arXiv subjects
Publications and source records attributed to Kevin Coulembier.
We determine the Jordan-Holder decomposition multiplicities of projective and cell modules over periplectic Brauer algebras in characteristic zero. These are obtained by developing the combinatorics of certain skew Young diagrams. We also establish a useful relationship with the Kazhdan-Lusztig multiplicities of the periplectic Lie superalgebra.
We compare a number of different definitions of structure algebras and TKK constructions for Jordan (super)algebras appearing in the literature. We demonstrate that, for unital superalgebras, all the definitions of the structure algebra and the TKK constructions fall apart into two cases. Moreover, one can be obtained as the Lie superalgebra of superderivations of the other. We also show that, for non-unital superalgebras, more definitions become non-equivalent. As an application, we obtain the corresponding Lie superalgebras for all simple finite dimensional Jordan superalgebras over an algebraically closed field of characteristic zero.
We study realisations of Lie (super)algebras in Weyl (super)algebras and connections with minimal representations. The main result is the construction of small realisations of Lie superalgebras, which we apply for two distinct purposes. Firstly it naturally introduces, and generalises, the Bessel operators for Jordan algebras in the study of minimal representations of simple Lie groups. These have already been applied very successfully by several authors, however an easy direct explanation for their relevance seemed still to be missing. Secondly, we work out the theoretical realisation concretely for the exceptional Lie superalgebra D(2,1;a), giving a useful hands-on realisation.
We determine the Ringel duals for all blocks in the parabolic versions of the BGG category O associated to a reductive finite dimensional Lie algebra. In particular we find that, contrary to the original category O and the specific previously known cases in the parabolic setting, the blocks are not necessarily Ringel self-dual. However, the parabolic category O as a whole is still Ringel self-dual. Furthermore, we use generalisations of the Ringel duality functor to obtain large classes of derived equivalences between blocks in parabolic and original category O. It seems that only some special cases were known thus far, where the proof relied purely on a geometric approach to category O. We subsequently classify all derived equivalence classes of blocks of category O in type A which preserve the Koszul grading.
We prove Conjecture 5.7 in [arXiv:1409.2532], describing all inclusions between primitive ideals for the general linear superalgebra in terms of the Ext1-quiver of simple highest weight modules. For arbitrary basic classical Lie superalgebras, we formulate two types of Kazhdan-Lusztig quasi-orders on the dual of the Cartan subalgebra, where one corresponds to the above conjecture. Both orders can be seen as generalisations of the left Kazhdan-Lusztig order on Hecke algebras and are related to categorical braid group actions. We prove that the primitive spectrum is always described by one of the orders, obtaining for the first time a description of the inclusions. We also prove that the two orders are identical if category O admits `enough' abstract Kazhdan-Lusztig theories. In particular, they are identical for the general linear superalgebra, concluding the proof of the conjecture.
Using deformation theory, Braverman and Joseph obtained an alternative characterisation of the Joseph ideal for simple Lie algebras, which included even type A. In this note we extend that characterisation to define a remarkable quadratic ideal for sl(m|n). When m-n>2 we prove the ideal is primitive and can also be characterised similarly to the construction of the Joseph ideal by Garfinkle.
We study three related homological properties of modules in the BGG category O for basic classical Lie superalgebras, with specific focus on the general linear superalgebra. These are the projective dimension, associated variety and complexity. We demonstrate connections between projective dimension and singularity of modules and blocks. Similarly we investigate the connection between complexity and atypicality. This creates concrete tools to describe singularity and atypicality as homological, and hence categorical, properties of a block. However, we also demonstrate how two integral blocks in category O with identical global categorical characteristics of singularity and atypicality will generally still be inequivalent. This principle implies that category O for gl(m|n) can contain infinitely many non-equivalent blocks, which we work out explicitly for gl(3|1). All of this is in sharp contrast with category O for Lie algebras, but also with the category of finite dimensional modules for superalgebras. Furthermore we characterise modules with finite projective dimension to be those with trivial associated variety. We also study the associated variety of Verma modules. To do this, we also classify the orbits in the cone of self-commuting odd elements under the action of an even Borel subgroup.
We study inclusions between primitive ideals in the universal enveloping algebra of general linear superalgebras. For classical Lie superalgebras, any primitive ideal is the annihilator of a simple highest weight module. It therefore suffices to study the quasi-order on highest weights determined by the relation of inclusion between primitive ideals. For the specific case of reductive Lie algebras, this quasi-order is essentially the left Kazhdan-Lusztig quasi-order. For Lie superalgebras, the classification is unknown in general, safe from some low dimensional specific cases. We derive an alternative definition of the left Kazhdan-Lusztig quasi-order which extends to classical Lie superalgebras. We show that a relation in this preorder implies an inclusion between primitive ideals. For gl(m|n) the new quasi-order is defined explicitly in terms of Brundan's Kazhdan-Lusztig theory. We prove that the quasi-order induces an actual partial order on the set of primitive ideals. We conjecture that this is the inclusion order. By the above paragraph one direction of this conjecture is true. We prove several consistency results concerning the conjecture and prove it for singly atypical and typical blocks of gl(m|n) and in general for gl(2|2). An important tool is a new translation principle for primitive ideals, based on the crystal structure for category O. Finally we focus on an interesting explicit example; the poset of primitive ideals contained in the augmentation ideal for gl(m|1).
We study projective dimension and graded length of structural modules in parabolic-singular blocks of the BGG category O. Some of these are calculated explicitly, others are expressed in terms of two functions. We also obtain several partial results and estimates for these two functions and relate them to monotonicity properties for quasi-hereditary algebras. The results are then applied to study blocks of O in the context of Guichardet categories, in particular, we show that blocks of O are not always weakly Guichardet.
We prove that thick category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra is extension full in the category of all modules. We also prove the weak Alexandru conjecture both for regular blocks of thick category $\mathcal{O}$ and the associated categories of Harish-Chandra bimodules, but disprove it for singular blocks.
We prove that the categories of Gelfand-Zeitlin modules of $\mathfrak{g}=\mathfrak{gl}_n$ and Whittaker modules associated with a semi-simple complex finite-dimensional algebra $\mathfrak{g}$ are extension full in the category of all $\mathfrak{g}$-modules. This is used to estimate and in some cases determine the global dimension of blocks of the categories of Gelfand-Zeitlin and Whittaker modules.
The main focus of this paper is Bott-Borel-Weil (BBW) theory for basic classical Lie superalgebras. We take a purely algebraic self-contained approach to the problem. A new element in this study is twisting functors, which we use in particular to prove that the top of the cohomology groups of BBW theory for generic weights is described by the recently introduced star action. We also study the algebra of regular functions, related to BBW theory. Then we introduce a weaker form of genericness, relative to the Borel subalgebra and show that the virtual BGG reciprocity of Gruson and Serganova becomes an actual reciprocity in the relatively generic region. We also obtain a complete solution of BBW theory for osp(m|2), D(2,1;a), F(4) and G(3) with distinguished Borel subalgebra. Furthermore, we derive information about the category of finite dimensional osp(m|2)-modules, such as BGG-type resolutions and Kostant homology of Kac modules and the structure of projective modules.
Pizzetti's formula explicitly shows the equivalence of the rotation invariant integration over a sphere and the action of rotation invariant differential operators. We generalize this idea to the integrals over real, complex, and quaternion Stiefel manifolds in a unifying way. In particular we propose a new way to calculate group integrals and try to uncover some algebraic structures which manifest themselves for some well-known cases like the Harish-Chandra integral. We apply a particular case of our formula to an Itzykson-Zuber integral for the coset SO(4)/[SO(2)xSO(2)]. This integral naturally appears in the calculation of the two-point correlation function in the transition of the statistics of the Poisson ensemble and the Gaussian orthogonal ensemble in random matrix theory.
We study three related topics in representation theory of classical Lie superalgebras. The first one is classification of primitive ideals, i.e. annihilator ideals of simple modules, and inclusions between them. The second topic concerns Arkhipov's twisting functors on the BGG category O. The third topic addresses deformed orbits of the Weyl group. These take over the role of the usual Weyl group orbits for Lie algebras, in the study of primitive ideals and twisting functors for Lie superalgebras.
We introduce the spinor representations for osp(m|2n). These generalize the spinors for so(m) and the symplectic spinors for sp(2n) and correspond to representations of the supergroup with supergroup pair (Spin(m) x Mp(2n),osp(m|2n)). We prove that these spinor spaces are uniquely characterized as the completely pointed osp(m|2n)-modules. Then the tensor product of this representation with irreducible finite dimensional osp(m|2n)-modules is studied. Therefore we derive a criterion for complete reducibility of tensor product representations. We calculate the decomposition into irreducible osp(m|2n)-representations of the tensor product of the super spinor space with an extensive class of such representations and also obtain cases where the tensor product is not completely reducible.
We study Kostant cohomology and Bernstein-Gelfand-Gelfand resolutions for finite dimensional representations of basic classical Lie superalgebras and reductive Lie superalgebras based on them. For each choice of parabolic subalgebra and irreducible representation of such a Lie superalgebra, there is a natural definition of the derivative and coderivative, which define the (co)homology groups. We prove that a necessary condition to have a resolution of an irreducible module in terms of Verma modules is complete reducibility of the cohomology groups. Essentially, if it exists, every such a resolution is then given by modules induced by these cohomology groups. We also prove that if these cohomology groups are completely reducible, a sufficient condition for the existence of such a resolution is that these groups are isomorphic to the kernel of the Kostant quabla operator, which is equivalent with disjointness of the derivative and coderivative. Then we use these results to derive very explicit criteria under which BGG resolutions exist, which are particularly useful for the superalgebras of type I. For the unitarisable representations of gl(m|n) and osp(2|2n) we derive conditions on the parabolic subalgebra under which the BGG resolutions exist. This extends the BGG resolutions for gl(m|n) previously obtained through superduality and leads to entirely new results for osp(2|2n). We also apply the obtained theory to construct specific examples of BGG resolutions for osp(m|2n).
We discuss several topics of homological algebra for the Lie superalgebra osp(1|2n). First we focus on Bott-Kostant cohomology, which yields classical results although the cohomology is not given by the kernel of the Kostant quabla operator. Based on this cohomology we can derive strong Bernstein-Gelfand-Gelfand resolutions for finite dimensional osp(1|2n)-modules. Then we state the Bott-Borel-Weil theorem which follows immediately from the Bott-Kostant cohomology by using the Peter-Weyl theorem for osp(1|2n). Finally we calculate the projective dimension of irreducible and Verma modules in the category O.
In this paper, the Dirac operator, acting on super functions with values in super spinor space, is defined along the lines of the construction of generalized Cauchy-Riemann operators by Stein and Weiss. The introduction of the superalgebra of symmetries osp(m|2n) is a new and essential feature in this approach. This algebra of symmetries is extended to the algebra of conformal symmetries osp(m + 1, 1|2n). The kernel of the Dirac operator is studied as a representation of both algebras. The construction also gives an explicit realization of the Howe dual pair osp(1|2) x osp(m|2n) < osp(m + 4n|2m + 2n). Finally, the super Dirac operator gives insight into the open problem of classifying invariant first order differential operators in super parabolic geometries.