The signed shell Lie algebra over a full Cartan lattice #
Let B : Matrix I I ℤ be the Gram matrix of the lattice Q = I → ℤ, with norm-two shell Φ.
The Lie ring Matrix.ShellLieAlgebra B has Cartan part Q itself. For the Lie algebra of a
reductive group whose roots are the vectors of Φ, the Cartan part should instead be the full
cocharacter lattice Y: it can be larger than the span of the coroots, as for GLₙ, and it need
not split off the coroot lattice integrally. This file replaces Q by an arbitrary abelian group
Y equipped with
- root weights
⟨a, -⟩ : Y →ₗ[ℤ] ℤ, linear ina : Q, and - a coroot map
a ↦ a∨ : Q →ₗ[ℤ] Y,
subject to ⟨a, b∨⟩ = ⟪a, b⟫, where ⟪a, b⟫ = a ⬝ᵥ B *ᵥ b. This is a
Matrix.ShellCartanDatum B Y. The carrier Matrix.FullShellLieAlgebra D is Y × (Φ →₀ ℤ), with
a Cartan vector H_y for each y : Y and a root vector x_a for each a ∈ Φ, and the bracket is
⁅H_y, H_z⁆ = 0and⁅H_y, x_a⁆ = ⟨a, y⟩ x_a,⁅x_a, x_b⁆ = ε(a, b) x_(a + b)ifa + b ∈ Φ,⁅x_a, x_b⁆ = ε(a, b) H_(a∨)ifa + b = 0,⁅x_a, x_b⁆ = 0otherwise,
with the ordered sign ε of Matrix.SignCover B, as in Matrix.ShellLieAlgebra. Only the
coroot of an opposite bracket is moved into Y; no central or derived summand of Y is chosen.
The map Matrix.FullShellLieAlgebra.toFull, H_h ↦ H_(h∨) and x_a ↦ x_a, from
Matrix.ShellLieAlgebra B preserves the bracket for symmetric B
(Matrix.FullShellLieAlgebra.toFull_lie). Every root bracket of the full algebra is the image of
the corresponding root bracket of the shell algebra, so the antisymmetry and the Jacobi identity
for root vectors are transported from Matrix.ShellLieAlgebra; the remaining cases only use that
⟨-, y⟩ is additive. For B positive definite with even diagonal the bracket is therefore a Lie
ring (Matrix.FullShellLieAlgebra.lieRing). As in the shell case nothing is divided by 2, and
Y may have torsion. For the datum Matrix.ShellCartanDatum.rootLattice B, with Y = Q, the
map toFull is a linear equivalence (Matrix.FullShellLieAlgebra.toFullEquiv), so for
symmetric B the shell algebra is the special case of the coroot lattice. When Φ is empty, as
for a torus, the full algebra is the abelian Lie ring Y.
Main definitions #
Matrix.ShellCartanDatum B Y: root weights and coroots inYcompatible withB.Matrix.ShellCartanDatum.rootLattice B: the datum withY = Qand identity coroot map.Matrix.FullShellLieAlgebra D: the carrierY × (Φ →₀ ℤ).Matrix.FullShellLieAlgebra.toFull: the mapH_h ↦ H_(h∨),x_a ↦ x_afrom the shell algebra.Matrix.FullShellLieAlgebra.toFullEquiv B:toFullas a linear equivalence for the datumMatrix.ShellCartanDatum.rootLattice B.Matrix.FullShellLieAlgebra.rootBracket: the bracket of two root vectors.Matrix.FullShellLieAlgebra.bracketBilin: the bracket, as a bilinear map.Matrix.FullShellLieAlgebra.lieRing: the Lie ring structure, for positive definiteBwith even diagonal.
Main results #
Matrix.FullShellLieAlgebra.cartan_lie_root,Matrix.FullShellLieAlgebra.root_lie_root: the bracket on Cartan and root vectors.Matrix.FullShellLieAlgebra.rootBracket_of_add_eq_zero:⁅x_a, x_(-a)⁆ = ε(a, -a) H_(a∨).Matrix.FullShellLieAlgebra.toFull_lie:toFullpreserves the bracket, for symmetricB.Matrix.FullShellLieAlgebra.lie_skew,Matrix.FullShellLieAlgebra.lie_self: antisymmetry and alternation, for symmetricBwith even diagonal.Matrix.FullShellLieAlgebra.lie_jacobi: the Jacobi identity, for positive definiteBwith even diagonal.
References #
- J. C. Jantzen, Representations of Algebraic Groups, second edition, American Mathematical Society (2003), Part II, Chapter 1, for the decomposition of the Lie algebra of a split reductive group into the Lie algebra of its maximal torus and the root spaces.
- V. G. Kac, Infinite dimensional Lie algebras, third edition, Cambridge University Press (1990), §7.8.
A full Cartan lattice for the norm-two shell of an integer Gram matrix B on Q = I → ℤ:
an abelian group Y with root weights ⟨a, -⟩ = weight a : Y →ₗ[ℤ] ℤ, linear in a : Q, and a
coroot map coroot : Q →ₗ[ℤ] Y, such that ⟨a, b∨⟩ = a ⬝ᵥ B *ᵥ b. For a reductive root datum
whose roots, in simple-root coordinates, are the norm-two shell of B, Y is the cocharacter
lattice and ⟨a, -⟩ is the pairing with the character of a.
The root weight
⟨a, -⟩ : Y →ₗ[ℤ] ℤof a lattice vectora.The coroot
a∨ : Yof a lattice vectora.Root weights of coroots are given by the Gram matrix:
⟨a, b∨⟩ = a ⬝ᵥ B *ᵥ b.
Instances For
The coroot lattice as a full Cartan lattice: Y = Q, the identity coroot map, and root
weights ⟨a, y⟩ = a ⬝ᵥ B *ᵥ y.
Instances For
The carrier Y × (Φ →₀ ℤ) of the signed Lie algebra of the norm-two shell Φ of B over the
full Cartan lattice Y of D. The bracket is Matrix.FullShellLieAlgebra.bracketBilin.
Instances For
The identification of FullShellLieAlgebra D with the product Y × (Φ →₀ ℤ) defining it.
Instances For
The Cartan vector H_y of y : Y, as a linear map Y →ₗ[ℤ] FullShellLieAlgebra D.
Instances For
The root vector x_a of a vector a of the norm-two shell.
Instances For
Two linear maps out of FullShellLieAlgebra D agree once they agree on all Cartan vectors and
on all root vectors.
The map H_h ↦ H_(h∨), x_a ↦ x_a from the signed Lie algebra of the norm-two shell, with
Cartan part Q, to the one over the full Cartan lattice of D.
Instances For
For the coroot lattice Y = Q, toFull is a linear equivalence
(Matrix.FullShellLieAlgebra.toFullEquiv_apply); for symmetric B it preserves the bracket by
Matrix.FullShellLieAlgebra.toFull_lie.
Instances For
The bracket ⁅x_a, x_b⁆ of two root vectors: ε(a, b) x_(a + b) if a + b is in the norm-two
shell, ε(a, b) H_(a∨) if a + b = 0, and 0 otherwise, where ε(a, b) is the ordered sign
(B.orderedSignForm a b).negOnePow. It is the image under toFull of the root bracket of the
shell algebra.
Instances For
The bracket of two root vectors whose sum is in the norm-two shell.
The bracket of two opposite root vectors is a multiple of the coroot of the first.
Two root vectors whose sum is neither in the norm-two shell nor zero have bracket zero.
The bracket of FullShellLieAlgebra D, as the bilinear map determined by ⁅H_y, H_z⁆ = 0,
⁅H_y, x_b⁆ = ⟨b, y⟩ x_b, ⁅x_a, H_z⁆ = -⟨a, z⟩ x_a and ⁅x_a, x_b⁆ = rootBracket D a b.
Instances For
The bracket of FullShellLieAlgebra D is Matrix.FullShellLieAlgebra.bracketBilin. It is a
Lie bracket for positive definite B with even diagonal, see Matrix.FullShellLieAlgebra.lieRing.
Cartan vectors commute.
The Cartan vector H_y acts on the root vector x_b by the root weight ⟨b, y⟩.
The bracket ⁅x_a, H_z⁆ = -⟨a, z⟩ x_a.
The bracket of two root vectors is their root bracket.
toFull preserves the bracket, for a symmetric B.
The Cartan weight of a root bracket. The bracket of two root vectors has root weight the sum of the weights of their roots.
Antisymmetry of the root bracket, for a symmetric B with even diagonal.
Antisymmetry of the bracket, for a symmetric B with even diagonal.
The bracket is alternating, for a symmetric B with even diagonal. The group Y may have
2-torsion, so this is not a consequence of antisymmetry alone.
The Jacobi identity #
The Jacobi identity, for a positive definite B with even diagonal.
The Leibniz form of the Jacobi identity, for a positive definite B with even
diagonal.
The signed Lie ring of the norm-two shell over a full Cartan lattice, for a positive
definite integer matrix B with even diagonal. It is not an instance, since it depends on these
two hypotheses; install it locally when needed.