Algebra Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Past events in this series
24 September 2019
14:15
Adam Brown
Abstract

In 1985, McDowell introduced a family of parabolically induced Whittaker modules over a complex semisimple Lie algebra, which includes both Verma modules and the nondegenerate Whittaker modules studied by Kostant. Many classical results for Verma modules and the Bernstein--Gelfand--Gelfand category O have been generalized to the category of Whittaker modules introduced by Milicic--Soergel, including the classification of irreducible objects and the Kazhdan--Lusztig conjectures. Contravariant forms on Verma modules are unique up to scaling and play a key role in the definition of the Jantzen filtration. In this talk I will discuss a classification of contravariant forms on parabolically induced Whittaker modules. In a recent result, joint with Anna Romanov, we show that the dimension of the space of contravariant forms on a parabolically induced Whittaker module is given by the cardinality of a Weyl group. This result illustrates a divergence from classical results for Verma modules, and gives insight to two significant open problems in the theory of Whittaker modules: the Jantzen conjecture and the absence of an algebraic definition of duality.

22 October 2019
14:15
Philippe Meyer
Abstract

The notion of colour Lie algebra, introduced by Ree (1960), generalises notions of Lie algebra and Lie superalgebra. From an orthogonal representation V of a quadratic colour Lie algebra g, we give various ways of constructing a colour Lie algebra g’ whose bracket extends the bracket of g and the action of g on V. A first possibility is to consider g’=g⊕V and requires the cancellation of an invariant studied by Kostant (1999). Another construction is possible when the representation is ``special’’ and in this case the extension is of the form g’=g⊕sl(2,k)⊕V⊗k^2. Covariants are associated to special representations and satisfy to particular identities generalising properties studied by Mathews (1911) on binary cubics. The 7-dimensional fundamental representation of a Lie algebra of type G_2 and the 8-dimensional spinor representation of a Lie algebra of type so(7) are examples of special representations.

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