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B/ / for all a 2 A and b 2 B. b/ / for all a 2 A; b 2 B, and the reverse implication follows similarly. Thus, one of these conditions may be omitted in the definition of a dual pairing of Hopf -algebras. Let us consider several examples of dual pairings. 6. C// ! C/. C/ ! C/ is identified with the space of n n-matrices with vanishing trace. 14. 7. Let G be a compact connected Lie group with Lie algebra g. g/, which can be described as follows. G/. 12], and therefore every continuous finite-dimensional representation and every representative function of G is smooth.

Then the following conditions are equivalent: i) The comultiplication  W A ! A ˝ A is an algebra homomorphism. ii) The multiplication m W A ˝ A ! A is a morphism of coalgebras. 7)  / A ˝ A. Proof. A˝A˝A˝A id ˝† ˝ id ! A ˝ A/ respectively. 6. 7) commutes. A; / consisting of the algebra A and the comultiplication  as a bialgebra. A bialgebra is called unital if it is unital as an algebra and the comultiplication is a unital algebra homomorphism; it is called counital if it is counital as a coalgebra and the multiplication is a counital morphism of coalgebras.

Denote by † W A ˝ A ! A ˝ A the flip map a ˝ b 7! b ˝ a. A; A /. Evidently, a linear map W A ! A; A /cop . A; A /, that is, if † ı A D A . Direct sum. Denote by A˚B the composition of the map A ˚B W A˚B ! B ˝ B/ ,! A ˚ B/. A ˚ B; A˚B / is a coalgebra. a; b/ 7! A ˚ B; A˚B /. Tensor product. Denote by A˝B the composition of the map A ˝B W A˝ Š B ! A˝A˝B ˝B with the isomorphism A˝A˝B ˝B ! A˝B ˝A˝B given by a1 ˝ a2 ˝ b1 ˝ b2 7! a1 ˝ b1 ˝ a2 ˝ b2 . A ˝ B; A˝B / is a coalgebra. If A and B possess counits A and B , respectively, then the map a ˝ b 7!

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### Physics. 195 Supplementary Notes Groups, Lie algebras, and Lie groups 020922 by F. Porter

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