Towers of Infinite-Dimensional Algebras

• Towers of infinite-dimensional algebras are structured graded sequences characterized by coherent induction, restriction, and anti-automorphism maps.
• Their Grothendieck groups form dual Hopf algebras, underpinning categorifications such as the Heisenberg double and quantum differential operators.
• This framework advances representation theory by providing categorical models for Fock space representations and quantum algebra operations.

A tower of infinite-dimensional algebras is a structured sequence of (typically graded) algebras $\{A_n\}_{n\ge0}$, assembled so that each level encodes algebraic and categorical operations compatible in a coherent, inductive manner. Such structures generalize classical settings where the constituent algebras $A_n$ may be either finite- or infinite-dimensional and are tightly linked by induction, restriction, and anti-automorphism maps. The resulting Grothendieck groups of projective and finite-dimensional modules form dual Hopf algebras, yielding a framework for categorifying algebraic entities such as the Heisenberg double and quantum differential operators. Towers of this type serve as foundational objects in the paper of representation theory, categorification, and quantum algebra, especially in contexts where infinite combinatorics and dualities play central roles (Lai et al., 24 Sep 2025).

1. Structural Axioms for Infinite-Dimensional Towers

To formalize towers of infinite-dimensional algebras, the following axioms are introduced for a graded tower $A = \bigoplus_{n=0}^\infty A_n$:

• Niceness of Graded Algebras: Each $A_n$ (with grading $A_n = \bigoplus_{i} A_{n,i}$) is "nice," meaning there exists a graded ideal $I \subset \bigoplus_{i>0} A_{n,i}$ annihilating all graded simples, and $A_n/I$ is finite-dimensional.
• Multiplicative Structure: A unital algebra morphism $A_n \otimes A_m \to A_{n+m}$ is required for all $n,m\ge0$, encoding an induction operation at the algebraic level.
• Projectivity: Each $A_{n+m}$ must be two-sided projective as an $A_n \otimes A_m$-module, ensuring that induction functors behave well on modules.
• Anti-Automorphism: An anti-involution $\psi:A_n\to A_n$ is present at every level, allowing transfer of structures between left and right modules and playing a pivotal role in duality.

These axioms guarantee that the module categories over $\{A_n\}$ can be organized into a categorical object (the tower) in which the combinatorics of infinite-dimensional representation theory are tractable and compatible with graded and inductive structures.

2. Hopf Duality of Grothendieck Groups

Each tower $A = \bigoplus_n A_n$ admits two fundamental Grothendieck groups:

• $K_+(A)$: Direct sum of Grothendieck groups of finitely generated projective graded $A_n$-modules.
• $\mathcal{G}^f(A)$: Direct sum of Grothendieck groups of finite-dimensional graded $A_n$-modules.

The tower axioms ensure the existence of a nondegenerate pairing: $$\langle [P], [N] \rangle_\psi = \text{gdim}_k(P^\psi \otimes_{A_n} N),$$ where $P^\psi$ denotes right $A_n$-action twisted by the anti-involution $\psi$, and gdim is the graded dimension over the field $k$. This pairing is compatible with the induction and restriction operations, and under suitable conditions forms a duality of Hopf algebras.

Induction and restriction manifest as multiplication and comultiplication operations:

• Induction: $IND(M \otimes N)$ induces a product structure.
• Restriction: $RES(L)$ (viewing $A_{n+m}$ as an $A_n \otimes A_m$-module) induces a coproduct structure.

The duality is crucial for analyzing categorical and algebraic structures arising from towers, allowing the Heisenberg double construction and Fock space representations (Savage et al., 2013).

3. The Heisenberg Double and Operator Actions

Given this Hopf duality, one constructs the Heisenberg double $h(A)$ as a subalgebra of operators on $\mathcal{G}^f(A)$:

• Creation Operators: Left multiplication by $[M]$, $l_{[M]}$.
• Annihilation Operators: Adjoint operators, $r^*_{[P]}$, satisfying $r^*_{[P]}([N]) = [\mathcal{T}_P(N)]$ for appropriately defined $[\mathcal{T}_P]$ (cf. next section).

This extends the classical Heisenberg algebra to a categorical setting (cf. categorification of the Weyl algebra in the nil-Coxeter case, and Heisenberg algebra's action on symmetric functions when the tower is the group algebra of symmetric groups).

4. Categorification: Quantum Differential Operators and Functors

The paper (Lai et al., 24 Sep 2025) establishes categorical versions of multiplication and differentiation:

• Induction Functor: For $M \in A_m\text{-mod}$,

$$\mathcal{J}_M(N) = IND_{A_n \otimes A_m}^{A_{n+m}}(N \otimes M)$$

categorifies left multiplication ($l_{[M]}$).

• Twisted Restriction Functor: For $P \in A_p\text{-pmod}$,

$$\mathcal{T}_P(N) = P^\psi \otimes_{A_p} RES_{A_n}^{A_{n-p} \otimes A_p}(N),$$

interpreted as zero for $n<p$, categorifies the adjoint ($r^*_{[P]}$).

In the nilHecke algebra case $A_n = NH_n$, the Grothendieck groups correspond (under isomorphism) to a quantum differential operator algebra on $\mathbb{A}^1_{\mathbb{Z}[v,v^{-1}]}$. Specifically, divided power operators and their relations are categorified: $$\partial^{(m)}\partial^{(n)} = \binom{m+n}{n}_v \partial^{(m+n)}, \quad \partial^{(m)}x = v^m x\partial^{(m)} + \partial^{(m-1)}.$$ $\mathcal{J}_M$ acts as multiplication, $\mathcal{T}_P$ as "quantum differentiation" on $\mathbb{A}^1_{\mathbb{Z}[v,v^{-1}]}$, and commutation isomorphisms interpolate categorical actions corresponding to binomial coefficients.

5. Applications to Representation Theory and Quantum Algebra

This framework generalizes key aspects of algebraic and categorical representation theory in infinite-dimensional contexts:

• Fock Space Representations: The Heisenberg double algebra acts on $\mathcal{G}^f(A)$, categorifying the notion of "creation" and "annihilation" operators. The resulting Fock spaces recover familiar structures appearing in combinatorics and quantum algebra.
• Quantum Differential Operators: Through the nilHecke example, quantum differential operators on $\mathbb{A}^1_{\mathbb{Z}[v,v^{-1}]}$ are realized categorically, including the divided power structure and quantum binomial relations.
• Extension Beyond Finite-Dimensional Algebras: Towers satisfying the outlined axioms encompass infinite-dimensional graded algebras (e.g., nilHecke, Hecke, or generalizations), extending classical theory and enabling new categorifications in quantum algebra.

6. Connections and Generalizations

The described formalism interacts deeply with adjacent domains:

• Hopf Algebras and Duality: The pairing induces Hopf algebra structures, allowing the formulation of Heisenberg doubles and connections to symmetric functions and quasisymmetric function rings (Savage et al., 2013).
• Categorification Program: The method generalizes categorifications for differential operators, Fock space, and Heisenberg actions, providing templates for advancing higher representation theory.
• Infinity-Categorical Approaches: The framework is amenable to extensions using infinity-categories and higher categorical structures, potentially connecting to geometric representation theory and quantum topology.

7. Summary Table: Key Structures in a Tower of Infinite-Dimensional Algebras

Structure	Level	Role/Significance
$A_n$	Algebra	Graded, nice; basis of inductive construction.
$K_+(A)$	Grothendieck	Projectives; forms one part of dual Hopf algebra.
$\mathcal{G}^f(A)$	Grothendieck	Finite-dimensional modules; dual to $K_+(A)$.
Heisenberg double $h(A)$	Operator Algebra	Categorifies creation/annihilation algebraic actions.
$\mathcal{J}_M,\mathcal{T}_P$	Functors	Induction and twisted restriction; quantum differential operator action

This comprehensive framework provides powerful tools for constructing, manipulating, and categorifying algebraic structures arising from towers of infinite-dimensional algebras. It enables algebraic dualities, Fock space actions, and the categorical lifting of differential operator algebras critical for contemporary research in quantum and representation theory.