Index of a subfactor

Definition

The index of a subfactor is a numerical invariant that measures the 'size' or 'dimension' of a subfactor relative to its parent factor. It quantifies the degree of separability between the two von Neumann algebras involved and plays a critical role in the classification of subfactors. The index is calculated as the minimal number of projections required to decompose the identity operator in the larger algebra into projections in the smaller algebra, revealing important structural properties.

5 Must Know Facts For Your Next Test

1. The index is typically denoted by the symbol $[M:N]$ where $M$ is the larger factor and $N$ is the subfactor.
2. Values of the index are often integers or rational numbers, which can provide insight into the relationship between the two factors.
3. For certain classes of subfactors, such as those arising from Bisch-Haagerup constructions, the index can take specific values that classify them within broader categories.
4. The index is closely related to properties like the dimension of Hilbert spaces and can help determine whether certain types of fusion rules apply.
5. Understanding the index helps to categorize subfactors and their corresponding modular inclusions in representation theory and operator algebras.

Review Questions

• How does the index of a subfactor inform our understanding of its relationship with its parent factor?
  • The index of a subfactor acts as a bridge to understand how 'large' or 'complex' the subfactor is compared to its parent factor. It quantifies the separability and structural features shared between them. By analyzing this numerical value, one can derive insights about the interactions and potential embeddings between these algebras, aiding in their classification.
• Discuss how Jones' Index Theorem contributes to calculating the index for specific subfactors.
  • Jones' Index Theorem offers crucial insights and conditions that facilitate calculating the index for particular types of subfactors. By applying this theorem, one can connect various mathematical structures such as planar algebras and modular tensor categories, ultimately providing explicit formulas or methods for determining indices. This connection deepens our comprehension of how indices relate to other invariants and classification schemes in operator algebras.
• Evaluate the significance of the index when exploring Bisch-Haagerup subfactors and their implications in modern mathematical research.
  • The significance of the index in Bisch-Haagerup subfactors lies in its ability to classify these constructions within operator algebras and their applications in quantum theory. By analyzing specific index values, researchers can identify unique properties, such as fusion rules and modular inclusions. This exploration has led to advancements in understanding quantum symmetry and contributes to developing broader theories in mathematical physics and representation theory, emphasizing how foundational concepts intersect across disciplines.

Related terms

Subfactor: A von Neumann algebra that is a subalgebra of another von Neumann algebra, possessing a non-trivial inclusion relation.

Bicharacteristic: An invariant associated with a subfactor that captures additional structural information beyond the index, often used in understanding fusion categories.

Jones' Index Theorem: A result that provides conditions under which the index of a subfactor can be determined, linking it to other mathematical objects such as planar algebras.