The dimension character and the nonvanishing of the global dimension #
Let C be a pivotal category which is k-linear, for an algebraically closed field k, whose
tensor unit is simple with finite-dimensional endomorphisms. Reading the left dimension in k
through TauCeti.endAlgEquivOfSimple k (𝟙_ C), it is additive on biproducts, 1 on the unit,
and multiplicative on tensor products, because every scalar acts on every object in the same way
from the left and from the right (TauCeti.leftDim_tensor_eq_comp_of_simple). So it is a ring
homomorphism from the Grothendieck ring, the left dimension character
TauCeti.SplitK0.leftDimChar k C, with a mirror TauCeti.SplitK0.rightDimChar k C; the duality
[X] ↦ [Xᘁ] exchanges them, and they coincide in a spherical category.
For a fusion category over a field of characteristic zero this gives the theorem of
Etingof–Nikshych–Ostrik that the global dimension
globalDim C = ∑ S, leftDim S ≫ leftDim Sᘁ is nonzero:
- the global dimension is the scalar
∑ S, χ [S] * χ [Sᘁ], whereχis the left dimension character (TauCeti.globalDim_eq_sum_leftDimChar_smul); - the values of
χare algebraic integers, since the Grothendieck ring is a finitely generated abelian group (TauCeti.SplitK0.isIntegral_ringHom_apply), so they lie in the algebraic closure ofℚink, which embeds intoℂby someσ; - every ring homomorphism
ψfrom the Grothendieck ring toℂsatisfiesψ [Sᘁ] = conj (ψ [S])(TauCeti.SplitK0.ringHom_apply_dual); - so
σsends the global dimension to∑ S, |σ (χ [S])|², which is at least|σ (χ [𝟙_ C])|² = 1.
Characteristic zero is used exactly in the second step. It cannot be dropped: over an
algebraically closed field of characteristic p > 0, the pointed fusion category of
ℤ/p-graded vector spaces has p simple objects, each of dimension 1 for its standard pivotal
structure, so its global dimension is p = 0.
Main declarations #
TauCeti.SplitK0.leftDimCharandTauCeti.SplitK0.rightDimChar: the left and right dimension characters of the Grothendieck ring;TauCeti.globalDim_eq_sum_leftDimChar_smul: the global dimension as a scalar;TauCeti.SplitK0.rightDimChar_apply_eq_conjandTauCeti.norm_rightDim_eq_norm_leftDim: overℂ, the right dimension character of a fusion category is the complex conjugate of the left one, so the two categorical dimensions of an object have the same absolute value;TauCeti.endAlgEquivOfSimple_globalDim: overℂ, the global dimension of a fusion category is∑ S, |dim_L S|²;TauCeti.globalDim_ne_zero: the global dimension of a pivotal fusion category over a field of characteristic zero is nonzero.
References #
- P. Etingof, D. Nikshych, V. Ostrik, On fusion categories, Ann. of Math. 162 (2005), Theorem 2.3
- [P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor categories][egno15], §§4.7, 7.21
The left dimension character of a pivotal category whose tensor unit is simple with
finite-dimensional endomorphisms over an algebraically closed field k: the ring homomorphism
from the Grothendieck ring to k sending the class of X to its left dimension, read in k.
Instances For
The right dimension character of a pivotal category whose tensor unit is simple with
finite-dimensional endomorphisms over an algebraically closed field k: the ring homomorphism
from the Grothendieck ring to k sending the class of X to its right dimension, read in k.
Instances For
The left dimension character sends the class of X to the left dimension of X, read
in k.
The right dimension character sends the class of X to the right dimension of X, read
in k.
The left dimension of X is the value of the left dimension character at [X] times the
identity of the unit.
The right dimension of X is the value of the right dimension character at [X] times the
identity of the unit.
The left dimension character of the dual class is the right dimension character, since
leftDim Xᘁ = rightDim X.
The right dimension character of the dual class is the left dimension character, since
rightDim Xᘁ = leftDim X.
In a spherical category the left and right dimension characters agree.
The global dimension is the scalar ∑ S, χ [S] * χ [Sᘁ] times the identity of the unit,
where χ is the left dimension character.
In a complex pivotal fusion category, the right dimension character is the complex conjugate of the left dimension character, since characters of the Grothendieck ring are ⋆-characters.
In a complex pivotal fusion category, the left and right categorical dimensions of an object have the same absolute value.
In a complex pivotal fusion category, the global dimension, read as a complex number, is the sum of the squared absolute values of the left dimensions of the simple classes.
The global dimension of a pivotal fusion category over a field of characteristic zero is
nonzero (Etingof–Nikshych–Ostrik). Here C is pivotal, k-linear abelian and finite semisimple
over an algebraically closed field k of characteristic zero, with finite-dimensional morphism
spaces, simple tensor unit, and finitely many simple classes.