Webs of type P (Q6650484)

The authors introduce certain diagrammatic supercategories via generators and relations. These supercategories provide a combinatorial model of certain monoidal supercategories of representations for the type P Lie superalgebra \(\mathfrak{p}(n)\). The prefix ``super'' means there is a \(\mathbb{Z}_2\)-grading and definitions include signs according to the grading. For example, a supercategory is a category enriched in the category of \(\mathbb{Z}_2\)-graded vector spaces, while a monoidal supercategory is additionally equipped with a monoidal structure satisfying a graded analogue of the interchange law. The type P Lie superalgebra is one of the so-called strange families which appears in Kac's classification of the simple complex Lie superalgebras. It has no direct analogue in the classical world. One reason for this is that many classical techniques used to study Lie algebras cannot be directly adapted to the study of \(\mathfrak{p}(n)\).\N\NThe authors introduce type P web supercategories. They are defined as diagrammatic monoidal \(k\)-linear supercategories via generators and relations. They study the structure of these categories and provide diagrammatic bases for their morphism spaces. They also prove that these supercategories provide combinatorial models for the monoidal supercategory generated by the symmetric powers of the natural module and their duals for the Lie superalgebra of type P.