[Paper Review] The degenerate Heisenberg category and its Grothendieck ring

Jonathan Brundan, Alistair Savage+1 more|arXiv (Cornell University)|Dec 8, 2018

Algebraic structures and combinatorial models13 references14 citations

TL;DR

This paper proves that the Grothendieck ring of the additive Karoubi envelope of the degenerate Heisenberg category is isomorphic to a Z-form of the universal enveloping algebra of the infinite-dimensional Heisenberg Lie algebra at any integer central charge $k \in \mathbb{Z}$, thereby confirming a conjecture by Khovanov and extending it to arbitrary $k$. The proof establishes a categorification of the algebraic structure via string diagrams and categorified comultiplication relations.

ABSTRACT

The degenerate Heisenberg category $\mathcal{H}eis_k$ is a strict monoidal category which was originally introduced in the special case $k=-1$ by Khovanov in 2010. Khovanov conjectured that the Grothendieck ring of the additive Karoubi envelope of his category is isomorphic to a certain $\mathbb{Z}$-form for the universal enveloping algebra of the infinite-dimensional Heisenberg Lie algebra specialized at central charge $-1$. We prove this conjecture and extend it to arbitrary central charge $k \in \mathbb{Z}$. We also explain how to categorify the comultiplication (generically).

Motivation & Objective

• To prove Khovanov's conjecture that the Grothendieck ring of the additive Karoubi envelope of the degenerate Heisenberg category is isomorphic to a Z-form of the universal enveloping algebra of the infinite-dimensional Heisenberg Lie algebra at central charge $k = -1$.
• To extend this isomorphism to arbitrary integer central charge $k \in \mathbb{Z}$, generalizing the original conjecture.
• To provide a categorification of the comultiplication on the universal enveloping algebra $U(\mathfrak{h})$ at generic central charge.
• To establish a canonical isomorphism between the Grothendieck ring and the ring of symmetric functions with a $k$-deformed pairing.
• To demonstrate that the category supports a strict pivotal structure and that its objects categorify Schur functions via Young symmetrizers.

Proposed method

• Construct the degenerate Heisenberg category $\mathrm{Heis}_k$ as a strict $k$-linear monoidal category with generators and relations modeled on the universal enveloping algebra $U(\mathfrak{h})/(c - k)$.
• Define the Grothendieck ring $K_0(\mathrm{Kar}(\mathrm{Heis}_k))$ as the split Grothendieck group of the additive Karoubi envelope of $\mathrm{Heis}_k$, with multiplication via tensor product.
• Use string diagram calculus to represent morphisms, including thick strands, crossings, cups, caps, and bubbles, with relations derived from symmetric functions and the $k$-deformed pairing.
• Establish a ring homomorphism $\gamma_k: \mathrm{Heis}_k \to K_0(\mathrm{Kar}(\mathrm{Heis}_k))$ sending generators to classes of categorified Schur functions.
• Prove surjectivity of $\gamma_k$ by constructing explicit isomorphisms between tensor products of categorified symmetric functions and direct sums indexed by partitions and $k$-bounded partitions.
• Use the involution $\Omega_k$ and the antipode to relate categories at $k$ and $-k$, and apply induction and diagrammatic identities to prove the key isomorphism theorems.

Experimental results

Research questions

• RQ1 Is the Grothendieck ring of the additive Karoubi envelope of the degenerate Heisenberg category isomorphic to a Z-form of $U(\mathfrak{h})/(c - k)$ for any $k \in \mathbb{Z}$?
• RQ2 Can the comultiplication on $U(\mathfrak{h})$ be categorified in the context of $\mathrm{Heis}_k$?
• RQ3 Does the category $\mathrm{Heis}_k$ support a strict pivotal structure compatible with its monoidal and duality structures?
• RQ4 Are the classes of categorified Schur functions $[S^\pm_\lambda]$ in $K_0(\mathrm{Kar}(\mathrm{Heis}_k))$ linearly independent and generate the ring?
• RQ5 Can the isomorphism $\gamma_k$ be extended to a categorification of the Hopf algebra structure, including comultiplication and antipode?

Key findings

• The Grothendieck ring $K_0(\mathrm{Kar}(\mathrm{Heis}_k))$ is isomorphic to the Z-form $\mathrm{Heis}_k$ of the universal enveloping algebra $U(\mathfrak{h})/(c - k)$, proving Khovanov's conjecture for all $k \in \mathbb{Z}$.
• The isomorphism $\gamma_k: \mathrm{Heis}_k \to K_0(\mathrm{Kar}(\mathrm{Heis}_k))$ is both injective and surjective, with $s^\pm_\lambda \mapsto [S^\pm_\lambda]$, and $[X] = 0$ implies $X = 0$ in $\mathrm{Kar}(\mathrm{Heis}_k)$.
• For $k \geq 0$, the morphism $\theta_{m,n}$ defined by a column vector of diagrams gives a canonical isomorphism between $H^+_m \otimes E^-_n$ and $\bigoplus_{r=0}^{\min(m,n)} \bigoplus_{\lambda \in P_{r,k}} E^-_{n-r} \otimes H^+_{m-r}$, proving the key relation in the Grothendieck ring.
• The comultiplication $\delta_{l|m}: \mathrm{Heis}_k \to \mathrm{Heis}_l \otimes \mathrm{Heis}_m$ for $k = l + m$ is categorified via a monoidal functor structure on the category, with explicit diagrammatic formulas.
• The involution $\Omega_k$ and the antipode $\sigma_k$ satisfy $\Omega_k: \mathrm{Heis}_k \to \mathrm{Heis}_{-k}$ and $\sigma_k: \mathrm{Heis}_k \to \mathrm{Heis}_{-k}^{\mathrm{op}}$, preserving the categorified algebraic structure.
• The category $\mathrm{Heis}_k$ is strictly pivotal, with duality given by 180-degree rotation of string diagrams, and the objects $\mathbf{1}$ and $\mathbf{1}^*$ are duals.