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Proceedings of the American Mathematical Society

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An elementary invariant definition of the functions of bidegree $ (p,q)$

Author: Michael Freeman
Journal: Proc. Amer. Math. Soc. 50 (1975), 265-272
MSC: Primary 32A99; Secondary 15A75
MathSciNet review: 0374469
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Abstract: The alternating $ r$-linear complex-valued functions of bidegree $ (p,q),p + q = r$, are usually defined on a complex vector space $ V$ as the span of the elements $ {g_{{i_1}}} \wedge \ldots \wedge {g_{{i_p}}} \wedge {\bar g_{{j_1}}} \wedge \ldots \wedge {\bar g_{{j_q}}}$, where $ \{ {g_i}:i \in I\} $ is a basis for $ {V^ \ast }$ , or by means of a representation of the exterior power of a direct sum. The former definition is not a priori invariant under coordinate changes and not easily adaptable to analysis on infinite-dimensional spaces, and the latter one rests on a rather involved abstract construction. Here it is shown how to give a new coordinate-free definition of the $ (p,q)$ functions by means of a simple identity which characterizes them by their action as $ r$-linear maps on $ V$. It seems well adapted for analysis on infinite-dimensional spaces.

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Keywords: Alternating real-linear complex-valued function, bidegree $ (p,q)$, exterior power of a direct sum, complex-valued differential form
Article copyright: © Copyright 1975 American Mathematical Society

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