Classification of linkage systems

Abstract

In 1972, R. Carter introduced admissible diagrams to classify conjugacy classes in a finite Weyl group W . For any two non-orthogonal roots α and β corresponding to vertices of admissible diagram, we draw the dotted (resp. solid) edge {α, β} if (α, β) > 0 (resp. (α, β) < 0). The diagram with properties of admissible diagram and possibly containing dotted edges are said to be Carter diagrams. For any Carter diagram Γ, we introduce the partial Cartan matrix BΓ, which is analogous to the Cartan matrix associated with a Dynkin diagram. A linkage diagram is obtained from Γ by adding an extra root γ, together with its bonds, so that the resulting subset of roots is linearly independent. With every linkage diagram we associate the linkage label vector γ ∇ , similar to the "numerical labels" introduced by Dynkin for the study of irreducible linear representations of the semisimple Lie algebras. The linkage diagrams connected under the action of dual partial Weyl group W ∨ S (associated with BΓ) constitute the set L (Γ) dubbed the linkage system. For any simply-laced Carter diagram Γ, the linkage system L (Γ) is explicitly constructed. To obtain linkage diagrams θ ∇ ∈ L (Γ), we use an easily verifiable criterion: B ∨ Γ (θ ∇ ) < 2, where B ∨ Γ is the quadratic form associated with B −1 Γ . A Dynkin diagram Γ ′ such that rank(Γ ′ ) = rank(Γ) + 1 and any Γ-associated root subset S lies in the root system Φ(Γ ′ ), will be called the Dynkin extension of the Carter diagram Γ and denoted by Γ <D Γ ′ . The linkage system L (Γ) is the union of Γi-components LΓ i (Γ) taken for all Dynkin extensions of Γ <D Γi. The subset Φ(S) of roots of Φ(Γ ′ ) linearly dependent on roots of S is said to be a partial root system. The size of L Γ ′ (Γ) is estimated as follows:Carter diagrams of the same type and the same index are said to be covalent. For any pair {Γ, Γ} of covalent Carter diagrams, where Γ is the Dynkin diagram, we explicitly construct the invertible linear map M : P −→ R, where R (resp. P) is the root system (resp. partial root system) corresponding to Γ (resp. Γ). This allows us to derive some results for the Carter diagrams from analogous results for the Dynkin diagrams. In particular, we have | L ( Γ) |=| L (Γ) |.For the simple-laced Dynkin diagram Γ, every A-D-or E-component of the linkage system L (Γ) coincides with a weight system of one of fundamental representations of the simple Lie algebra g associated with Γ.The 8-cell "spindle-like" subsystems in L (Γ), called loctets, play the essential role in describing the linkage systems. 1.3.4. Dynkin extensions 1.3.5. Root stratum 1.3.6. Linkage system components 2. Introduction and main results 2.1. The partial Cartan matrix, inverse quadratic form and linkages 2.1.1. Linkages and linkage diagrams 2.1.2. The projection of the linkage root 2.2. The linkage systems and loctets 2.2.1. The starlike numbering of vertices 2.2.2. Loctets 2.3. The Carter diagrams and connection diagrams 2.3.1. The Dynkin diagrams 2.3.2. The connection diagrams 2.4. The main results 2.4.1. Partial Weyl...

Inverses of Cartan matrices of Lie algebras and Lie superalgebras

“…• For inverses of partial Cartan matrices, and some examples of their usage, see [Stk,]; the inverses of Cartan matrices are also recalled there (Tables A.14-A.15).…”

“…We also corrected and widened the list of references (and the range of applications) of inverses of Cartan matrices as compared with those given in [WZ], e.g., the explicit form of inverse Cartan matrices of finite-dimensional simple Lie algebras is reproduced in [WZ] as if new, though well-known, see [D1,OV,Stk]. These omissions and the desire to solve the problem considered, but not solved, in [WZ], except for occasional coincidences of Cartan matrices with Gram matrices of non-degenerate invariant symmetric bilinear forms (NISes) on the (super)algebras considered, is what prompted our work.…”