We construct a new family of graded representations W lambda for the positive elliptic Hall algebra E+ indexed by Young diagrams lambda which generalize the standard E+ action on symmetric functions. These representations have homogeneous bases of eigenvectors for the action of the Macdonald element P-0,P-1 is an element of E+ with distinct Q(q, t)-rational spectrum generalizing the symmetric Macdonald functions. The analysis of the structure of these representations exhibits interesting combinatorics arising from the limits of periodic standard Young tableaux. We find an explicit combinatorial rule for the action of the multiplication operators e(r)(center dot) generalizing the Pieri rule for symmetric Macdonald functions.
We analyze three related families of q-series arising from characters of vertex (super)algebras. More precisely, we consider: (i) graph series and characters of principal subspaces associated with arc algebras; (ii) characters, or indices, of class-S vertex operator algebras in the context of 4d N = 2 SOFT; and (iii) supercharacters of the U family of vertex superalgebras and their quasi-modular properties. Along the way, we introduce a family of multiple q-zeta values associated with simple Lie algebras and present their conjectural properties.
In [Internat. J. Theoret. Phys. 64 (2025), 73, 10 pages], Fritz was motivated by the structure of Hamiltonian/Heisenberg mechanics to define the notion of ''Lie quandle'', which he argued is a nonlinear generalization of finite-dimensional real Lie algebras. In this article, we will investigate a linear/nonlinear correspondence to which Fritz's is a special case, classify a family of generalizations of these objects, as well as describe some results in the direction of a nonlinear analogue of Noether's first theorem first described by Fritz.
We show that circle-valued spectral values in Tate Rabinowitz Floer homology give rise to quantized energy levels, which recover the quantum mechanical spectrum of a system of uncoupled harmonic oscillators.
We give a direct differential-geometric proof of a result of Crainic on the cohomology of pullbacks of Lie algebroids under surjective submersions: If the fibers are connected and have vanishing cohomology through degree $n$, pullback with coefficients in any representation is an isomorphism through degree $n$ and is injective in degree $n+1$. The proof reduces to a vanishing result for leafwise cohomology and uses an exhaustion-and-patching argument due to Buchdahl. No local triviality or finite-type assumption on the fibers is required. We also prove the analogous result for holomorphic Lie algebroids.
