Under certain conditions, a binary linear code can be lifted to a lattice, which can in turn be lifted to a vertex operator algebra (VOA). For example, the extended Hamming code lifts to the E_8 lattice, which lifts to a lattice VOA. In a somewhat more complicated construction, the extended Golay code lifts to the Leech lattice, which can be used to construct the monster module, an orbifold of the Leech lattice VOA. We investigate such towers by considering the associated sub-objects fixed by lifts of code automorphisms. We identify replicable functions which occur as quotients of theta functions of fixed sublattices and consider further ways in which the structure at the code (resp. lattice) level determines behavior at the lattice (resp. VOA) level of the tower. This is joint work with Lea Beneish, Jen Berg, Hussain Kadhem, Eva Goedhart, and Allechar Serrano Lopez.
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Abstract: (Universal) enveloping algebras of finite-dimensional Lie algebras are some of the most well-understood examples in noncommutative ring theory. On the other hand, enveloping algebras of infinite-dimensional Lie algebras are much more mysterious. This talk will survey what's known about these rings, focussing on noetherianity questions and applications to representation theory.
Let $W$ be the Witt algebra of vector fields on the punctured complex plane, and let $\text{Vir}$ be the Virasoro algebra, the unique nontrivial central extension of $W$. $\text{Vir}$ is an infinite-dimensional Lie algebra that is ubiquitous in representation theory and important in physics. The symmetric algebras of $\text{Vir}$ and $W$ are polynomial rings in infinitely many variables which also have a Poisson structure induced from the bracket on the Lie algebras. Poisson ideals of $\text{Sym(Vir)}$ and $\text{Sym(W)}$ approximate two-sided ideals in the corresponding universal enveloping algebras and give evidence for the two-sided structure of the enveloping algebras.
We study prime Poisson ideals in these symmetric algebras, focussing on understanding Poisson primitive ideals, which are the Poisson cores of maximal ideals of $\text{Sym(Vir)}$ and of $\text{Sym(W)}$. Surprisingly, these give a way to do finite-dimensional algebraic geometry with these infinite polynomial rings! We give a complete classification of maximal ideals of $\text{Sym(W)}$ which have nontrivial Poisson cores. We then lift this classification to $\text{Sym(Vir)}$, and use it to show that if $q$ is a nonzero complex number, then $(z-q)$ is a maximal Poisson ideal in $\text{Sym(Vir)}$.
This is joint work with Alexey Petukhov.
Under certain conditions, a binary linear code can be lifted to a lattice, which can in turn be lifted to a vertex operator algebra (VOA). For example, the extended Hamming code lifts to the E_8 lattice, which lifts to a lattice VOA. In a somewhat more complicated construction, the extended Golay code lifts to the Leech lattice, which can be used to construct the monster module, an orbifold of the Leech lattice VOA. We investigate such towers by considering the associated sub-objects fixed by lifts of code automorphisms. We identify replicable functions which occur as quotients of theta functions of fixed sublattices and consider further ways in which the structure at the code (resp. lattice) level determines behavior at the lattice (resp. VOA) level of the tower. This is joint work with Lea Beneish, Jen Berg, Hussain Kadhem, Eva Goedhart, and Allechar Serrano Lopez.