Fall, 2022

• Speaker Chris Sadowski, Ursinus College
  • Title Weight-one elements of vertex operator algebras and automorphisms of categories of generalized twisted modules
  • Time/place 10/28/2022, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract Given a weight-one elements u of a vertex operator algebra V, we construct an automorphism of the category of generalized g-twisted modules for automorphisms g of V fixing u. We apply these results to the case that V is an affine vertex algebra to obtain explicit results on these automorphisms of categories. In particular, we give explicit constructions of certain generalized twisted modules from generalized twisted modules associated to diagram automorphisms of finite-dimensional simple Lie algebras and generalized (untwisted) modules. This talk is based on a joint work with Yi-Zhi Huang.

• Speaker Filip Dul, Rutgers University
  • Title An Introduction to the Batalin-Vilkovisky Formalism
  • Time/place 11/4/2022, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract The (classical) BV formalism uses homological algebra to define classical field theories. I will present a number of examples which showcase the power of the formalism when dealing with gauge invariance and general covariance: this is where the homological methods truly shine. Finally, I will briefly describe how the observables of such a field theory define a factorization algebra on the underlying space: factorization algebras in, for example, the context of holomorphic theories provide derived enrichments of vertex operator algebras.

• Speaker Darlayne Addabbo, University of Arizona
  • Title Vertex operators for imaginary gl_2-subalgebras in the Monster Lie Algebra
  • Time/place 11/11/2022, Friday, 12:10 pm (Eastern Time), Zoom link above (online)
  • Abstract The Monster Lie algebra m is a quotient of the physical space of the vertex algebra V=V^\natural\otimes V_{1,1}, where V^\natural is the Moonshine module of Frenkel, Lepowsky, and Meurman, and V_{1,1} is the vertex algebra corresponding to the rank 2 even unimodular lattice II_{1,1}. It is known that m has gl_2-subalgebras generated by both real and imaginary root vectors and that the Monster simple group M acts trivially on the gl_2-subalgebra corresponding to the unique real simple root. We construct elements of V that project under the quotient map to the Serre-Chevalley generators of families of gl_2-subalgebras corresponding to the imaginary simple roots (1,n) of m for 0< n< 100. We prove the existence of primary vectors in V^\natural of each homogeneous weight n and, for 0< n< 100, we show that there exist primary vectors that can be used to construct the elements in $V$ corresponding to the generators of our gl_2-subalgebras. We show that the action of M on V^\natural induces an action on the Serre-Chevalley generators of the aforementioned subalgebras. We conjecture that this M-action is non-trivial. This talk is based on joint work with Lisa Carbone, Elizabeth Jurisich, Maryam Khaqan, and Scott H. Murray.

• Speaker Sven Möller, University of Hamburg
  • Title On the Classification of Holomorphic Vertex Operator Superalgebras
  • Time/place 11/18/2022, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract I will discuss the classification of holomorphic vertex operator superalgebras of central charge between 0 and 24 using the 2-neighbourhood method. This is joint work with Gerald Höhn.

• Speaker Daniel Soskin, University of Notre Dame
  • Title Determinantal inequalities for totally positive matrices
  • Time/place 12/2/2022, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract Totally positive matrices are matrices in which each minor is positive. Lusztig extended the notion to reductive Lie groups. He also proved that specialization of elements of the dual canonical basis in representation theory of quantum groups at q=1 are totally non-negative polynomials. Thus, it is important to investigate classes of functions on matrices that are positive on totally positive matrices. I will discuss two sourses of such functions. One has to do with multiplicative determinantal inequalities (joint work with M.Gekhtman). Another deals with majorizing monotonicity of symmetrized Fischer's products which are known for hermitian positive semidefinite case which brings additional motivation to verify if they hold for totally positive matrices as well (joint work with M.Skandera). The main tools we employed are network parametrization, Temperley-Lieb and monomial trace immanants.

• Speaker Abid Ali, Rutgers University
  • Title Strong integrality of inversion subgroups of Kac-Moody groups
  • Time/place 12/9/2022, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract Let g be a symmetrizable Kac-Moody algebra over Q. Let V be an integrable highest weight g-module and let V_Z be a Z-form of V. Let G(Q) be an associated minimal representation-theoretic Kac-Moody group and let G(Z) be its integral subgroup. Let Gamma(Z) be the Chevalley subgroup of G, that is, the subgroup that stabilizes the lattice V_Z in V. It is a difficult question to determine if G(Z)=Gamma(Z). We establish this equality for inversion subgroups U_w of G where, for an element w of the Weyl group, U_w is the group generated by positive real root groups that are flipped to negative roots by w^{-1}. This result extends to other subgroups of G, particularly when G has rank 2. This is joint work with Lisa Carbone, Dongwen Liu and Scott H. Murray.