notatio

Coproduct ​

The coproduct Δ\Delta: takes one element to a sum of HopfTensor pairs. Deconcatenation for QSym; the multiplicative extension of splitting a part for NSym.

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Coproduct(element)Δ(x)\Delta(x), as a sum of tensor pairs
Typed signatures
TypePackageApplies
(number) -> numberhopfalways

Domain: Hopf algebras

Details
  • This is what makes these Hopf algebras rather than merely algebras — the product puts things together, the coproduct pulls them apart
  • The two are tied by the bialgebra axiom Δ(xy)=Δ(x)Δ(y)\Delta(xy) = \Delta(x)\Delta(y), checked across every pair of basis elements up to degree 3 in both algebras
  • Both coproducts are coassociative
  • Tensor pairs use HopfTensor(left, right), NOT CircleTimes — that head is already the shared ordered product
  • Graded: the two halves' degrees sum to the degree of the input

Plot

Examples

Basic #

deconcatenation: cut at each gap

live ↗
InCoproduct(QSymM([1,2]))\mathrm{Coproduct}(\mathrm{QSymM}(\bigl\lbrack1, 2\bigr\rbrack))OutHopfTensor(QSymM([]),QSymM([1,2]))+HopfTensor(QSymM([1,2]),QSymM([]))+HopfTensor(QSymM([1]),QSymM([2]))\mathrm{HopfTensor}(\mathrm{QSymM}(\bigl\lbrack\bigr\rbrack), \mathrm{QSymM}(\bigl\lbrack1, 2\bigr\rbrack))+\mathrm{HopfTensor}(\mathrm{QSymM}(\bigl\lbrack1, 2\bigr\rbrack), \mathrm{QSymM}(\bigl\lbrack\bigr\rbrack))+\mathrm{HopfTensor}(\mathrm{QSymM}(\bigl\lbrack1\bigr\rbrack), \mathrm{QSymM}(\bigl\lbrack2\bigr\rbrack))

Δ(H2)=1⊗H2+H1⊗H1+H2⊗1\Delta(H_2) = 1\otimes H_2 + H_1\otimes H_1 + H_2\otimes 1

live ↗
InCoproduct(NSymH([2]))\mathrm{Coproduct}(\mathrm{NSymH}(\bigl\lbrack2\bigr\rbrack))OutHopfTensor(NSymH([]),NSymH([2]))+HopfTensor(NSymH([2]),NSymH([]))+HopfTensor(NSymH([1]),NSymH([1]))\mathrm{HopfTensor}(\mathrm{NSymH}(\bigl\lbrack\bigr\rbrack), \mathrm{NSymH}(\bigl\lbrack2\bigr\rbrack))+\mathrm{HopfTensor}(\mathrm{NSymH}(\bigl\lbrack2\bigr\rbrack), \mathrm{NSymH}(\bigl\lbrack\bigr\rbrack))+\mathrm{HopfTensor}(\mathrm{NSymH}(\bigl\lbrack1\bigr\rbrack), \mathrm{NSymH}(\bigl\lbrack1\bigr\rbrack))
Properties #

the counit picks out the constant term

live ↗
InCounit(NSymH([]))\mathrm{Counit}(\mathrm{NSymH}(\bigl\lbrack\bigr\rbrack))Out11
About
Examples3
CrosswalkWikipediaMathWorldWikidataEncyclopedia of MathematicsnLabWolfram Language

See also: QSymM, Antipode, Counit