Two-cocycles on loop algebras and their cohomology classes #
Let L be a Lie algebra over an additively torsion-free commutative ring K, such as a
characteristic-zero domain, and write x_m = t^m ⊗ x for the modes of the polynomial loop algebra
loopAlgebra K ℤ L = K[t, t⁻¹] ⊗ L. Mathlib attaches to an invariant symmetric form Ψ on L the
residue 2-cocycle
LieAlgebra.LoopAlgebra.twoCocycleOfBilinear, whose value on modes is
(x_m, y_n) ↦ n Ψ(x, y) δ_{m + n, 0}, the residue of x_m d y_n paired by Ψ.
This file proves the mode-recurrence step of the classification of central extensions of loop
algebras: if L is perfect, then a 2-cocycle with trivial coefficients is the residue cocycle of
an invariant symmetric form on L exactly when it vanishes whenever its first argument is a
constant loop x_0 = 1 ⊗ x.
The argument runs as follows. For a 2-cocycle c vanishing on constant loops, the cocycle
identity with a constant first argument says that each mode block B_{m,n} = (x, y) ↦ c(x_m, y_n)
is an invariant form on L. The cocycle identity on three modes, rewritten by invariance, says
that B_{k + m, n} and B_{k, m + n} + B_{m, k + n} agree on brackets, hence everywhere because
the brackets of a perfect Lie algebra span it. On each pair of vectors this is the recurrence
λ(k + m, n) = λ(k, m + n) + λ(m, k + n). Along each line m + n = s it makes a ↦ λ(a, s - a)
additive, hence linear in a; since it vanishes at a = s, it vanishes identically unless
s = 0, and on the line s = 0 it is a λ(1, -1). So c is the residue cocycle of -B_{1,-1}.
As a corollary, if L is perfect and every invariant bilinear form on L is a multiple of
Ψ, then a 2-cocycle is a scalar multiple of the residue cocycle of Ψ exactly when it vanishes
on constant loops. A finite-dimensional Lie algebra over an algebraically closed field of
characteristic zero with irreducible adjoint module and nondegenerate Killing form, such as a
simple Lie algebra, satisfies these hypotheses with Ψ its Killing form.
The vanishing on constant loops is a normalisation that every 2-cocycle admits up to a coboundary
when L is finite-dimensional with nondegenerate Killing form over a field of characteristic zero.
For each mode n, the cocycle identity says that x ↦ (y ↦ c(x_0, y_n)) is a 1-cocycle of L
with values in the coadjoint module Module.Dual K L. This module is finite-dimensional, so
Whitehead's first lemma writes the block as c(x_0, y_n) = -φ_n ⁅x, y⁆, and subtracting the
coboundary of the 1-cochain y_n ↦ φ_n y removes all values on constant loops. Working one mode
at a time avoids any Whitehead lemma for the infinite-dimensional dual of the loop algebra.
Combining the two steps, for L finite-dimensional with nondegenerate Killing form over a field of
characteristic zero, that is for L semisimple, every 2-cocycle on loopAlgebra K ℤ L is the sum
of the residue cocycle of an invariant symmetric form and a coboundary; when one form Ψ spans the
invariant forms, the 2-cocycles are the sums of a multiple of the residue cocycle of Ψ and a
coboundary. Conversely, over a characteristic-zero domain, a nonzero multiple of the residue
2-cochain of a nonzero symmetric form is never a coboundary: on (x_1, y_{-1}) and (x_{-1}, y_1)
it takes opposite values ∓Ψ(x, y), while a coboundary takes the same value -f(⁅x, y⁆_0) on both.
Thus for a simple Lie algebra over an algebraically closed field of characteristic zero the second
cohomology of its loop algebra with trivial coefficients is one-dimensional, spanned by the residue
class of the Killing form.
Main results #
TauCeti.LieAlgebra.LoopAlgebra.residuePairing_single_tmul: the residue pairing on modes.TauCeti.LieAlgebra.LoopAlgebra.residuePairing_one_tmul: the residue pairing vanishes when its first argument is a constant loop.TauCeti.LieAlgebra.LoopAlgebra.single_add_tmul_mem_span_lie: for a perfect Lie algebra, every modex_{a + b}is a linear combination of brackets⁅y_a, z_b⁆.TauCeti.LieAlgebra.LoopAlgebra.twoCochainOfBilinear_smul: rescaling the form rescales the residue 2-cochain.TauCeti.LieAlgebra.LoopAlgebra.exists_eq_twoCocycleOfBilinear_iff: for a perfect Lie algebra, a 2-cocycle is a residue cocycle if and only if it vanishes on constant loops.TauCeti.LieAlgebra.LoopAlgebra.exists_eq_smul_twoCocycleOfBilinear_iff: a 2-cocycle is a multiple of the residue cocycle ofΨif and only if it vanishes on constant loops, for a perfect Lie algebra and a formΨspanning the invariant forms.TauCeti.LieAlgebra.LoopAlgebra.exists_eq_smul_twoCocycleOfBilinear_killingForm_iff: the case of the Killing form of a simple Lie algebra.TauCeti.LieAlgebra.LoopAlgebra.exists_apply_one_tmul_eq_d₁₂_apply: every 2-cocycle agrees with a coboundary on constant loops, for any exponent monoid.TauCeti.LieAlgebra.LoopAlgebra.smul_twoCochainOfBilinear_mem_range_d₁₂_iff: over a characteristic-zero domain, a multiple of the residue 2-cochain of a nonzero symmetric form is a coboundary only when the multiple is zero.TauCeti.LieAlgebra.LoopAlgebra.exists_eq_twoCochainOfBilinear_add_d₁₂: for a semisimple Lie algebra, every 2-cocycle is a residue cocycle plus a coboundary.TauCeti.LieAlgebra.LoopAlgebra.twoCocycle_eq_span_sup_range_d₁₂: the 2-cocycles are the multiples of the residue cocycle of a spanning form plus coboundaries.TauCeti.LieAlgebra.LoopAlgebra.twoCocycle_eq_span_killingForm_sup_range_d₁₂: the case of the Killing form of a simple Lie algebra.
References #
- V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed., Chapter 7, for loop algebras and their central extensions by the residue cocycle.
- N. Jacobson, Lie Algebras, Chapter III, for Whitehead's first lemma.
- C. Kassel, Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra, J. Pure Appl. Algebra 34 (1984), for the universal central extension of current algebras.
The residue pairing on modes: Res(x_m d y_n) paired by Φ is n Φ(x, y) when
m + n = 0, and zero otherwise.
The residue pairing is linear in the form: rescaling the form rescales the pairing.
Rescaling the form rescales the residue 2-cochain.
The residue pairing vanishes when its first argument is a constant loop 1 ⊗ x.
Modes of a perfect Lie algebra are sums of brackets of modes. If L is perfect, then for
all exponents a and b every mode x_{a + b} is a linear combination of brackets
⁅y_a, z_b⁆ of modes y_a = t^a ⊗ y and z_b = t^b ⊗ z.
The 2-cocycles on the loop algebra of a perfect Lie algebra vanishing on constant loops are
the residue cocycles. Let L be a perfect Lie algebra over an additively torsion-free commutative
ring. A 2-cocycle with trivial coefficients on the polynomial loop algebra loopAlgebra K ℤ L is
the residue cocycle of an invariant symmetric form on L if and only if it vanishes whenever its
first argument is a constant loop 1 ⊗ x.
The 2-cocycles on a loop algebra vanishing on constant loops are the multiples of the
residue cocycle. Let L be a perfect Lie algebra over an additively torsion-free commutative
ring and Ψ an invariant symmetric form on L such that every invariant bilinear form on L is a
multiple of Ψ. A 2-cocycle with trivial coefficients on the polynomial loop algebra
loopAlgebra K ℤ L is a scalar multiple of the residue cocycle of Ψ if and only if it vanishes
whenever its first argument is a constant loop 1 ⊗ x.
The 2-cocycles on the loop algebra of a simple Lie algebra vanishing on constant loops are
the multiples of the residue cocycle of the Killing form. Here L is a finite-dimensional Lie
algebra over an algebraically closed field of characteristic zero whose adjoint module is
irreducible and whose Killing form is nondegenerate, for instance a simple Lie algebra.
Every 2-cocycle on a loop algebra agrees with a coboundary on constant loops. Let L be a
finite-dimensional Lie algebra with nondegenerate Killing form over a field of characteristic
zero. For every 2-cocycle c with trivial coefficients on loopAlgebra K A L there is a 1-cochain
f with c(1 ⊗ x, a) = d₁₂ f (1 ⊗ x, a) for all x and a, so that c - d₁₂ f vanishes
whenever its first argument is a constant loop.
The residue 2-cochain of a nonzero symmetric form is not a coboundary. Over a
characteristic-zero domain, a multiple t • c_Ψ of the residue 2-cochain of a nonzero symmetric
form Ψ on L is a coboundary on the loop algebra loopAlgebra K ℤ L only for t = 0.
Every 2-cocycle on a loop algebra is a residue cocycle up to a coboundary. Let L be a
finite-dimensional Lie algebra with nondegenerate Killing form over a field of characteristic zero,
for instance a semisimple Lie algebra. Every 2-cocycle c with trivial coefficients on
loopAlgebra K ℤ L is c_Ψ + d₁₂ f for an invariant symmetric form Ψ on L and a 1-cochain
f.
The second cohomology of a loop algebra is spanned by the residue class. Let L be a
finite-dimensional Lie algebra with nondegenerate Killing form over a field of characteristic
zero, and Ψ an invariant symmetric form on L spanning the invariant forms. Then the 2-cocycles
with trivial coefficients on loopAlgebra K ℤ L are exactly the sums of a multiple of the residue
cocycle of Ψ and a coboundary.
The second cohomology of the loop algebra of a simple Lie algebra is spanned by the residue
class of the Killing form. Here L is a finite-dimensional Lie algebra over an algebraically
closed field of characteristic zero whose adjoint module is irreducible and whose Killing form is
nondegenerate, for instance a simple Lie algebra. Together with
smul_twoCochainOfBilinear_mem_range_d₁₂_iff, this says that the second cohomology of
loopAlgebra K ℤ L with trivial coefficients is one-dimensional.