Invariant Theory for the Periplectic Lie Superalgebra and Ramifications:

We will start the mini course with an elementary introduction to an interesting Lie superalgebra appearing in the Kac classification of classical Lie superalgebras, known as the ‘strange’ or ‘periplectic’ Lie superalgebra. Its invariant theory has been studied by D. Moon and by P. Deligne, G. Lehrer and R.B. Zhang. In this invariant theory a new algebra A, for which J. Kujawa and B. Tharp obtained a diagram calculus, takes over the role played by the symmetric group or Brauer algebra in invariant theory for general linear (super)algebras or orthosymplectic Lie (super)algebras.

The main part of the mini course will deal with results obtained during the last year, partly in collaboration with M. Ehrig, following the preprints arXiv:1609.06760, arXiv:1701.04606 and arXiv:1704.07547. First we will discuss a number of fundamental results on A, such as the homological properties and the Jordan-Hölder decomposition multiplicities of projective modules. Then we will discuss a number of applications of the former results, including:

• A proof that the existing linkage result for modules over the periplectic Lie supergroup, obtained by C.-W.Chen are exhaustive, hence obtaining a complete block decomposition.
• The classification of indecomposable summands in tensor products of the periplectic Lie superalgebra, with a description of which summands are projective.
• Introduction and categorification of a Fock type representation of an infinitely generated Temperley-Lieb algebra, using the periplectic analogue of Deligne’s category.

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