By Salzmann H.

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**Example text**

We must show yi ≥ n − 1. The proof proceeds in 3 steps. Step 1 y1 ≥ √ 3 2 for i = 1, 2, . . , √ 3 . 2 ⎛ ⎞ In−2 0 −1 ⎠ on γ ◦ z. Here In−2 1 0 denotes the identity (n − 2) × (n − 2)–matrix. First of all This follows from the action of α := ⎝ φ(α ◦ γ ◦ z) = ||en · α ◦ γ ◦ z|| = ||en−1 · x · y|| = ||(en−1 + xn−1,n en ) · y|| 2 . = d y12 + xn−1,n Since |xn−1,n | ≤ 12 we see that φ(αγ z)2 ≤ d 2 (y12 + 41 ). On the other hand, the assumption of minimality forces φ(γ z)2 = d 2 ≤ d 2 y12 + 14 . This implies that y1 ≥ √ 3 .

Consequently, 0 = φ(u) = D f (g · exp(uh)) − (D f ) g · exp(uh) = D f (g) − (D f ) g · exp(uh) . Thus, D f is invariant on the right by O(n, R)+ . 6 that D f is invariant on the right by the special element δ1 of determinant −1. It follows that D f must be right–invariant by the entire orthogonal group O(n, R). This proves the proposition. We now show how to explicitly construct certain differential operators (called Casimir operators) that lie in Dn , the center of the universal enveloping algebra of gl(n, R).

Similarly, if W is another vector space with basis vectors w1 , w2 , . . such that wi ∈ V for i = 1, 2, . . 1 Lie algebras 41 (defined over K ) with basis vectors v1 , w1 , v2 , w2 , . . Similarly, we can also define higher direct sums ⊕ Vℓ ℓ of a set of linearly independent vector spaces Vℓ . We shall also consider the tensor product V ⊗ W which is the vector space with basis vectors vi ⊗ w j , (i = 1, 2, . . , j = 1, 2, . ) and the higher tensor products ⊗k V (for k = 0, 1, 2, 3, . ) where ⊗0 V = K and for k ≥ 1, ⊗k V denotes the vector space with basis vectors vi1 ⊗ vi2 ⊗ · · · ⊗ vik where i j = 1, 2, .