Operator growth in 2d CFT

We investigate and characterize the dynamics of operator growth in irrational two-dimensional conformal field theories. By employing the oscillator realization of the Virasoro algebra and CFT states, we systematically implement the Lanczos algorithm and evaluate the Krylov complexity of simple operators (primaries and the stress tensor) under a unitary evolution protocol. Evolution of primary operators proceeds as a flow into the ‘bath of descendants’ of the Verma module. These descendants are labeled by integer partitions and have a one-to-one map to Young diagrams. This relationship allows us to rigorously formulate operator growth as paths spreading along the Young’s lattice. We extract quantitative features of these paths and also identify the one that saturates the conjectured upper bound on operator growth.

Affinoid Dixmier Modules and the Deformed Dixmier-Moeglin Equivalence

The affinoid enveloping algebra $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K of a free, finitely generated $\mathbb {Z}_{p}$ ℤ p -Lie algebra ${\mathscr{L}}$ L has proven to be useful within the representation theory of compact p-adic Lie groups, and we aim to further understand its algebraic structure. To this end, we define the notion of a Dixmier module over $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K , a generalisation of the Verma module, and we prove that when ${\mathscr{L}}$ L is nilpotent, all primitive ideals of $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K can be described in terms of annihilator ideals of Dixmier modules. Using this, we take steps towards proving that this algebra satisfies a version of the classical Dixmier-Moeglin equivalence.

Bigrassmannian permutations and Verma modules

We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type A. All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by $$w\in S_n$$ w ∈ S n into the dominant Verma module are shown to be determined by the essential set of w and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from a simple module to a Verma module.

Classification of Degenerate Verma Modules for E(5, 10)

Given a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

Overconvergent modular forms are highest-weight vectors in the Hodge-Tate weight zero part of completed cohomology

We construct a $(\mathfrak {gl}_2, B(\mathbb {Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $0$ of a sheaf on $\mathbb {P}^1$ , landing in the compactly supported completed $\mathbb {C}_p$ -cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest-weight vector for any overconvergent modular form of infinitesimal weight not equal to $1$ . For classical weight $k\geq 2$ , the Verma has an algebraic quotient $H^1(\mathbb {P}^1, \mathcal {O}(-k))$ , and on classical forms, the pairing factors through this quotient, giving a geometric description of ‘half’ of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $\mathbb {P}^1$ . Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan, but the perspective here is different, and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology.

Support varieties of baby Verma modules

Let [Formula: see text] be a connected reductive algebraic group over an algebraically closed field [Formula: see text] of prime characteristic [Formula: see text] and [Formula: see text]. For a given nilpotent [Formula: see text]-character [Formula: see text], let [Formula: see text] be a baby Verma module associated with a restricted weight [Formula: see text]. A conjecture describing the support variety of [Formula: see text] via that of its restricted counterpart is given: [Formula: see text]. Under the assumption of [Formula: see text](the Coxeter number) and [Formula: see text] [Formula: see text]-regular, this conjecture is proved when [Formula: see text] falls in the regular nilpotent orbit for any [Formula: see text] and the subregular nilpotent orbit for [Formula: see text] being of type [Formula: see text]. We also verify this conjecture whenever [Formula: see text] is of type [Formula: see text] and [Formula: see text] falls in the minimal nilpotent orbit.

Automorphisms and Verma Modules for Generalized 2-dim Affine-Virasoro Algebra

We study the structure of the generalized 2-dim affine-Virasoro algebra, and describe its automorphism group. Furthermore, we also determine the irreducibility of a Verma module over the generalized 2-dim affine-Virasoro algebra.

Orthogonal basis for the Shapovalov form on $U_q(\mathfrak{sl}(n + 1))$

Let U be either the classical or quantized universal enveloping algebra of the Lie algebra [Formula: see text] extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in U over the extended Cartan subalgebra diagonalizing the contravariant Shapovalov form on generic Verma module. The matrix coefficients of the form are calculated and the inverse form is explicitly constructed.