The objects that occur in this chapter (vector spaces, algebras, algebraic varieties, etc.) are considered over a fixed ground field K. In subsections 1.5–3.3 it is assumed to be algebraically closed1. Sometimes we require that it be of zero characteristic. The reader, however, would not lose much by restricting himself to the cases K = ℂ or (where the algebraic closedness is not required) K = ℝ. Only these cases are needed for future applications to the Lie group theory and we only consider more general fields in order to elucidate the algebraic nature of the theory discussed.
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