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Physlib.Mathematics.Groups.SpecialUnitary.LieAlgebra.KillingForm

The Killing form of `su(n)`

The Killing form of `su(n)` and its complexification, and its relation to the adjoint contraction.

i. Overview

The Killing form `K(A, B) = tr (ad A ∘ ad B)` of the complexification is computed by transporting `ad A ∘ ad B` to the matrices, where it is `M ↦ -[A, [B, M]]`, the sign coming from the factor `i` in the bracket. Its trace is `-2n tr (A B)` (A).

The contractions `adjointℂContr` and `adjointContr` of adjoint indices are therefore `-1 / (2n)` times the Killing forms of the complexification and of `su(n)` (B).

ii. Key results

- `SULieAlgebra.killingForm_eq_trace` : the Killing form of the complexification is `-2n` times the trace form. - `SULieAlgebra.adjointℂContr_eq_killingForm` : the complex contraction is `-1 / (2n)` times the Killing form. - `SULieAlgebra.adjointContr_eq_killingForm` : the real contraction is `-1 / (2n)` times the Killing form.

iii. Table of contents

  • A. The Killing form as a trace
  • B. The contraction as the Killing form

iv. References

* None.

A. The Killing form as a trace

B. The contraction as the Killing form

3 declarations

theorem

K(A,B)=−2ntr⁡(AB)K(A, B) = -2n \operatorname{tr}(AB) for the complexification of su(n)\mathfrak{su}(n)

For any natural number nn, let AA and BB be elements of the complexification of the Lie algebra su(n)\mathfrak{su}(n) (which is isomorphic to sln(C)\mathfrak{sl}_n(\mathbb{C})). The Killing form K(A,B)K(A, B) of this complexified Lie algebra is given by K(A,B)=−2ntr⁡(AB)K(A, B) = -2n \operatorname{tr}(AB) where AA and BB are viewed as n×nn \times n complex matrices and tr⁡\operatorname{tr} denotes the standard matrix trace.

theorem

The complexified adjoint contraction of su(n)\mathfrak{su}(n) is −12n-\frac{1}{2n} times its Killing form

For any natural number nn, let su(n)C=C⊗Rsu(n)\mathfrak{su}(n)_{\mathbb{C}} = \mathbb{C} \otimes_{\mathbb{R}} \mathfrak{su}(n) be the complexification of the Lie algebra su(n)\mathfrak{su}(n). For any elements A,B∈su(n)CA, B \in \mathfrak{su}(n)_{\mathbb{C}}, the complexified adjoint contraction (the invariant trace form Tr(AB)\text{Tr}(AB)) and the Killing form K(A,B)K(A, B) of the complexified Lie algebra are related by the identity: Tr(AB)=−12nK(A,B)\text{Tr}(AB) = -\frac{1}{2n} K(A, B) where nn is viewed as a complex number in the denominator.

theorem

The adjoint contraction of su(n)\mathfrak{su}(n) is −12n-\frac{1}{2n} times its Killing form.

For any natural number nn, let su(n)\mathfrak{su}(n) be the Lie algebra of n×nn \times n complex traceless hermitian matrices. For any elements x,y∈su(n)x, y \in \mathfrak{su}(n), the adjoint contraction (the invariant trace form Tr(xy)\text{Tr}(xy)) and the Killing form B(x,y)=Tr(adx∘ady)B(x, y) = \text{Tr}(\text{ad}_x \circ \text{ad}_y) are related by the identity: Tr(xy)=−12nB(x,y). \text{Tr}(xy) = -\frac{1}{2n} B(x, y).