koszul property of the homogeneous coordinate ring R of a set of m lines in complex projective space. Kempf proved that the coordinate ring of s points in general linear position is Koszul if s is less than or equal to 2n. Further, Conca, Trung and Valla showed that if the points are algebraically independent over Q, then the coordinate ring is Koszul if and only if s is less than or equal to 1+n+n^2/4. We expand on Kempfs Theorem with the exception we consider lines in projective space.
October 5, 2021
More on $$\mathrm{DR}(\mathfrak{osp}(1|2))$$
Jonas Hartwig (ISU)
I will report on recent progress of our investigation into the diagonal reduction algebra for the orthosymplectic Lie superalgebra $$\mathfrak{osp}(1|2)$$, including presentation, PBW basis and ghost center. Joint work with Dwight Williams II.