The recognition theorem for graded Lie algebras in prime by Georgia Benkart
By Georgia Benkart
Quantity 197, quantity 920 (second of five numbers).
Read or Download The recognition theorem for graded Lie algebras in prime characteristic PDF
Similar algebra & trigonometry books
During this creation to commutative algebra, the writer choses a course that leads the reader during the crucial principles, with out getting embroiled in technicalities. he's taking the reader quick to the basics of advanced projective geometry, requiring just a simple wisdom of linear and multilinear algebra and a few user-friendly workforce concept.
This publication is meant for the Mathematical Olympiad scholars who desire to arrange for the research of inequalities, a subject matter now of widespread use at a number of degrees of mathematical competitions. during this quantity we current either vintage inequalities and the extra valuable inequalities for confronting and fixing optimization difficulties.
This quantity provides the court cases of the foreign convention on ""Recent development in Algebra"" that was once held on the Korea complicated Institute of technological know-how and know-how (KAIST) and Korea Institute for complex examine (KIAS). It introduced jointly specialists within the box to debate development in algebra, combinatorics, algebraic geometry and quantity thought.
- Polynomials: a problem book
- Algebra and Tiling: Homomorphisms in the Service of Geometry
- Ockham Algebras (Oxford Science Publications)
- Matrix Theory: A Second Course
- Degree Theory for Equivariant Maps, the General S1-Action
Additional resources for The recognition theorem for graded Lie algebras in prime characteristic
Hence if P ⊆ t, then −γ P contains a root vector, eγ say. In this case e−γ ∈ [g−γ 0 , [g0 , eγ ]] is in P as well. So it can be assumed further that γ > 0. As the Lie algebra n+ is nilpotent, the ad n+ -module generated by eγ contains a root vector eδ (with δ > 0) such that [n+ , eδ ] = 0. 12. 63 IN THE DEPTH-ONE CASE 21 root α > 0. As t is abelian, this shows that P is contained in the center Z(g0 ) of g0 . Since dim Z(g0 ) ≤ 1, it must be that P = Z(g0 ) in this case. If P ⊆ t, we set e = eδ .
Thus, any rational G-module L(λ) can be regarded, canonically, as a restricted Lie(G)-module. 13. (Compare [Ja, Part II, Prop. 10 and Prop. ) Any irreducible restricted Lie(G)-module is isomorphic to exactly one module L(λ) with λ ∈ X1 (T ). Conversely, any G-module L(λ) with λ ∈ X1 (T ) is irreducible as a Lie(G)-module. The module V (λ) := H 0 (−w0 λ)∗ which is dual to H 0 (−w0 λ) is called the Weyl module corresponding to λ. There is an antiautomorphism τ of order 2 on G which acts as the identity on T and sends the root subgroup Uα to U−α , for each α ∈ Φ (as in [Ja, Part II, Sec.
For k = 0, 1, . . , p − 1 we set xk = [(ad eα )k (w), v] and observe that xk ∈ gη+kα 0 where η = β − λ + ν. Since e commutes with eα and (ad eα )p [w, e] = χ(eα )p [w, e], a nonzero multiple of [w, e], it must be that [u, xk ] = [(ad eα )k (w), [u, v]] = (ad eα )k [w, e] = 0 for all k ≥ 0. But then gη+kα = 0 for k = 0, 1, . . , p − 1. If η is not a 0 multiple of α, this is impossible, since root strings in classical Lie algebras have length at most 4. If η is a multiple of α, then all multiples kα for k = 1, .