# Thomas Ihringer's Allgemeine Algebra (Skript) PDF By Thomas Ihringer

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For any supervector space V we denote by V˜ the new supervector space given by V˜0 = V1 , V˜1 = V0 . Clearly, we have a natural isomorphism Det V˜ = (Det V )∗ . 2) Now let (V, ∂) be a finite-dimensional supervector space with a differential ∂ such that ∂V0 ⊂ V1 , ∂V1 ⊂ V0 , ∂ 2 = 0. Then Ker ∂, Im ∂, and 44 3. The Cayley Method for Studying Discriminants the cohomology space H(V ) = Ker ∂/ Im ∂ are again supervector spaces. We claim that there exists a natural isomorphism Det V Det H(V ). 3) Indeed, from the exact sequence ∂ 0 → Ker ∂ → V → Im ∂ → 0 we see that Det V Det(Ker ∂) ⊗ Det(Im ∂)∗ .

R are eigenvalues of A (counted with multiplicities) then the set of eigenvalues of ad(A) consists of n − 1 zeros and λi − λj , i, j = 1, . . , n, i = j. Therefore, D0 (A) = . . = Dn−2 (A) = 0 and Dn−1 (A) coincides with D(A) up to a non-zero scalar. Suppose now that g is a simple Lie algebra of rank r. Let dim g = n. For any x ∈ g let n ti Di (x) Qx = det(t Id − ad(x)) = i=0 be the characteristic polynomial of the adjoint operator. Then D(x) = Dr (x) is called the discriminant of x. Clearly, D is a homogeneous Ad-invariant polynomial on g of degree n − r.

But l is simple. Contradiction. Therefore, the center of g is one-dimensional, spanned by ξ, P is a maximal parabolic subgroup, and #Π0 = #Π − 1. Thus, there is a unique simple root in ∆1 . Call it β. 14 (a) β is long. e. the coefficient nβ in the sum α0 = nβ β + α∈Π0 nα α, is equal to 1. (c) For any α, α ∈ ∆1 , (α, α ) ≥ 0. (d) W0 acts transitively on long roots in ∆1 . Proof. (a) Indeed, since l1 is an irreducible g-module, β is the unique lowest weight of the g-module l1 . e. to γ. Hence β is long.

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### Allgemeine Algebra (Skript) by Thomas Ihringer

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