Tag - Cohomology theory

David Galban: Cohomology and Representation Theory for Lie Superalgebras

14th March, 2022

This talk will consist of two parts. In the first, I will describe the cohomology groups for the subalgebra 𝔫+ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu, essentially with these calculations essentially providing the first steps towards an analogue of Kostant’s theorem for Lie superalgebras. In the second part, based on joint work with Nakano, I will analyze the sheaf cohomology groups RI indBG L𝔣(λ), where L𝔣(λ) is an irreducible representation for the detecting subalgebra 𝔣, providing analogues for the BBW theorem and Kempf’s vanishing theorem for sufficiently large λ.

Jon Carlson: Idempotent modules and endomorphisms

14th February, 2022

We work in the group algebra kG of a finite group scheme defined over a field of characteristic p>0. The stable category stmod(kG) of finitely generated kG-modules is a tensor triangulated category of compact objects. Associated to any thick tensor ideal subcategory C is a distinguished triangle

→ E → k → F →

where E and F are idempotent modules in stable category StMod(kG) of all kG-modules. Tensoring with F is the localization functor associated to C. Indeed, the endomorphism ring of F is the endomorphism ring of the trivial module in the localized category. In this lecture we will discuss some results on the nature of the endomorphism ring of the module F and some strange variant support varieties for modules.

Aleksander Horawa: Motivic action on coherent cohomology of Hilbert modular varieties

3rd February, 2022

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.

Kent Vashaw: Can a Zhang twist be a cocycle twist?

26th October, 2021

The Zhang twist of a graded algebra was defined by J. Zhang in 1996, and has provend an important tool in non-commutative algebra and non-commutative algebraic geometry. On the other hand, in the world of Hopf algebras and quantum groups, the 2-cocycle twist of a Hopf algebra gives a new Hopf algebra which is Morita-Takeuchi equivalent to the original Hopf algebra. We provide sufficient conditions for a Zhang twist of a graded Hopf algebra H to be again a Hopf algebra, to be an H-cleft object, or a 2-cocycle twist of H. In particular, we introduce the notion of a twisting pair for H such that the Zhang twist of H by such a pair is a 2-cocycle twist. This new notion is investigated in the context of various examples of Hopf algebras including Manin's universal quantum groups, and the quantized coordinate rings of general linear groups.

Milen Yakimov: Non-commutative Tensor Triangular Geometry and Support Varieties for Hopf Algebras

16th October, 2021

We will describe a theory of noncommutative tensor triangular geometry for monoidal triangulated categories. It is aimed at investigating support varieties for finite dimensional Hopf algebras via non-commutative Balmer spectra. We will state effective reconstruction theorems for these spectra and an intrinsic characterization of those categories whose support variety maps satisfy the tensor product property. As an application, we obtain a treatment of the Benson-Witherspoon Hopf algebras, which previously eluded approaches of this kind, and a proof of a recent conjecture of Negron and Pevtsova that the cohomological support maps of the Borel subalgebras of all Lusztig small quantum groups possess the tensor product property. This is joint work with Daniel Nakano (University of Georgia) and Kent Vashaw (MIT).

Jon Carlson: Nilpotence and Duality in the Complete Cohomology of a Module

4th October, 2021

Suppose that G is a finite group and k is a field of characteristic p>0. Let M be a finitely generated kG-module. We consider the complete cohomology ring ℰ̂M∗=Σn∈ℤ Ex̂tkGn(M, M). We show that the ring has two distinguished ideals I∗ ⊆ J∗ ⊆ ℰ̂M∗ such that I∗ is bounded above in degrees, ℰ̂M∗/J∗ is bounded below in degree and J∗/I∗ is eventually periodic with terms of bounded dimension. We prove that if M is neither projective nor periodic, then the subring of all elements in negative degrees in ℰ̂M∗ is a nilpotent algebra.

Jon Carlson: The endomorphism ring of the trivial module

19th April, 2021

Let k be a field of characteristic p and let H be a finite group or group scheme. We show that the negative Tate cohomology ring Ĥ≤0(H,k) can be realized as the endomorphism ring of the trivial module in a Verdier localization of the stable category of kG-modules for G an extension of H. This means in some cases that the endomorphism of the trivial module is a local ring with infinitely generated radical with square zero. This stands in stark contrast to some known calculations in which the endomorphism ring of the trivial module is the degree zero component of a localization of the cohomology ring of the group.

Brian Boe: Complexity and Support Varieties for Type P Lie Superalgebras

5th April, 2021

We compute the complexity, z-complexity, and support varieties of the (thick) Kac modules for the Lie superalgebras of type P. We also show the complexity and the z-complexity have geometric interpretations in terms of support and associated varieties; these results are in agreement with formulas previously discovered for other classes of Lie superalgebras. Our main technical tool is a recursive algorithm for constructing projective resolutions for the Kac modules. The indecomposable projective summands which appear in a given degree of the resolution are explicitly described using the combinatorics of weight diagrams. Surprisingly, the number of indecomposable summands in each degree can be computed exactly: we give an explicit formula for the corresponding generating function. I wrote an iOS app to implement the combinatorics quickly and graphically, and I’ll be demoing live some of the interesting features of these resolutions.

David Galban: First and Second Cohomology Groups for BBW Parabolics for Lie Superalgebras

1st March, 2021

For semisimple Lie algebras, a well-known theorem of Kostant computes the cohomology groups of parabolic subalgebras, but it is unknown whether an analogue of Kostant’s theorem exists for Lie superalgebras. Seeking to provide the first calculations in this direction, in this talk, I will describe the cohomology groups for the subalgebra 𝔫+ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu. These classical Lie superalgebras have a triangular decomposition 𝔤 = 𝔫- + 𝔣 + 𝔫+, where 𝔣 is a detecting subalgebra as introduced by Boe, Kujawa and Nakano. I will show that there exists a Hochschild-Serre spectral sequence that collapses for all infinite families of classical simple Lie superalgebras. Using this, I will provide examples of computation of the first and second cohomologies for various 𝔫+.