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The Shuffle Hopf Algebra and Noncommutative Full Completeness - Blute, Scott (1996) (5 citations) (Correct)
....action of a general Hopf algebra on a tensor product is not the the tensor of actions. In other words, if H is a Hopf algebra, and h 2 H then h(v Omega w) will generally not be hv Omega hw. This leads one to define the notion of a grouplike element of H, which will satisfy this equation. See [35]. We now reformulate the results of [13] in light of this new observation. It is straightforward to verify that Dinat(F; F 0 ) is a vector space, under pointwise operations. We call it the space of proofs associated to the sequent F F 0 . Note that we identify formulas with definable ....
....from sequentialization for Yetter s nets, cf. 31] and the planarity condition described above. 2 5 Hopf algebras and Representations 5.1 Algebras and Coalgebras In this section we give a quick summary of the necessary background in bialgebras and Hopf algebras. For suitable introductions, see [1, 35, 21]. Definition 5.1 A Hopf algebra is a k vector space y , H, equipped with an algebra structure and a compatible coalgebra structure ( bialgebra) and an antipode satisfying the appropriate equations [21, 35] The following chart summarizes the necessary structure. All maps shown are linear. y ....
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M. Sweedler, Hopf Algebras, Benjamin Press, (1969).
Linear Läuchli Semantics - Blute, Scott (Correct)
....algebras provides a natural generalization of that of groups and may ultimately allow us to generalize the previous results to the noncommutative, braided and cyclic settings. We briefly review the basic theory before stating our conservativity result. For a more complete discussion, see [1] or [45]. 11.1 Definition and Categorical Structure Definition 11.1 A Hopf algebra is a vector space, H, equipped with an algebra structure, a compatible coalgebra structure and an antipode. These must satisfy equations as outlined in [45] The following chart summarizes the necessary structure. 2 In ....
....conservativity result. For a more complete discussion, see [1] or [45] 11.1 Definition and Categorical Structure Definition 11.1 A Hopf algebra is a vector space, H, equipped with an algebra structure, a compatible coalgebra structure and an antipode. These must satisfy equations as outlined in [45]. The following chart summarizes the necessary structure. 2 In fact, we obtain an indexed autonomous category in the sense of [42] Structure k[G] Algebra m: H Omega H H m(g 1 Omega g 2 ) g 1 g 2 j: k H j(1 k ) 1 G Coalgebra Delta: H H Omega H Delta(g) g Omega g ffl: H k ....
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M. Sweedler, Hopf Algebras, Benjamin Press, (1969).
Differential Geometry on Hopf Algebras and Quantum Groups - Watts (1994) Self-citation (Algebras) (Correct)
....is not only to introduce the concepts needed in the manipulations of these mathematical objects, but to also establish much of the notation which will appear throughout this work. For the interested reader, much more information about Hopf algebras and their properties is readily available in [5, 6, 7]. 2.1 Basic Definitions An algebra is a vector space A over a field k such that the algebra multiplication m : A Omega A A is a bilinear map satisfying m(a Omega (b c) m(a Omega b) m(a Omega c) m( a b) Omega c) m(a Omega c) m(b Omega c) 2.1) for all a; b; c 2 A. In ....
....is the quantum determinant of A, and is central within the algebra. 2.5 Sweedler s Notation We end this chapter with a discussion of an extremely useful notation which we will use for the remainder of this work. It is referred to as Sweedler s notation after the man who first introduced it in [5], and is a way of easing the computations involved in dealing with Hopf algebras. If A is a Hopf algebra, then the coproduct Delta(a) of an element a 2 A will in general consist of a sum of elements in A Omega A; the examples presented in the previous section show this fact explicitly. Thus, ....
M. E. Sweedler, Hopf Algebras, Benjamin Press, 1969
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