l2mathbbZ

The term **l₂ℤ** refers to the set of all sequences of integers that are square-summable, equipped with a specific algebraic structure. This concept is primarily studied in the context of **Banach algebras** and **operator algebras**, particularly within the field of **noncommutative geometry** and **quantum groups**.

Formally, l₂ℤ consists of all sequences (xₙ)ₙ∈ℤ where each xₙ is an integer, and the sum of the squares of the absolute values of the terms converges, i.e., ∑ₙ∈ℤ |xₙ|² < ∞. This space is a **Banach space** under the norm ∥(xₙ)∥₂ = (∑ₙ∈ℤ |xₙ|²)^(1/2), making it isomorphic to the ℓ² space over the integers. It also forms a **unital algebra** under pointwise multiplication, where the unit element is the sequence (1, 0, 0, ...).

In the study of **quantum groups**, particularly **Kac–Moody algebras** and their representations, l₂ℤ arises as a natural space for certain **positive-energy representations**. It plays a role in the construction of **positive-energy representations of affine Kac–Moody algebras**, where it serves as a model for the **positive-energy subspace** of the Verma module.

The space l₂ℤ is also connected to **discrete series representations** in the context of **loop groups** and their associated algebras. It provides a framework for understanding **discrete spectra** in noncommutative harmonic analysis, particularly in the study of **quantum tori** and **quantum modular forms**.

While l₂ℤ is closely related to the more familiar ℓ² space over the integers, its algebraic structure and role in representation theory make it a distinct object of interest in advanced mathematical research. The study of such spaces contributes to deeper insights into the interplay between analysis, algebra, and geometry in modern mathematics.