The regular trace form and semisimplicity

A finite-dimensional algebra A over a field K, not necessarily commutative, carries the regular trace form Algebra.regularTraceForm K A, (x, y) ↦ tr_K(z ↦ x y z), the trace of left multiplication by the product. This file proves the trace-form criterion for semisimplicity:

• over any field, if the regular trace form is nondegenerate then A is semisimple;
• over a field of characteristic zero the converse holds, so there semisimplicity is exactly the nondegeneracy of the regular trace form.

The first statement holds because an element x of the Jacobson radical makes every y x nilpotent, the radical being a nilpotent ideal, so the trace of left multiplication by y x vanishes for every y. The second uses that the elements x with tr(L_{yx}) = 0 for all y form a left ideal, which in a semisimple ring is generated by an idempotent e. Left multiplication by e is then an idempotent operator of trace 0; in characteristic zero its trace is its rank, so it is zero, and so is e.

In characteristic p the converse fails: the matrix algebra Matrix (Fin p) (Fin p) 𝔽_p is simple, yet its regular trace form (x, y) ↦ p · tr(x y) vanishes identically.

Expanded in a basis, nondegeneracy is the nonvanishing of the Gram determinant of the trace form, the discriminant of the algebra. This is how the criterion is used for algebras whose structure constants depend on a parameter, such as the Brauer algebras: the discriminant is then a polynomial in the parameter, and the algebra is semisimple wherever that polynomial does not vanish.

Main results

• TauCeti.regularTraceForm_eq_zero_of_mem_jacobson: an element of the Jacobson radical is in the kernel of the regular trace form.
• TauCeti.isSemisimpleRing_of_nondegenerate_regularTraceForm: a finite-dimensional algebra with nondegenerate regular trace form is semisimple.
• TauCeti.nondegenerate_regularTraceForm: in characteristic zero, the regular trace form of a finite-dimensional semisimple algebra is nondegenerate.
• TauCeti.isSemisimpleRing_iff_nondegenerate_regularTraceForm: in characteristic zero, a finite-dimensional algebra is semisimple exactly when its regular trace form is nondegenerate.

Combined with LinearMap.BilinForm.nondegenerate_iff_det_ne_zero, these give the criteria in terms of the Gram determinant (LinearMap.BilinForm.toMatrix b (Algebra.regularTraceForm K A)).det in a basis b.

References

• C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Interscience (1962), §26.
• R. S. Pierce, Associative algebras, Graduate Texts in Mathematics 88, Springer (1982), §10.2.

theorem TauCeti.regularTraceForm_eq_zero_of_mem_jacobson {K : Type u_1} {A : Type u_2} [Field K] [Ring A] [Algebra K A] [FiniteDimensional K A] {x : A} (hx : x ∈ Ring.jacobson A) (y : A) : ((Algebra.regularTraceForm K A) y) x = 0

An element of the Jacobson radical lies in the kernel of the regular trace form: if x is in the Jacobson radical of a finite-dimensional algebra, then tr(L_{yx}) = 0 for every y.

theorem TauCeti.isSemisimpleRing_of_nondegenerate_regularTraceForm {K : Type u_1} {A : Type u_2} [Field K] [Ring A] [Algebra K A] [FiniteDimensional K A] (h : (Algebra.regularTraceForm K A).Nondegenerate) : IsSemisimpleRing A

A finite-dimensional algebra whose regular trace form is nondegenerate is semisimple. This holds over every field.

theorem TauCeti.nondegenerate_regularTraceForm {K : Type u_1} {A : Type u_2} [Field K] [Ring A] [Algebra K A] [FiniteDimensional K A] [CharZero K] [IsSemisimpleRing A] : (Algebra.regularTraceForm K A).Nondegenerate

In characteristic zero, the regular trace form of a semisimple algebra is nondegenerate.

theorem TauCeti.isSemisimpleRing_iff_nondegenerate_regularTraceForm {K : Type u_1} {A : Type u_2} [Field K] [Ring A] [Algebra K A] [FiniteDimensional K A] [CharZero K] : IsSemisimpleRing A ↔ (Algebra.regularTraceForm K A).Nondegenerate

The trace-form criterion for semisimplicity: over a field of characteristic zero, a finite-dimensional algebra is semisimple exactly when its regular trace form is nondegenerate.