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The PBW model of a Verma module
Statement
Multiplication gives a vector-space isomorphism
Consequently, ordered negative-root PBW monomials from PBW gives an ordered monomial basis for the enveloping algebra applied to form a basis.
Facts & Assumptions
Given: The triangular decomposition and the induced-module relations defining .
Proof
Triangular PBW from Triangular decomposition from a chosen positive root system identifies with . Tensoring over with therefore gives .
Under that identification, maps to , so the map is bijective. Applying the ordered PBW basis in the negative factor proves the asserted basis statement.
Depends on
Used by
- The BGG resolution has length the number of positive roots Corollary
- Equal central character does not give every Verma embedding direction Counterexample
- The Jantzen deformation and filtration of a Verma module Definition
- A two-dimensional A2 Verma weight space Example
- Nonintegral sl2 central characters split into two simple blocks Example
- Reducible and generic sl2 Verma modules Example
- The A2 regular integral-dominant Verma embedding poset Example
- The finite-dimensional sl2 quotient of a Verma module Example
- The regular integral sl2 block Example
- The sl2 Verma action in the PBW basis Example
- The sl2 Verma embedding chain Example
- Translation between two regular integral blocks Example
- Translation to a singular wall in type A₂ Example
- A nonzero homomorphism between Verma modules is injective Lemma
- Dimension of the kernel modulo n-minus equals the next term (BGG 10.7) Lemma
- Preliminary factorization of a Shapovalov determinant Lemma
- Surjectivity modulo n-minus for free weight-generated modules (BGG 10.5) Lemma
- Tensoring a Verma module by a finite-dimensional module shifts the type Lemma
- The augmentation kernel is the sum of the simple-reflection Verma submodules Lemma
- The BGG differential induces an injection into kernel coinvariants (BGG 10.6) Lemma
- The first transverse Shapovalov pairing is perfect Lemma
- The simple-root singular vector in a Verma module Lemma
- Tor with the trivial module is computed by the weak BGG resolution Lemma
- Verma-filtered objects are acyclic for n-minus coinvariants Lemma
- Simple-reflection embeddings of Verma modules Proposition
- The formal character of a Verma module Proposition
- Weights of a Verma module lie below lambda Proposition
- Duflo–Joseph g-finite endomorphisms of every Verma module Theorem
- Existence and uniqueness of the Shapovalov form Theorem
- Parabolic category O multiplicities are parabolic Kazhdan–Lusztig values Theorem
- Soergel's Struktursatz: the functor V is fully faithful on projectives Theorem
- The BGG resolution of a finite-dimensional simple module Theorem
Dependency tree · two levels
8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Pavel Etingof, Lie Groups and Lie Algebras I & II, Proposition 25.7 (standard reference, not scraped)