Mathematisches Institut WS 2001/02 der Universität München Set 3 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(9) (Tensors in physics: # 9,10,11) Let V be a finite dimensional vector space over the field 𝕂 and let V^{∗} be its dual space. Let t be a tensor in V ⊗ … ⊗ V ⊗ V^{∗} ⊗ … ⊗ V^{∗} = V^{⊗r} ⊗ (V^{∗})^{⊗s}.

Show that for each basis B = (b_{1}, …, b_{n}) and dual basis B^{∗} = (b^{1}, …, b^{n}) there is a uniquely determined scheme (a family or an (r + s)-dimensional matrix) of coefficients (a(B)_{j_{1},...,j_{s}}^{i_{1},...,i_{r}}) with a(B)_{j_{1},...,j_{s}}^{i_{1},...,i_{r}} ∈ 𝕂 such that

(1)  t = ∑_{i_{1}=1}^{n} … ∑_{i_{r}=1}^{n} ∑_{j_{1}=1}^{n} … ∑_{j_{s}=1}^{n} a(B)_{j_{1},...,j_{s}}^{i_{1},...,i_{r}}b_{i_{1}} ⊗ … ⊗ b_{i_{r}} ⊗ b^{j_{1}} ⊗ … ⊗ b^{j_{s}}.(1)

(10) Show that for each change of bases L : B ⟶ C with c_{j} = ∑ λ_{j}^{i}b_{i} (with inverse matrix (μ_{j}^{i})) the following transformation formula holds

(2)  a(B)_{j_{1},...,j_{s}}^{i_{1},...,i_{r}} = ∑_{k_{1}=1}^{n} … ∑_{k_{r}=1}^{n} ∑_{l_{1}=1}^{n} … ∑_{l_{s}=1}^{n} λ_{k_{1}}^{i_{1}} … λ_{k_{r}}^{i_{r}}μ_{j_{1}}^{l_{1}} … μ_{j_{s}}^{l_{s}}a(C)_{l_{1},...,l_{s}}^{k_{1},...,k_{r}}

(11) Show that every family of schemes of coefficients

(a(B)|B basis of V)

with a(B) = (a(B)_{j_{1},...,j_{s}}^{i_{1},...,i_{r}}) and a(B)_{j_{1},...,j_{s}}^{i_{1},...,i_{r}} ∈ 𝕂 satisfying the transformation formula (2) defines a unique tensor (independent of the choice of the basis) t ∈ V^{⊗r} ⊗ (V^{∗})^{⊗s} such that (1) holds. Rule for physicists: A tensor is a collection of schemes of coefficients that transform according to the transformation formula for tensors.

(12) Show that (A, ∇ : A ⊗ A ⟶ A, η : 𝕂 ⟶ A) is a 𝕂-algebra if and only if A with the multiplication A × A ⟶^{⊗} A ⊗ A ⟶^{∇} A and the unit η(1) is a ring and η : 𝕂 ⟶ Cent(A) is a ring homomorphism into the center of A, where Cent(A) := {a ∈ A|∀b ∈ A : ab = ba}.

Due date: Tuesday, 06.11.2001, 16:15 in Lecture Hall 138