Jantzen filtration - Alchetron, The Free Social Encyclopedia

In algebra, a Jantzen filtration is a filtration of a Verma module of a semisimple Lie algebra, or a Weyl module of a reductive algebraic group of positive characteristic. Jantzen filtrations were introduced by Jantzen (1979).

Jantzen filtration for Verma modules

If M(λ) is a Verma module of a semisimple Lie algebra with highest weight λ, then the Janzen filtration is a decreasing filtration

M ( λ ) = M ( λ ) 0 ⊇ M ( λ ) 1 ⊇ M ( λ ) 2 ⊇ ⋯ .

It has the following properties:

M(λ)¹ is the maximal proper submodule of M(λ)

The quotients M(λ)^k/M(λ)^k+1 have non-degenerate contravariant bilinear forms.

∑ i > 0 Ch ( M ( λ ) i ) = ∑ α > 0 , s α ( λ ) < λ Ch ( M ( s α ( λ ) ) )

(the Jantzen sum formula)

References

Jantzen filtration Wikipedia

(Text) CC BY-SA