CUBIC DIRAC OPERATORS AND THE STRANGE FREUDENTHAL–DE VRIES FORMULA FOR COLOUR LIE ALGEBRAS

The aim of this paper is to define cubic Dirac operators for colour Lie algebras. We give a necessary and sufficient condition to construct a colour Lie algebra from an ϵ-orthogonal representation of an ϵ-quadratic colour Lie algebra. This is used to prove a strange Freudenthal–de Vries formula for basic colour Lie algebras as well as a Parthasarathy formula for cubic Dirac operators of colour Lie algebras. We calculate the cohomology induced by this Dirac operator, analogously to the algebraic Vogan conjecture proved by Huang and Pandžić.

INTERNAL NATURAL TRANSFORMATIONS AND FROBENIUS ALGEBRAS IN THE DRINFELD CENTER

For ℳ and $$ \mathcal{N} $$ N finite module categories over a finite tensor category $$ \mathcal{C} $$ C , the category $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) of right exact module functors is a finite module category over the Drinfeld center $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map $$ \mathcal{C} $$ C -$$ \mathcal{C} $$ C -bimodule functors to objects of $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if ℳ and $$ \mathcal{N} $$ N are exact $$ \mathcal{C} $$ C -modules and $$ \mathcal{C} $$ C is pivotal, then the $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C )-module $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) is exact. We compute its relative Serre functor and show that if ℳ and $$ \mathcal{N} $$ N are even pivotal module categories, then $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ).

THE ASYMPTOTIC SEMIGROUP OF A SUBSEMIGROUP OF A NILPOTENT LIE GROUP

Let S be a subsemigroup of a simply connected nilpotent Lie group G. We construct an asymptotic semigroup S0 in the associated graded Lie group G0 of G. We can compute the image of S0 in the abelianization $$ {G}_0^{\mathrm{ab}}={G}^{\mathrm{ab}}. $$ G 0 ab = G ab . This gives useful information about S. As an application, we obtain a transparent proof of the following result of E. B. Vinberg and the author: either there is an epimorphism f : G → ℝ such that f (s) ≥ 0 for every s in S or the closure $$ \overline{S} $$ S ¯ of S is a subgroup of G and $$ G/\overline{S} $$ G / S ¯ is compact.