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Series of Lie groups. (English) Zbl 1165.17302

From the introduction: One way to define a collection of Lie algebras \({\mathfrak g}(t)\), parameterized by \(t\) and each equipped with a representation \(V(t)\), as forming a “series” is to require (following Deligne) that the tensor powers of \(V(t)\) decompose into irreducible \({\mathfrak g}(t)\)-modules in a manner independent of \(t\), with formulas for the dimensions of the irreducible components of the form \(P(t)/Q(t)\) where \(P\), \(Q\) are polynomials that decompose into products of integral linear factors. We study such decomposition formulas in this paper, which provides a companion to [J. M. Landsberg and L. Manivel, Adv. Math. 171, No. 1, 59–85 (2002; Zbl 1035.17016)], where we study the corresponding dimension formulas. We connect the formulas to the geometry of the closed orbits \(X(t)\subset\mathbb PV(t)\) and their unirulings by homogeneous subvarieties. We relate the linear unirulings to work of B. Kostant [Topology 3, Suppl. 2, dedicated to Arnold Shapiro, 147–159 (1965; Zbl 0134.03504)]. By studying such series, we determine new modules that, appropriately viewed, are exceptional in the sense of M. Brion [Ann. Inst. Fourier 33, No. 1, 1–27 (1983; Zbl 0475.14038) (see, e.g., Theorem 6.2).
The starting point of this paper was the work of Deligne et al. containing uniform decomposition and dimension formulas for the tensor powers of the adjoint representations of the exceptional simple Lie algebras up to \({\mathfrak g}^{\otimes 5}\). Deligne’s method for the decomposition formulas was based on comparing Casimir eigenvalues, and he offered a conjectural explanation for the formulas via a categorical model based on bordisms between finite sets. P. Vogel [The universal Lie algebra, preprint (1999)] obtained similar formulas for all simple Lie superalgebras based on his universal Lie algebra. We show that all primitive factors in the decomposition formulas of Deligne and Vogel can be accounted for using a pictorial procedure with Dynkin diagrams. (The nonprimitive factors are those either inherited from lower degrees or arising from a bilinear form, so knowledge of the primitive factors gives the full decomposition.) We also derive new decomposition formulas for other series of Lie algebras.
In Section 2, we describe a pictorial procedure using Dynkin diagrams for determining the decomposition of \(V^{\otimes k}\) In Sections 3 and 4 we distinguish and interpret the primitive components in the decomposition formulas of Deligne and Vogel.
The exceptional series of Lie algebras occurs as a line in Freudenthal’s magic square. The three other lines each come with preferred representations. Dimension formulas for all representations supported on the cone in the weight lattice generated by the weights of the preferred representations, similar to those of the exceptional series, were obtained in the authors’ paper (loc. cit.).
Sections 5–7 we obtain the companion decomposition formulas. A nice property shared by many of these preferred representations is that they are exceptional , in the sense of Brion (loc. cit.) that is, their covariant algebras are polynomial algebras. We prove that, in some cases where this is not naively true, it becomes so when we take the symmetry group of the associated marked Dynkin diagram into account.
In the course of revising the exposition of this paper, we ran across the closely related article by P. Deligne and B. H. Gross [C. R., Math., Acad. Sci. Paris 335, No. 11, 877–881 (2002; Zbl 1017.22008)].

MSC:

17B25 Exceptional (super)algebras
14M15 Grassmannians, Schubert varieties, flag manifolds
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Software:

LiE
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Full Text: DOI arXiv Euclid

References:

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