A basis for the right quantum algebra and the “1 = q” by Foata D., Han G.-N.

By Foata D., Han G.-N.

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Given a permutation σ ∈ Sk and T ∈ I k , show that T σ ∈ I k . 8. Let W be a subspace of Lk having the following two properties. (a) For S ∈ S 2 (V ) and T ∈ Lk−2 , S ⊗ T is in W. (b) For T in W and σ ∈ Sk , T σ is in W. 1 and by the author of these notes in his book with Alan Pollack, “Differential Topology” 30 Chapter 1. Multilinear algebra Show that W has to contain I k and conclude that I k is the smallest subspace of Lk having properties a and b. 9. 5) απ(T ) = 1 Alt (T ) k! 2, exercise 8).

3. 16). In particular conclude that iv T ∈ Ak−1 . ) 4. Assume the dimension of V is n and let Ω be a non-zero element of the one dimensional vector space Λ n . 15) ρ : V → Λn−1 , v → ιv Ω , is a bijective linear map. Hint: One can assume Ω = e ∗1 ∧ · · · ∧ e∗n where e1 , . . , en is a basis of V . 14) to compute this map on basis elements. 5. ) Let V be a 3-dimensional vector space, B an inner product on V and Ω a non-zero element of Λ 3 . 9). Show that this map is linear in v1 , with v2 fixed and linear in v2 with v1 fixed, and show that v1 × v2 = −v2 × v1 .

12) dfp : Tp Rn → Tc R and by making the identification, Tc R = {c, R} = R 54 Chapter 2. , as an element of (Tp Rn )∗ . 13) defines a one-form on U which we’ll denote by df . 2. Given a one-form ω and a function, ϕ : U → R the product of ϕ with ω is the one-form, p ∈ U → ϕ(p)ω p . 3. Given two one-forms ω1 and ω2 their sum, ω1 + ω2 is the one-form, p ∈ U → ω1 (p) + ω2 (p). 4. The one-forms dx1 , . . , dxn play a particularly important role. , is equal to 1 if i = j and zero if i = j. Thus (dx 1 )p , .

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