A lot of combinatorial objects have a natural bialgebra structure. In this paper, we prove that the vector space spanned by labeled simple graphs is a bialgebra with the conjunction product and the unshuffle coproduct. In fact, it is a Hopf algebra since it is graded connected. The main conclusions are that the vector space spanned by labeled simple graphs arising from the unshuffle coproduct is a Hopf algebra and that there is a Hopf homomorphism from permutations to label simple graphs.
The basic structure of Hopf algebra was first proposed by Hopf to study algebraic topology and the properties of algebraic groups in 1941 [1]. In 1960’s, Milnor, Moore, Chase and Sweedler systematically developed the theory of Hopf algebra and gave the explicit definition, basic properties and common symbols of Hopf algebra [2] [3] [4]. In 1979, Joni and Rota constructed Hopf algebras on polynomials and on puzzles [5]. After that, Hopf algebra has been used to study a lot of objects, such as posets [6], symmetric functions [7] [8], quantum groups [9] [10], Clifford algebras [11] [12] and Lie superalgebras [13] [14].
Graphs are important combinatorial objects, on which there are rich Hopf algebra structures. In 1994, Schmitt studied incidence of Hopf algebras and gave Hopf algebras on many objects, such as Hopf algebras on permutations, matroids and graphs [15]. Schmitt also studied invariants of graphs through the properties of a Hopf algebra on graphs in 1995 [16]. Later, Connes and Kreimer established connections between quantum physics and Hopf algebras on rooted trees and on rooted forests [17] [18].
Permutations are related to graphs closely. In 1995, Malvenuto and Reutenauer first gave a classical Hopf algebra on permutations by shuffle product ш [19]. On this basis, Aguiar and Sottile considered the concept of global descents on permutations in 2005 [20]. In 2004, Novelli, Thibon and Thiéry studied Hopf algebras with bases labeled by graphs and hypergraphs, which are graded by the number of edges [21]. In 2010, Forcey, Lauve and Sottile studied Hopf algebras on binary trees [22]. In 2014, Vargas defined the super-shuffle product ш _ and the cut-box coproduct Δ ⋄ on permutations by global ascents of permutations [23]. In 2016, Giraudo and Vialette defined the unshuffle coproduct Δ ∗ on permutations [24]. In 2020, Zhao and Li derived a new Hopf algebra on permutations with another shuffle product ш _ G ∗ from the classical one [25]. In the same year, Guo, Thibon and Yu introduced the Hopf algebra of signed permutations and established its relationship with the Hopf algebras of permutations and weak quasi-symmetric functions [26]. In 2021, Liu and Li proved that the vector space spanned by permutations arising from the super-shuffle product ш _ and the cut-box coproduct Δ ⋄ is a Hopf algebra [27]. It is well-known that permutations are elements of symmetric groups, which are widely used in various fields, such as algebraic number theory [28] and substochastic matrices [29] [30] [31] [32].
In 2020, Aval, Bergeron and Machacek gave a Hopf algebra on labeled simple graphs with the conjunction product and the unshuffle coproduct without a proof. Here the Hopf algebra is graded by the number of vertices. They also introduced a mapping, named g in this paper, from permutations to labeled simple graphs and claimed that it is a Hopf homomorphism [33]. In this paper, we will prove these conclusions.
This paper is organized as follows. We review basic concepts of Hopf algebra in Section 2. In Section 3, we define the vector space H spanned by labeled simple graphs and give the definitions of the conjunction product ⋄ and the unshuffle coproduct Δ * on the vector space. In Section 4, we prove ( H , ⋄ , μ ) is a graded algebra, ( H , Δ ∗ , ν ) is a graded coalgebra and the compatibility between the conjunction product and the unshuffle coproduct. So ( H , ⋄ , μ , Δ ∗ , ν ) is a Hopf algebra from that H is graded connected. In Section 5, we recall a mapping g from permutations to labeled simple graphs and prove it is a Hopf homomorphism. Lastly, we summarize our main conclusions in Section 6.
In this paper, we not only prove the conclusions but also provide a lot of examples to help people to understand the operation rules and the Hopf structure on labled simple graphs. Furthermore, the Hopf homomorphism from permutations to labled simple graphs idicates the closed relations between them.
2. Hopf Algebra
Firstly, we introduce some basic definitions of Hopf algebra. For more details see [2] [3]. Let K be a commutative ring and A , B be K -modules. Denote id : A → A to be the identity mapping of A. The notation A → ≅ B means that A is isomorphic to B as K -modules.
If there are linear mappings m from A ⊗ A to A and μ from K to A such that the diagrams in Figure 1 are commutative, then we say that ( A , m , μ ) is an algebra, m is a product and μ is a unit.
If there are linear mappings Δ from A to A ⊗ A and ν from A to K such that the diagrams in Figure 2 are commutative, then we say that ( A , Δ , ν ) is a coalgebra, Δ is a coproduct and ν is a counit.
We call ( A , m , μ , Δ , ν ) a bialgebra, if ( A , m , μ ) is an algebra, ( A , Δ , ν ) is a coalgebra and A satisfies compatibility, i.e.,
Δ ( m ( x ⊗ y ) ) = m ( Δ ( x ) ⊗ Δ ( y ) )
and
ν ( m ( x ⊗ y ) ) = m ( ν ( x ) ⊗ ν ( y ) ) ,
for any x and y in A.
The bialgebra ( A , m , μ , Δ , ν ) is a Hopf algebra, if there is a linear mapping S from A to A satisfies
m ∘ ( id ⊗ S ) ∘ Δ ( x ) = μ ( ν ( x ) ) = m ∘ ( S ⊗ id ) ∘ Δ ( x )
for any x in A, i.e., the diagram in Figure 3 is commutative. We call S an antipode.
The vector space A is graded if A = ⊕ n ≥ 0 A n and A is connected if K A 0 ≅ K .
The algebra ( A , m , μ ) is a graded algebra if A is a graded vector space, m satisfies m ( A i ⊗ A j ) ⊆ A i + j for i , j ≥ 0 and μ satisfies μ ( K ) ⊆ A 0 . Similarly, the
coalgebra ( A , Δ , ν ) is a graded coalgebra if Δ satisfies Δ ( A n ) ⊆ ⊕ i ≥ 0 A i ⊗ A n − i
and ν satisfies ν ( A n ) = 0 for any n ≥ 1 . The bialgebra ( A , m , μ , Δ , ν ) is a graded bialgebra when ( A , m , μ ) is a graded algebra and ( A , Δ , ν ) is a graded coalgebra. In fact, a graded connected bialgebra is always a Hopf algebra ( [34], Proposition 1.4.14.).
3. Basic Definitions3.1. Labeled Simple Graph
A graph can generally be represented as Γ = ( V , E ) , where V is the vertex set and E is the edge set. We call a grpah Γ = ( V , E ) a labeled simple graph if it does not have cycles and multiple edges, V is a set of positive integers, also denoted by V ( Γ ) , and E is the set of all edges of Γ , also denoted by E ( Γ ) . Obviously, E ⊆ V × V and if ( i 1 , i 2 ) ∈ E , then i 1 ≠ i 2 and ( i 2 , i 1 ) ∉ E since simple graphs do not have cycles and multiple edges. In particular, Γ is the empty graph when V is empty, denoted by ε .
Let Γ = ( V , E ) and I ⊆ V . Define the restriction of Γ on I by Γ I = ( I , E I ) , where E I = { ( i , j ) | i , j ∈ I , ( i , j ) ∈ E } , and we call Γ I a subgraph of Γ . If Γ 1 = ( V 1 , E 1 ) , Γ 2 = ( V 2 , E 2 ) and V 1 ∩ V 2 = ∅ , then the union graph of Γ 1 and Γ 2 is defined by Γ 1 ∪ Γ 2 = ( V 1 ∪ V 2 , E 1 ∪ E 2 ) . For more details, see [35].
Define
[ n ] = ( { 1 , 2 , ⋯ , n } , n > 0 , ∅ , n = 0 ,
and
[ i , j ] = ( { i , i + 1 , ⋯ , j } , i ≤ j , ∅ , i > j .
Example 1. The graph Γ = ( { 1,2,3,4,6,7 } , { ( 1,2 ) , ( 4,6 ) } ) is a labeled simple graph and can be represented by
We have
and
Let H n = { Γ | Γ = ( [ n ] , E ) is alabeledsimplegraph } and H n be the linear space spanned by H n over field K , for any non-negative integer n. For example,
In particular, H 0 = { ε } and H 0 = K H 0 . Denote
H = ∪ n = 0 ∞ H n and H = ⊕ n = 0 ∞ H n .
Let I = { i 1 , i 2 , ⋯ , i n } be a set of positive integers where i 1 < i 2 < ⋯ < i n and denote | I | = n , the cardinality of I” at the end of the first sentence. Define a mapping st I from I to [ | I | ] by st I ( i a ) = a for 1 ≤ a ≤ n , and call it the standardization of I. For x , y in I, st I ( x ) < st I ( y ) if and only if x < y . Sometimes, we omit the subscript of the standardization when the set is obvious. Let T be a subset of I, then st I ( T ) = { st I ( x ) | x ∈ T } .
For any labeled simple graph Γ = ( V , E ) , define the standard form of Γ by
st ( Γ ) = ( st V ( V ) , st V ( E ) ) = ( st ( V ) , st ( E ) ) ,
where st ( V ) = [ | V | ] and st ( E ) satisfies
( st ( v 1 ) , st ( v 2 ) ) ∈ st ( E ) ⇔ ( v 1 , v 2 ) ∈ E ,
for v 1 and v 2 in V. In particular, st ( ε ) = ε . That means, the standardization maintains the edge relationships of the vertices in Γ . For any labeled simple graph Γ = ( V , E ) , there is a graph in H n which is the standard form of Γ , where n = | V | . In addition, for a non-negative integer n, let Γ ↑ n be the graph by raising each vertex in Γ by n and maintaining its edge relationships. Similarly, let Γ ↓ n be the graph by reducing each vertex in Γ by n and maintaining its edge relationships.
Example 2. For labeled simple graph
we have
and
3.2. Conjunction Product and Unshuffle Coproduct
Define the conjunction product ⋄ on H [33] by
Γ 1 ⋄ Γ 2 = Γ 1 ∪ Γ 2 ↑ m ,
for Γ 1 in H m and Γ 2 in H n , and the unit μ from K to H by μ ( 1 ) = ε .
Example 3. For and , the conjunction product of Γ 1 and Γ 2 is
and the conjunction product of Γ 2 and Γ 3 is
Define the unshuffle coproduct Δ ∗ on H [33] by
Δ ∗ ( Γ ) = ∑ I ⊆ [ n ] st ( Γ I ) ⊗ st ( Γ [ n ] \ I ) ,
for Γ = ( [ n ] , E ) in H n , and the counit ν from H to K by
ν ( Γ ) = ( 1, Γ = ε , 0, otherwise .
In particular, Δ ∗ ( ε ) = ε ⊗ ε .
Example 4. For and , the unshuffle coproduct of Γ 1 is
and the unshuffle coproduct of Γ 2 is
4. Main Theorem
Theorem 2. The vector space H with the conjunction product ⋄ and the unit μ is a graded algebra.
Proof. For any Γ 1 in H m , Γ 2 in H n and Γ 3 in H k , we have
( Γ 1 ⋄ Γ 2 ) ⋄ Γ 3 = ( Γ 1 ∪ Γ 2 ↑ m ) ⋄ Γ 3 = ( Γ 1 ∪ Γ 2 ↑ m ) ∪ Γ 3 ↑ m + n = Γ 1 ∪ Γ 2 ↑ m ∪ Γ 3 ↑ m + n = Γ 1 ∪ ( Γ 2 ∪ Γ 3 ↑ n ) ↑ m = Γ 1 ⋄ ( Γ 2 ⋄ Γ 3 ) .
So, ⋄ is associative. It is easy to prove that the μ is a unit. Then ( H , ⋄ , μ ) is an algebra. Obviously, by the definitions of ⋄ and μ , we have H i ⋄ H j ⊆ H i + j for i , j ≥ 0 and μ ( K ) ⊆ H 0 . So ( H , ⋄ , μ ) is a graded algebra. □
Example 5. For and , we have
Lemma 1. Assume I is a set of positive integers, J ⊆ I and K = st I ( J ) .Then
st K ( st I ( i ) ) = st J ( i ) ,
forany i in J. □
Proof. Denote I = { i 1 , i 2 , ⋯ , i n } and J = { i j 1 , i j 2 , ⋯ , i j t } , where t ≤ n . Suppose i 1 < i 2 < ⋯ < i n and 1 ≤ j 1 < j 2 < ⋯ < j t ≤ n . Obviously, K = st I ( J ) = { j 1 , j 2 , ⋯ , j t } and st K ( j m ) = m for any i j m ∈ J . Then
st K ( st I ( i j s ) ) = s = st J ( i j s ) ,
for any 1 ≤ s ≤ t . □
Example 6. If I = { 3,5,7,8,9 } and J = { 3,7,8 } , then st J ( 3 ) = 1 , st J ( 7 ) = 2 and st J ( 8 ) = 3 . On the other hand, st I ( 3 ) = 1 , st I ( 7 ) = 3 , st I ( 8 ) = 4 and K = st I ( { 3,7,8 } ) = { 1,3,4 } . So st K ( st I ( 3 ) ) = 1 , st K ( st I ( 7 ) ) = 2 , st K ( st I ( 8 ) ) = 3 .
Lemma 2. Assume Γ = ( V , E ) is a labeled simple graph, J ⊆ I ⊆ V and K = st I ( J ) .Then
st ( st ( Γ I ) K ) = st ( Γ J ) .
Proof. For convenience, we denote st ( st ( Γ I ) K ) is Γ 1 and st ( Γ J ) is Γ 2 . Obviously,
V ( Γ 1 ) = | K | = | J | = V ( Γ 2 ) .
We just need to show that their edges are the same. For ( i , j ) in E ( Γ 1 ) , there must exist i ″ and j ″ in K such that st K ( i ″ ) = i and st K ( j ″ ) = j . Meanwhile, there must exist i ′ and j ′ in J such that st I ( i ′ ) = i ″ and st I ( j ′ ) = j ″ , i.e., st K ( st I ( i ′ ) ) = i and st K ( st I ( j ′ ) ) = j . Then ( i ′ , j ′ ) in E ( Γ ) . By the definition of st ( Γ J ) , ( st J ( i ′ ) , st J ( j ′ ) ) is in E ( Γ 2 ) . By Lemma 1, st J ( i ′ ) = i and st J ( j ′ ) = j , then ( i , j ) ∈ E ( Γ 2 ) . Similarly, we can prove if ( i , j ) ∈ E ( Γ 2 ) then ( i , j ) ∈ E ( Γ 1 ) . So
st ( st ( Γ I ) K ) = st ( Γ J ) . □
Example 7. Let , I = { 2,3,4,6,7,9 } and J = { 2,3,6 } then K = st I ( J ) = { 1,2,4 } . We have
and
On the other hand,
Theorem 2. The vector space H with the unshuffle coproduct Δ ∗ and the counit ν is a graded coalgebra.
For any Γ = ( [ n ] , E ) in H n where n ≥ 1 , we have
( Δ ∗ ⊗ id ) ∘ Δ ∗ ( Γ ) = ( Δ ∗ ⊗ id ) ( ∑ I ⊆ [ n ] st ( Γ I ) ⊗ st ( Γ [ n ] \ I ) ) . (1)
Denote st ( Γ I ) by Θ , then
Δ ∗ ( Θ ) = ∑ K ⊆ [ | I | ] st ( Θ K ) ⊗ st ( Θ [ | I | ] \ K ) . (2)
Denote J as a subset in I such that st I ( J ) = K , then st I ( I \ J ) = [ | I | ] \ K . By Lemma 2,
st ( st ( Γ I ) K ) = st ( Γ J )
and
st ( st ( Γ I ) [ | I | ] \ K ) = st ( Γ I \ J ) .
Since K in (2) traverses all subsets of [ | I | ] , the corresponding J also traverses all subsets of I. Then (2) can be rewritten as
∑ J ⊆ I st ( Γ J ) ⊗ st ( Γ I \ J ) . (3)
Then (1) can be rewritten as
∑ I ⊆ [ n ] , J ⊆ I st ( Γ J ) ⊗ st ( Γ I \ J ) ⊗ st ( Γ [ n ] \ I ) . (4)
By the arbitrariness of I and J, (4) can be rewritten as
∑ I , J , K ⊆ [ n ] I ∪ J ∪ K = [ n ] | I | + | J | + | K | = n st ( Γ I ) ⊗ st ( Γ J ) ⊗ st ( Γ K ) . (5)
Similarly, we can get ( id ⊗ Δ ∗ ) ∘ Δ ∗ ( Γ ) is also equal to (5). Then Δ ∗ satisfies coassociativity. It is easy to prove that ν is a counit. So, ( H , Δ ∗ , ν ) is a coalgebra.
Obviously, by the definition of Δ ∗ and ν , we have Δ ∗ ( H n ) ⊆ ⊕ 0 ≤ i ≤ n H i ⊗ H n − i and μ ( H n ) = 0 for n > 0 . So ( H , Δ ∗ , ν ) is a graded coalgebra. □
Next we prove the compatibility between the conjunction product and the unshuffle coproduct.
Lemma 3. Assume Γ 1 = ( [ m ] , E 1 ) , Γ 2 = ( [ n ] , E 2 ) , I ⊂ [ m ] and J ⊆ [ m + 1, m + n ] .Then
st ( ( Γ 1 ∪ Γ 2 ↑ m ) I ∪ J ) = st ( ( Γ 1 ) I ) ∪ st ( ( Γ 2 ↑ m ) J ) ↑ | I | .
Proof. Denote Θ = Γ 1 ∪ Γ 2 ↑ m . Since I ⊂ [ m ] , J ⊆ [ m + 1, m + n ] , there are no edges between Θ I and Θ J in Θ , i.e., Θ I ∪ J = Θ I ∪ Θ J . Then, st ( Θ I ∪ J ) = st ( Θ I ∪ Θ J ) .
Since max { I } < min { J } , st I ∪ J ( I ) = [ | I | ] and st I ∪ J ( J ) = [ | I | + 1, | I | + | J | ] . By Lemma 2,
st ( st ( Θ I ∪ J ) [ | I | ] ) = st ( Θ I )
and
st ( st ( Θ I ∪ J ) [ | I | + 1, | I | + | J | ] ) = st ( Θ J ) .
Since there are no edges between Θ I and Θ J , there are no edges between st ( Θ I ∪ J ) [ | I | ] and st ( Θ I ∪ J ) [ | I | + 1, | I | + | J | ] . So, st ( Θ I ∪ J ) = st ( Θ I ) ∪ st ( Θ J ) ↑ | I | .
Since Θ = Γ 1 ∪ Γ 2 ↑ m , I ⊆ [ m ] and J ⊆ [ m + 1, m + n ] , we have Θ I = ( Γ 1 ) I and Θ J = ( Γ 2 ↑ m ) J . Then
st ( ( Γ 1 ∪ Γ 2 ↑ m ) I ∪ J ) = st ( ( Γ 1 ) I ) ∪ st ( ( Γ 2 ↑ m ) J ) ↑ | I | . □
Corollary 1. Assume Γ 1 = ( [ m ] , E 1 ) , Γ 2 = ( [ n ] , E 2 ) , I ⊂ [ m ] and J ⊆ [ m + 1, m + n ] .Then
st ( ( Γ 1 ⋄ Γ 2 ) I ∪ J ) = st ( ( Γ 1 ) I ) ⋄ st ( ( Γ 2 ) J ′ ) , (6)
where J ′ = st [ m + 1, m + n ] ( J ) .
Proof. By the definition of ⋄ and Lemma 3,
st ( ( Γ 1 ⋄ Γ 2 ) I ∪ J ) = st ( ( Γ 1 ) I ) ⋄ st ( ( Γ 2 ↑ m ) J ) . (7)
Obviously, by the definition of J ′ and Lemma 2, we have
st ( ( Γ 2 ↑ m ) J ) = st ( st ( Γ 2 ↑ m ) J ′ ) = st ( ( Γ 2 ) J ′ ) .
Then (7) can be rewritten as (6). □
Theorem 3. ( H , ⋄ , μ , Δ ∗ , ν ) is a bialgebra.
Proof. Obviously, the counit μ is an algebra homomorphism. We only need to prove the Δ ∗ is an algebra homomorphism, i.e.,
for any Γ 1 and Γ 2 in H. If Γ 1 = ε or Γ 2 = ε then (8) holds. If Γ 1 = ( [ m ] , E 1 ) and Γ 2 = ( [ n ] , E 2 ) are non-empty, we denote Θ = Γ 1 ⋄ Γ 2 = Γ 1 ∪ Γ 2 ↑ m and
Δ ∗ ( Θ ) = ∑ I ⊆ [ m + n ] st ( Θ I ) ⊗ st ( Θ [ m + n ] \ I ) . (9)
Denote I 11 = I ∩ [ m ] , I 12 = I ∩ [ m + 1 , m + n ] , I 21 = ( [ m + n ] \ I ) ∩ [ m ] and I 22 = ( [ m + n ] \ I ) ∩ [ m + 1, m + n ] . Furthermore, denote st [ m + 1, m + n ] ( I 12 ) by J 12 and denote st [ m + 1, m + n ] ( I 22 ) by J 22 . We have I 11 ⊆ [ m ] , I 12 ⊆ [ m + 1, m + n ] and J 12 = st [ m + 1, m + n ] ( I 12 ) . By Corollary 1,
st ( Θ I ) = st ( Θ I 11 ∪ I 12 ) = st ( ( Γ 1 ) I 11 ) ⋄ st ( ( Γ 2 ) J 12 ) .
Similarly,
st ( Θ [ m + n ] \ I ) = st ( Θ I 21 ∪ I 22 ) = st ( ( Γ 1 ) I 21 ) ⋄ st ( ( Γ 2 ) J 22 ) .
Then (9) can be rewritten as
∑ I ⊆ [ m + n ] st ( ( Γ 1 ) I 11 ) ⋄ st ( ( Γ 2 ) J 12 ) ⊗ st ( ( Γ 1 ) I 21 ) ⋄ st ( ( Γ 2 ) J 22 ) . (10)
Obviously, when I traverses all subsets of [ m + n ] , I 12 and I 21 traverse all disjoint subsets of [ m ] , I 12 and I 22 traverse all disjoint subsets of [ m + 1, m + n ] , and meanwhile J 12 and J 22 travese all disjoint subsets of [ n ] .
Then we rewrite (10) as
∑ I 11 ∩ I 21 = ∅ I 11 ∪ I 21 = [ m ] st ( ( Γ 1 ) I 11 ) ⊗ st ( ( Γ 1 ) I 21 ) ⋄ ∑ J 12 ∩ J 22 = ∅ J 12 ∪ J 22 = [ n ] st ( ( Γ 1 ) J 12 ) ⊗ st ( ( Γ 2 ) J 22 ) . (11)
By the definition of Δ ∗ , (11) is equal to
Δ ∗ ( Γ 1 ) ⋄ Δ ∗ ( Γ 2 ) .
Therefore,
Δ ∗ ( Γ 1 ⋄ Γ 2 ) = Δ ∗ ( Γ 1 ) ⋄ Δ ∗ ( Γ 2 ) .
So ( H , ⋄ , μ , Δ ∗ , ν ) is a bialgebra. □
Corollary 2. ( H , ⋄ , μ , Δ ∗ , ν ) is a Hopf algebra.
Proof. By Theorem 1, Theorem 2 and Theorem 3, ( H , ⋄ , μ , Δ ∗ , ν ) is a graded connected bialgebra. So it is a Hopf algebra. □
Example 8. For and , we have
5. Hopf Homomorphism from Permutations to Labeled Simple Graphs
In 2020, Aval, Bergeron and Machacek gave a Hopf homomorphism from permutations to labeled simple graphs without a proof [33]. Next, we prove this mapping is a Hopf homomorphism.
Firstly, we review some basic concepts as follows. For more details, see [27] [33].
Denote S n as the set of all permutations of degree n. For any permutation α in S n , denote α = α 1 α 2 ⋯ α n , where α ( i ) = α i for 1 ≤ i ≤ n . For example, S 3 = { 123,132,213,231,312,321 } . In particular, S 0 = { ε p } , where ε p is the empty permutation. If I = { α i 1 , α i 2 , ⋯ , α i s } ⊆ [ n ] and i 1 < i 2 < ⋯ < i s , then we define the restriction of α on I by α I = α i 1 α i 2 ⋯ α i s , and call it a subsequence of α .
More generally, for a set B = { β 1 , β 2 , ⋯ , β n } of positive integers, we call β = β 1 β 2 ⋯ β n a sequence on B with length n. Define the standard form of β by st ( β ) = st B ( β 1 ) st B ( β 2 ) ⋯ st B ( β n ) . In fact, any permutation is a sequence. Define S n to be the vector space spanned by S n over field K . Denote
S = ∪ n = 1 ∞ S n and S = ⊕ n = 1 ∞ S n .
Next, we review definitions of the conjunction product • and the unshuffle coproduct Δ ∗ on permutations [24] [33].
Define the conjunction product • on S by
α • β = α 1 α 2 ⋯ α m ( β 1 + m ) ( β 2 + m ) ⋯ ( β n + m ) ,
and the unshuffle coproduct on S by
Δ ∗ ( α ) = ∑ I ⊆ [ m ] st ( α I ) ⊗ st ( α [ m ] \ I ) ,
for α = α 1 α 2 ⋯ α m in S m and β = β 1 β 2 ⋯ β n in S n .
Obviously, we have α • ε p = ε p • α = α for any α ∈ S and Δ ( ε p ) = ε p ⊗ ε p . Define the unit μ p by μ p ( 1 ) = ε p and the counit ν p by
ν p ( α ) = ( 1 α = ε p , 0 otherwise ,
for α in S. Then ( S , • , μ p , Δ ∗ , ν p ) is a Hopf algebra [33].
For α = α 1 α 2 ⋯ α n in S n , we call ( α i , α j ) an inversion of α if i < j and α i > α j . Define a linear mapping g from S to H by g ( α ) = ( [ n ] , E α ) , where
E α = { ( α i , α j ) | i < j and α i > α j } , (12)
for α = α 1 α 2 ⋯ α n in S n . In particular, g ( ε p ) = ε . Obviously, each edge of g ( α ) connects an inversion of α . In fact, we can define the mapping g sending a sequence on B to a labeled simple graph on B by (12).
Example 10. For α = 13425 and β = 3465 ,
and
Next, we prove g is a Hopf homomorphism from S to H , that means it is an algebra homomorphism and a coalgebra homomorphism.
Lemma 4.The mapping g is an algebra homomorphism.
Proof. Suppose α = α 1 α 2 ⋯ α m in S m and β = β 1 β 2 ⋯ β n in S n . If α or β is an empty permutation, then obviously
g ( α • β ) = g ( α ) ⋄ g ( β ) .
If they are non-empty permutations, then
σ : = α • β = α 1 α 2 ⋯ α m ( β 1 + m ) ( β 2 + m ) ⋯ ( β n + m ) .
Since
max 1 ≤ i ≤ m { α i } < min 1 ≤ j ≤ n { β j + m } ,
there are no edges between { α 1 , ⋯ , α m } and { β 1 + m , ⋯ , β n + m } . If we denote g ( σ ) = ( [ m + n ] , E σ ) , then
E σ = { ( α i , α j ) | i < j and α i > α j } ∪ { ( β i + m , β j + m ) | i < j and β i + m > β j + m }
Denote E 1 = { ( α i , α j ) | i < j and α i > α j } and E 2 = { ( β i + m , β j + m ) | i < j and β i + m > β j + m } , then
( [ m + n ] , E σ ) = ( [ m ] , E 1 ) ∪ ( [ m + 1, m + n ] , E 2 ) .
Obviously, g ( α ) = ( [ m ] , E 1 ) by the definition of g . On the other hand, g ( β ) = ( [ n ] , E β ) , where
E β = { ( β i , β j ) | i < j and β i > β j } .
Then g ( β ) ↑ m = ( [ m + 1, m + n ] , E ′ 2 ) , where
E ′ 2 = { ( β i + m , β j + m ) | i < j and β i > β j } ,
which is equal to E 2 . So g ( β ) ↑ m = ( [ m + 1, m + n ] , E 2 ) . Then
g ( α • β ) = g ( α ) ∪ g ( β ) ↑ m = g ( α ) ⋄ g ( β ) .
So g is an algebra homomorphism. □
Lemma 5. Assume α is a non-empty permutation in S n and I is a subset of [ n ] .Then
g ( st ( α I ) ) = st ( g ( α ) I ) .
Proof. Let α = α 1 α 2 ⋯ α n ∈ S n , I = { α i 1 , α i 2 , ⋯ , α i s } and i 1 < i 2 < ⋯ < i s . Then α I = α i 1 α i 2 ⋯ α i s and
st ( α I ) = st I ( α i 1 ) st ( α i 2 ) ⋯ st ( α i s ) .
Denote γ k = st I ( α i k ) for 1 ≤ k ≤ s , then st ( α I ) = γ 1 γ 2 ⋯ γ s . If g ( st ( α I ) ) = ( [ | I | ] , E 1 ) , then
E 1 = [ ( γ j , γ k ) | j < k and γ j > γ k ] = { ( st I ( α i j ) , st I ( α i k ) ) | i j < i k and st I ( α i j ) > st I ( α i k ) } .
On the other hand, g ( α ) = ( [ n ] , E 2 ) , where
E 2 = { ( α j , α k ) | j < k and α j > α k } .
If we have g ( α ) I = ( I , E 3 ) then
E 3 = ( E 2 ) I = { ( α i j , α i k ) | i j < i k and α i j > α i k } .
And st ( g ( α ) I ) = st ( I , E 3 ) = ( [ | I | ] , st ( E 3 ) ) , where
st ( E 3 ) = { ( st I ( α i j ) , st I ( α i k ) ) | i j < i k and α i j > α i k } = { ( st I ( α i j ) , st I ( α i k ) ) | i j < i k and st I ( α i j ) > st I ( α i k ) } = E 1 .
So g ( st ( α I ) ) = st ( g ( α ) I ) . □
Lemma 6. The mapping g is a coalgebra homomorphism.
Proof. Obviously, for empty permutation ε p , we have
g ( Δ ∗ ( ε p ) ) = ε ⊗ ε = Δ ∗ ( g ( ε p ) ) .
For any non-empty permutation α ∈ S n , by Lemma 5 we have
g ( Δ ∗ ( α ) ) = g ( ∑ I ⊆ [ n ] st ( α I ) ⊗ st ( α [ n ] \ I ) ) = ∑ I ⊆ [ n ] g ( st ( α I ) ) ⊗ g ( st ( α [ n ] \ I ) )
= ∑ I ⊆ [ n ] st ( g ( α ) I ) ⊗ st ( g ( α ) [ n ] \ I ) = Δ ∗ ( g ( α ) ) .
So g is a coalgebra homomorphism. □
Corollary 3. The mapping g is a Hopf homomorphism.
Proof. By Lemma 4 and Lemma 6, g is a Hopf homomorphism. □
6. Conclusion
Let H be the vector space spanned by labeled simple graphs. Firstly, we give the definitions of the conjunction product ⋄ and the unshuffle coproduct Δ ∗ on H . Then we prove the conjunction product ⋄ satisfies associativity and the unshuffle coproduct Δ ∗ satisfies coassociativity, i.e., ( H , ⋄ , μ ) is an algebra and ( H , Δ ∗ , ν ) is a coalgebra. We prove the compatibility between ⋄ and Δ ∗ , and ( H , ⋄ , μ , Δ ∗ , ν ) is a graded connected bialgebra. So ( H , ⋄ , μ , Δ ∗ , ν ) is Hopf algebra. Lastly, let S be the vector space spanned by permutations. We recall a mapping g from S to H and prove it is a Hopf homomorphism. In the future, we will study the duality of the Hopf algebra ( H , ⋄ , μ , Δ ∗ , ν ) .
Acknowledgements
This work is supported by National Natural Science Foundation of China (Nos. 11701339 and 12071265).
Conflicts of Interest
The author declares that there are no conflicts of interest in this paper.
Cite this paper
Dong, J.M. and Li, H.L. (2023) Hopf Algebra of Labeled Simple Graphs. Open Journal of Applied Sciences, 13, 120-135. https://doi.org/10.4236/ojapps.2023.131011
NOTESReferencesHopf, H. (1964) über die topologie der gruppen-mannigfaltigkeiten und ihrer verallgemeinerungen. In: Hopf, H., Ed., Selecta Heinz Hopf, Springer, Berlin, 119-151. https://doi.org/10.1007/978-3-662-25046-4_9Milnor, J.W. and Moore, J.C. (1965) On the Structure of Hopf Algebras. Journal Annals of Mathematics, 81, 211-264. https://doi.org/10.2307/1970615Sweedler, M.E. (1969) Hopf Algebras. Benjamin, New York.Chase, S.U. and Sweedler, M.E. (1969) Hopf Algebras and Galois Theory. In: Chase, S.U. and Sweedler, M.E., Eds., Hopf Algebras and Galois Theory, Springer, Berlin, 52-83. https://doi.org/10.1007/BFb0101435Joni, S.A. and Rota, G.-C. (1979) Coalgebras and Bialgebras in Combinatorics. Studies in Applied Mathematics, 61, 93-139. https://doi.org/10.1002/sapm197961293Ehrenborg, R. (1996) On Posets and Hopf Algebras. Advances in Mathematics, 119, 1-25. https://doi.org/10.1006/aima.1996.0026Gelfand, I., Krob, D., Lascoux, A., et al. (1995) Noncommutative Symmetric Functions. Advances in Mathematics, 112, 218-348. https://doi.org/10.1006/aima.1995.1032Li, H., Morse, J. and Shields, P. (2016) Structure Constants for K-Theory of Grassmannians. Journal of Combinatorial Theory, Series A, 144, 306-325. https://doi.org/10.1016/j.jcta.2016.06.016Kassel, C. (2012) Quantum Groups. Springer Science & Business Media, Berlin.Cheng, J., Wang, Y. and Zhang, R. (2020) Degenerate Quantum General Linear Groups. Advances in Theoretical and Mathematical Physics, 24, 1371-1418. https://doi.org/10.4310/ATMP.2020.v24.n6.a2Cheng, T., Huang, H. and Yang, Y. (2015) Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-Categories. Symmetry Integrability and Geometry: Methods and Applications, 12, 4. https://doi.org/10.3842/SIGMA.2016.004Cheng, T., Li, H. and Yang, Y. (2019) A New View of Generalized Clifford Algebras. Advances in Applied Clifford Algebras, 29, 42. https://doi.org/10.1007/s00006-019-0966-zCheng, J. and Zeng, Z. (2018) Irreducible Wakimoto-Like Modules for the Lie Superalgebra D(2, 1, α). Acta Mathematica Sinica, English Series, 34, 1578-1588. https://doi.org/10.1007/s10114-018-7265-9Cheng, J. and Gao, Y. (2021) Generalized P(N)-Graded Lie Superalgebras. Frontiers of Mathematics in China, 16, 647-687. https://doi.org/10.1007/s11464-021-0888-7Schmitt, W.R. (1994) Incidence Hopf Algebras. Journal of Pure and Applied Algebra, 96, 299-330. https://doi.org/10.1016/0022-4049(94)90105-8Schmitt, W.R. (1995) Hopf Algebra Methods in Graph Theory. Journal of Pure and Applied Algebra, 101, 77-90. https://doi.org/10.1016/0022-4049(95)90925-BConnes, A. and Kreimer, D. (1999) Hopf Algebras, Renormalization and Noncommutative Geometry. In: Quantum Field Theory: Perspective and Prospective, Springer, Berlin, 59-109. https://doi.org/10.1007/978-94-011-4542-8_4Kreimer, D. (1997) On the Hopf Algebra Structure of Perturbative Quantum Field Theories. Advances in Theoretical and Mathematical Physics, 2, 303-334. https://doi.org/10.4310/ATMP.1998.v2.n2.a4Malvenuto, C. and Reutenauer, C. (1995) Duality between Quasi-Symmetrical Functions and the Solomon Descent Algebra. Journal of Algebra, 177, 967-982. https://doi.org/10.1006/jabr.1995.1336Aguiar, M. and Sottile, F. (2005) Cocommutative Hopf Algebras of Permutations and Trees. Journal of Algebraic Combinatorics, 22, 451-470. https://doi.org/10.1007/s10801-005-4628-yNovelli, J.-C., Thibon, J.-Y. and Thiéry, N.M. (2004) Algèbres de Hopf de graphes. Comptes Rendus Mathematique, 333, 607-610. https://doi.org/10.1016/j.crma.2004.09.012Forcey, S., Lauve, A. and Sottile, F. (2010) Hopf Structures on the Multiplihedra. SIAM Journal on Discrete Mathematics, 24, 1250-1271. https://doi.org/10.1137/090776834Vargas, Y. (2014) Hopf Algebra of Permutation Pattern Functions. Discrete Mathematics & Theoretical Computer Science, Nancy. https://doi.org/10.46298/dmtcs.2446Giraudo, S. and Vialette, S. (2016) Unshuffling Permutations. Latin American Theoretical Informatics Symposium, Vol. 9644, 509-521. https://doi.org/10.1007/978-3-662-49529-2_38Zhao, M. and Li, H. (2021) A Pair of Dual Hopf Algebras on Permutations. AIMS Mathematics, 6, 5106-5123. https://doi.org/10.3934/math.2021302Guo, L., Thibon, J.-Y. and Yu, H. (2020) The Hopf Algebras of Signed Permutations, of Weak Quasi-Symmetric Functions and of Malvenuto-Reutenauer. Advances in Mathematics, 374, Article ID: 107341. https://doi.org/10.1016/j.aim.2020.107341Liu, M. and Li, H. (2021) A Hopf Algebra on Permutations Arising from Super-Shuffle Product. Symmetry, 13, 1010. https://doi.org/10.3390/sym13061010Conci, A., Li, H. and MacHenry, T. (2014) Rational Convolution Roots of Isobaric Polynomials. Rocky Mountain Journal of Mathematics, 47, 1259-1275.Cao, L. (2018) A Minimal Completion of (0, 1)-Matrices without Total Support. Linear Multilinear Algebra, 67, 1522-1530. https://doi.org/10.1080/03081087.2018.1460793Cao, L., Chen, Z., Duan, X., et al. (2019) Diagonal Sums of Doubly Substochastic Matrices. Electronic Journal of Linear Algebra, 35, 42-52. https://doi.org/10.13001/1081-3810.3760Cao, L. and Chen, Z. (2019) Partitions of the Polytope of Doubly Substochastic Matrices. Linear Algebra and Its Applications, 563, 98-122. https://doi.org/10.1016/j.laa.2018.10.024Cao, L., Chen, Z., Koyuncu, S. and Li, H. (2020) Permanents of Doubly Substochastic Matrices. Linear and Multilinear Algebra, 68, 594-605. https://doi.org/10.1080/03081087.2018.1513448Aval, J.-C., Bergeron, N. and Machacek, J. (2020) New Invariants for Permutations, Orders and Graphs. Journal of the American Mathematical Society, 10, Article ID: 102080. https://doi.org/10.1016/j.aam.2020.102080Grinberg, D. and Reiner, V. (2020) Hopf Algebras in Combinatorics. Mathematisches Forschungsinstitut Oberwolfach gGmbH, Bibliothek.West, D.B. (2001) Introduction to Graph Theory. Prentice Hall, Upper Saddle River.
