Fully Exact Module Categories Advance Stability under Deligne Product for Finite Braided Tensor Categories

Researchers are developing a refined understanding of module categories, fundamental structures in representation theory, with Azat M. Gainutdinov and Robert Laugwitz leading this investigation? Their work defines and explores ‘fully exact’ module categories, a specific subclass exhibiting stability under a key operation called the relative Deligne product, a property not universally shared by all exact module categories? This research is significant because it establishes a hierarchy within module categories, demonstrating that fully exact categories strictly encompass known classes like invertible and separable ones, and introduces the concept of ‘perfect’ categories as a potential model for finite tensor 2-categories? Furthermore, Gainutdinov, Laugwitz et al. provide detailed classifications for specific cases, including those arising from Sweedler’s Hopf algebra, offering insights into the boundaries of this new framework and its implications for broader mathematical structures.

This research demonstrates, through examples in both zero and non-zero characteristic fields, that the broader class of exact module categories does not maintain this stability under the same product. Observations reveal that fully exact module categories densely populate the class of all exact module categories, suggesting a close relationship between the two. The monoidal 2-category constructed from fully exact module categories strictly encompasses those of invertible and separable module categories, expanding the scope of established mathematical structures.

This breakthrough establishes that every internal algebra within a fully exact module category is projectively separable, a significant generalization of separable algebras involving projective objects. In the specific case of semisimple categories, the study proves that a module category is fully exact if and only if it is separable, providing a clear and concise criterion for this condition. While fully exact module categories are not always dualizable within their own class, the research shows that when they are, they become fully dualizable objects within the larger monoidal 2-category of finite module categories. These specialized module categories are termed ‘perfect’ by the researchers.
The team achieved a crucial result by demonstrating that perfect module categories form a rigid monoidal 2-subcategory, containing all fully dualizable objects, and proposes this structure as a potential model for finite tensor 2-categories. If the braiding within the tensor category is symmetric, the study reveals a direct equivalence: module categories are fully exact if and only if they are perfect. As a detailed illustration, scientists classified fully exact, and consequently perfect, module categories over the symmetric tensor category derived from modules over Sweedler’s four-dimensional Hopf algebra, meticulously computing their relative Deligne products and analyzing the categories of 1-morphisms. Further analysis extends to general quasi-triangular Hopf algebras, where the researchers investigate the conditions under which the category of finite-dimensional vector spaces is fully exact. Their findings indicate that this is not the case for both Sweedler’s Hopf algebra and Lusztig’s factorizable small group of type at an odd root of unity, suggesting limitations to the applicability of this property. The work concludes with a conjecture proposing a similar outcome for small quantum groups of any simple Lie algebra, opening avenues for future investigation and potentially refining our understanding of these complex mathematical structures.

Fully Exact Module Category Stability

Scientists investigated fully exact module categories, a specific subclass within exact module categories defined over finite braided tensor categories, establishing their stability under the relative Deligne product. The research demonstrated, through examples in both zero and non-zero characteristic base fields, that the broader class of exact module categories does not share this stability. Detailed analysis revealed that fully exact module categories densely populate the space of all exact module categories. The monoidal 2-category encompassing fully exact module categories strictly includes those of invertible and separable module categories, a significant expansion of established classifications.

Researchers proved that every internal algebra within a fully exact module category is projectively separable, extending the concept of separable algebras to encompass projective objects. In the semisimple scenario, a module category is fully exact if and only if it is separable, providing a clear criterion for identification. While fully exact module categories are not generally dualizable within their class, the study showed that if dualizable, they are fully dualizable objects within the monoidal 2-category of finite module categories, termed ‘perfect’ module categories. These perfect module categories were then proposed as a model for finite tensor 2-categories.

The team engineered a classification of fully exact, and consequently perfect, module categories over the symmetric tensor category of modules derived from Sweedler’s four-dimensional Hopf algebra, alongside an analysis of their relative Deligne products and 1-morphisms. For a general quasi-triangular Hopf algebra, scientists analysed the conditions under which the category of finite-dimensional vector spaces becomes fully exact, demonstrating this is not the case for Sweedler’s Hopf algebra or Lusztig’s factorizable small group at an odd root of unity. Experiments employed functorial computations, specifically FunC(Vλ, Vμ) ≃vect(μ−λ)−1 = Vμ−λ, to establish relationships between module categories. The study pioneered the use of centralizer functors, denoted C →C∗ Vλ, to demonstrate the invertibility of C-modules Vλ, leveraging Lemma 0.7.1 and Theorem 0.7.2. Calculations of Vλ ⊠C Vμ ≃Vλ+μ were performed using these equivalences, with more complex methods detailed in a forthcoming publication [GL]. Corollary 8.24 established that all indecomposable fully exact C-module categories, Vλ and Sλ, are invertible, building upon Proposition 0.8.23.

Fully Exact Module Categories and Deligne Stability

Scientists have defined fully exact module categories, a subclass of exact module categories existing over a finite braided tensor category, demonstrating stability under the relative Deligne product. However, experiments reveal that the broader class of exact module categories is not stable under this product, with examples observed in both zero and non-zero characteristic base fields. Observations also indicate that fully exact module categories densely populate the class of exact ones, and the monoidal 2-category of fully exact module categories strictly encompasses those of invertible and separable module categories. The team measured that each internal algebra within a fully exact module category is projectively separable, a generalization of separable algebras involving projective objects.

In the semisimple case, results demonstrate a module category is fully exact if and only if it is separable. Tests prove that fully exact module categories are not generally dualizable within their class, but when they are, they are fully dualizable objects in the monoidal 2-category of finite module categories, termed ‘perfect’ module categories. Scientists recorded that perfect module categories form a rigid monoidal 2-subcategory containing all fully dualizable objects, proposing them as a model for finite tensor 2-categories. For a general quasi-triangular Hopf algebra, the study analyzed when the category of finite-dimensional vector spaces is fully exact, showing this is not the case for Sweedler’s Hopf algebra or Lusztig’s factorizable small group at an odd root of unity. The double centralizer theorem holds, implying Morita equivalence between the tensor category C and its centralizer C∗M = FunC(M, M). Proposition A establishes that the full 2-subcategory of fully exact C-modules is a monoidal 2-subcategory of C-mod. Theorem B demonstrates that a finite left C-module category M is fully exact if and only if its centralizer functor AM preserves projective objects, with equivalent conditions also identified. Examples in both zero and non-zero characteristic fields show that the class of exact module categories does not remain stable under this product, although fully exact module categories appear as a dense subset within the broader class of exact ones. This research establishes that the monoidal 2-category of fully exact module categories strictly encompasses those of invertible and separable module categories, with each internal algebra within a fully exact module category being projectively separable. In the semisimple scenario, a module category is fully exact if and only if it is separable, but generally, fully exact module categories lack dualizability within their class, becoming fully dualizable only when they form perfect module categories.

These perfect module categories constitute a rigid monoidal 2-subcategory, containing all fully dualizable objects and offering a potential model for finite tensor 2-categories. The authors acknowledge a limitation in that fully exact module categories are not always dualizable, and further research is needed to fully understand their properties in broader contexts. Future work could explore symmetric Z2-graded extensions of certain categories, potentially offering insights into interesting symmetric extensions. The detailed classification of fully exact, and therefore perfect, module categories over the symmetric tensor category of modules over Sweedler’s four-dimensional Hopf algebra, alongside the analysis of when the category of finite-dimensional vector spaces is fully exact for a general quasi-triangular Hopf algebra, provides a concrete example and direction for future investigation.