Definition 1.1.1(alphabet, letter, word, concatenation, free monoid, word algebra). LetXbe a set. We callXthealphabetand its elementsletters.1 Then-tuples of letters are calledwords (of length n), and form the setXn, where n ∈ N0. We callX∗ := S
n∈N0Xntheset of words, noting that it includes theempty word, denoted by 1∅(which is the only element ofX0), and we define theconcatenationof words,
X∗×X∗ → X∗,
(x1. . .xn) _ (y1. . .ym) := (x1. . .xny1. . .ym), (1.1) which equipsX∗with the structure of a monoid, called thefree monoid onX. Its unit is the empty word 1∅. Since the concatenation is associative, the parentheses delimiting the words
1In the literature, frequently only finite alphabets such asX={0, . . .d−1}are used.
and the concatenation symbol_are often omitted. We denote repetitions of letters or other words by
xn := x| {z }. . .x
n
(1.2) forx∈X∗andn∈N0.
Let furtherKbe a field of characteristic 0, then theK-vector spaceKhXiis defined by the span ofX∗overK. The concatenation can be extended bilinearly to an associative linear map KhXi ×KhXi →KhXi, which makes (KhXi, _,1∅) an associative unital algebra, called the free associative unital algebraor theword algebraoverK. It is obviously graded by word length.
In addition to the concatenation, we can endowKhXiwith a further multiplication.
Definition 1.1.2(shuffle permutations, shuffle product). 1. Letn,m∈N0. We call the set Sn,m :=
ρ∈Sn+m
ρ−1(1)< . . . < ρ−1(n) andρ−1(n+1)< . . . < ρ−1(n+m) (1.3) (whereSndesignates the set of permutations ofnelements) the set ofshuffle permutations.
2. The shuffle product is the map µ : KhXi ⊗KhXi → KhXi defined by extending the following mapµ:X∗×X∗→KhXibilinearly:
letx = x1. . .xn∈ Xn ⊂ X∗, y =xn+1. . .xn+m ∈ Xm ⊂ X∗be words. Then their shuffle product is defined as
x#y := xy := µ(x⊗y) := X
π∈Sn,m
xπ(1). . .xπ(n+m) ∈KhXin+m. (1.4)
3. Define theunityη:K→KhXiby
η(k) := k·1∅for allk∈K. (1.5)
Remark 1.1.3(notation of the shuffle product). To understand the term “shuffle product”, it is illustrative to imagine the wordsx, yabove as decks of playing cards, and a single shuffle permutation as one particular way to shuffle the two decks together; the shuffle product is the sum over all possible shuffles. In the mathematical literature the notationsx#yandxxy (the latter from the cyrillic letterx, pronounced “sha”) are frequently used for the shuffle product. The box notation xy is due to Pohlmeyer, nicely conveys the image of two decks of cards being shuffled into each other from above and below, and is very well suited to situations where shuffles and concatenations are being used together, in particular when individual letters are being considered. It does, however, become cumbersome when shuffle products of many factors are used. Consider the examples of both notations
e
x1. . .xi−1
xn. . .xi+1
yj−1. . .y1
yj+1. . .ym
=e x1. . .xi−1#xn. . .xi+1_yj−1. . .y1#yj+1. . .ym
, w1#. . .#wk =
w1
... wk
.
In the sequel, we will use both notations whereever they are more convenient.
There is also a widely known recursive identity for the shuffle product that provides an equivalent, iterative, definition and is often convenient to use in proofs.
Lemma 1.1.4(recursive formula for the shuffle product). Let x = x1. . .xn∈Xnand y =y1. . .ym ∈Xmbe words. Then:
x1. . .xn
y1. . .ym = yx1. . .xn
1. . .ym−1ym+xy1. . .xn−1
1. . .ym xn = x1yx2. . .xn
1. . .ym+y1yx1. . .xn
2. . .ym. (1.6)
Example 1.1.5. We calculate the shuffle product of the two wordsabandacin the alphabet X={a,b,c}:
ab
ac = abac + 2aabc + 2aacb + acab.
For easy reference, we recapitulate some very basic definitions that will be used through-out this thesis.
Definition 1.1.6.(algebra, associative, commutative, abelian, unital, homo-/epi-/mono-/isomorphism, subalgebra, ideal, gradation, derivation (along a homomorphism))
1. LetAbe a vector space over a fieldKand letµ:A×A→Abe aK-bilinear map. We writea·a0 forµ(a,a0). Then (A, µ) or (A,·) is called analgebra overK.
2. An algebra (A,·) that satisfiesa·b=b·afor alla,b∈Ais calledcommutativeorabelian.
3. An algebra (A,·) that satisfiesa·(b·c)=(a·b)·cfor alla,b,c∈Ais calledassociative.
4. An algebra (A,·) over a fieldKequipped with aK-linear mapη : K →A(called the unity) such thatη(1K)·a=a=a·η(1K) for alla∈Ais calledunital.
5. Let (A,·) and (B,∗) be algebras over a field K. A homomorphism / epimorphism / monomorphism/isomorphism ofK-vector spacesϕ:A→Bis called ahomomorphism /epimorphism/monomorphism/isomorphism of (K-)algebrasif
ϕ(a·a0) = ϕ(a)∗ϕ(a0) for alla,a0 ∈A. (1.7) If an isomorphism between two algebras exists, they are calledisomorphic.
6. Let (A,·) be an algebra and let Bbe a vector subspace of A. IfB·B ⊂ B, we callB a subalgebraofA, and ifA·B ⊂BandB·A⊂ B(these conditions are the same ifAis a commutative or Lie algebra), we callBanidealofA. Thekernelkerϕ:=ϕ−1(0)⊂Aof any homomorphism of algebrasϕ:A→Cis an ideal ofA.
7. Let (A,·) be an algebra over a field K with idealB ⊂ A. We can then define the set A/B:={a+B|a∈A}, obtain thecanonical projectionπ:A→A/B,a7→a+Band define the multiplication
∗:A/B×A/B → A/B,
(a+B, a0+B) 7→ a·a0+B. (1.8) Now (A/B,∗) is an algebra overK called the quotient algebra A mod B and we have B=kerπ.
8. Let (I,+) be a semigroup. An algebraAis calledI-gradedorequipped with a I-gradationif
A = M
i∈I
Ai (1.9)
(direct sum of vector spaces; the summandsAiare calledstrata) and
Ai·Ai0 ⊂ Ai+i0 for alli,i0∈I. (1.10) 9. Let (A,·) be an algebra over a fieldK. An endomorphism ofK-vector spaces∂:A→A
is called aderivation of Aif
∂(a·a0) = a·∂(a0) + ∂(a)·a0for alla,a0 ∈A. (1.11) If (B,∗) is another algebra overKandϕ : A →B is a homomorphism, then a homo-morphism ofK-vector spaces∂:A→Bis called aderivation along the homomorphismϕ if
∂(a·a0) = ϕ(a)∗∂(a0) + ∂(a)∗ϕ(a0) for alla,a0 ∈A. (1.12) Theorem 1.1.7. (KhXi, µ, η) is an associative, commutative, unital algebra, called the shuffle algebra Sh(X). It is graded by word length.
Proof. This immediately follows from the definition.
In addition to being an algebra, the shuffle algebra is equipped with more algebraic stucture. We quickly recapitulate2some further commonly found definitions.
Definition 1.1.8. (coassociative counital coalgebra, bialgebra, convolution product, antipode, Hopf algebra, gradation, connectedness)
1. LetHbe a vector space over a fieldKand let∆:H→H⊗Hand:H→KbeK-linear maps with the property that the diagrams
H H⊗H
H⊗H H⊗H⊗H
∆
∆ id⊗∆
∆⊗id
(1.13)
(coassociativity) and
K⊗H H⊗H H⊗K H
⊗id id⊗
∆
(1.14)
(counitality) commute. Then the triple (H,∆, ) is called acoassociative counital coalgebra with thecoproduct∆andcounit.
2. Let (H, µ, η) be an associative unital algebra and (H,∆, ) be a coassociative counital coalgebra. If∆andare morphisms of algebras, then (H, µ, η, ∆, ) is called abialgebra.
2See [HGK10, Chapters 2 and 3] or [Kas95, Chapter III] for a comprehensive introduction to Hopf algebras.
3. Let (H, µ, η, ∆, ) be a bialgebra. Theconvolution productis defined by
∗: EndK(H)×EndK(H) →EndK(H),
f∗g :=µ◦(f⊗g)◦∆. (1.15) It follows fom the definitions of (associative unital) algebra and (coassociative counital) coalgebra that∗is associative andηis its identity element.
4. Let (H, µ, η, ∆, ) be a bialgebra. An endomorphismS∈EndK(H) is calledantipodeif S∗idH = idH∗S = η◦:H→H. (1.16) The tuple (H, µ, η, ∆, , S) is then called aHopf algebra.
5. IfIis a semigoup (usually written as an addition), a Hopf algebraHis calledI-graded ifH=L
i∈IHi(iis called the degree) and both multiplication and comultiplication are additive with respect to the degree:
µ(Hi×Hj) ⊂Hi+j, (1.17)
∆(Hk) ⊂ M
i+j=k
Hi+j. (1.18)
6. IfMis a monoid, aM-graded Hopf algebra is calledconnectedif its grade 0 component is the ground fieldK.
For the shuffle Hopf algebra, these general objects can be defined as follows:
Definition 1.1.9(deconcatenation coproduct, counit, antipode).
1. Thedeconcatenation coproductis defined as the map
∆:KhXi → KhXi ⊗KhXi, x1. . .xn 7→
n
X
k=0
x1. . .xk⊗xk+1. . .xn. (1.19)
2. Thecounitis defined as the map:KhXi →Kobtained by linear extension of the map
:X∗→K,
(x) :=
(1K ifx=1∅
0 ifx∈X∗\X0 . (1.20)
3. Finally, theantipodeis defined as the map
S:KhXi →KhXi,
x1. . .xn 7→ (−1)nxn. . .x1. (1.21)
The names given to the objects we just defined are justified by the following theorem.
In fact, the shuffle Hopf algebra is an example of a Hopf algebra commonly encountered in books about Hopf algebras, for instance [HGK10, Example 3.4.6].
Theorem 1.1.10(shuffle algebra is a Hopf algebra). (KhXi, µ, η, ∆, , S)is a commutative, but not cocommutative Hopf algebra. It isN0-graded by word length and connected.
Proof. A proof of this result can be found in [Lod94] or [HGK10].