Abstract
Let k be a field. For any k-vector space V the symbol 1dv (or simply, 1d) denotes the identity map on V. An associative k-algebra (or simply, algebra) is a vector space A together with a linear map \(\mu :A \otimes A \to A\) such that the diagram
is commutative. Writing \(\mu (a \otimes b) = ab\) this may be expressed as (ab)c = a(bc) for all a, b, c, \( \in \) A. An identity of an algebra A is a k-linear map \(\eta :k \to A\) such that the diagrams
are commutative. By (ii)ℓ and (ii)r it follows that \(\eta (1) = :{1_A}\) (or simply, 1) satisfies la = al = a for all \(a \in A\) Moreover \(\eta (\alpha )a = \alpha a\) and \(\eta (\alpha \beta ) = \alpha \eta (\beta ) = \eta (\alpha )\eta (\beta )\) so \(\eta \) is an algebra homomorphism. It is often convenient to omit \(\eta \). Unless otherwise specified the term algebra will mean associative algebra.
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© 1995 Springer-Verlag Berlin Heidelberg
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Joseph, A. (1995). Hopf Algebras. In: Quantum Groups and Their Primitive Ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-78400-2_2
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DOI: https://doi.org/10.1007/978-3-642-78400-2_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-78402-6
Online ISBN: 978-3-642-78400-2
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