By Padberg M.W. (ed.)

ISBN-10: 1101091061

ISBN-13: 9781101091067

ISBN-10: 3642008011

ISBN-13: 9783642008016

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Are given by (1⊗)(c) = 1 ⊗ c and (⊗1)(c) = c ⊗ 1 for c ∈ C. We may omit the indexing required to express a coproduct Δ (c) as an element of C ⊗ C and use Sweedler notation to write Δ (c) = ∑ c1 ⊗ c2 . Thus, in Sweedler notation, diagrams i. and ii. 1) ∑ ε (c1 )c2 = c = ∑ ε (c2 )c1 . 2) We say the coproduct Δ is cocommutative if c ⊗ c is a term of Δ (c) whenever c ⊗ c is. A submodule I ⊆ C is a coideal if Δ (I ) ⊆ I ⊗ C + C ⊗ I and ε (I ) = {0}. 22 3 Hopf algebras A map f : C → C , where (C , Δ , ε ) is another coalgebra over R, is a coalgebra morphism if Δ ◦ f = ( f ⊗ f ) ◦ Δ and ε = ε ◦ f .

12) P where the sum is over all paths P in the (x, y) plane from (0, 0) to ( (α ), (β )) with steps (1, 0), (0, 1) and (1, 1). Given two permutations σ = σ (1) · · · σ (n) ∈ Sn and τ = τ (1) · · · τ (m) ∈ Sm , we say a shuffle of σ and τ is a permutation in Sn+m such that σ (i) appears to the right of σ (i − 1) and to the left of σ (i + 1) for all 2 i n − 1 and similarly, τ (i) + n appears to the right of τ (i − 1) + n and to the left of τ (i + 1) + n for all 2 i m − 1. We denote by σ τ the set of all shuffles of σ and τ .

Our presentation is based on that in [69]. Other references are [68] and [84]. Throughout this section, let R be a commutative ring with identity element. We remind the reader that an R-module is defined in the same way as a vector space, except that the field of scalars is replaced by the ring R. 1. An algebra over R is an R-module A together with R-linear maps product or multiplication m : A ⊗ A → A and unit u : R → A , such that the following diagrams commute. K. 1007/978-1-4614-7300-8 3, © Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg 2013 19 20 3 Hopf algebras i.

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