An alternating multilinear map vanishes whenever two arguments are equal. Swapping two arguments reverses its sign.
A multilinear map is antisymmetric when swapping two arguments negates its value. Over a field of characteristic different from , this is equivalent to being an alternating multilinear map: repeated arguments give , hence ; conversely expand to get the sign change. In characteristic , the repeated-argument condition is stronger than the sign-change condition alone.
An alternating trilinear form is a scalar-valued map, linear in three arguments, which vanishes whenever two arguments agree. In characteristic zero, interchanging two arguments negates its value, while cyclic permutation preserves it. For a symmetric bilinear form on a Lie algebra, the tensor is alternating exactly when is an invariant bilinear form on a Lie algebra.
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An alternating multilinear map is a special type of multilinear function that takes several input arguments from a vector space and has the property of being alternating. Here's a more detailed breakdown of what this means: 1. **Multilinear Map**: A function \( f: V_1 \times V_2 \times \ldots \times V_n \to W \) is called multilinear if it is linear in each of its \( n \) arguments.
