Free modified Rota-Baxter algebras and Hopf algebras
Xigou Zhang, Xing Gao, Li Guo

TL;DR
This paper constructs free modified Rota-Baxter algebras and demonstrates how to equip them with bialgebra and Hopf algebra structures using cocycle methods, advancing algebraic theory.
Contribution
It introduces a method to construct free modified Rota-Baxter algebras and establishes their bialgebra and Hopf algebra structures under certain conditions.
Findings
Constructed free modified Rota-Baxter algebras.
Established bialgebra structures via cocycle construction.
Proved Hopf algebra structures for connected cases.
Abstract
The notion of a modified Rota-Baxter algebra comes from the combination of those of a Rota-Baxter algebra and a modified Yang-Baxter equation. In this paper, we first construct free modified Rota-Baxter algebras. We then equip a free modified Rota-Baxter algebra with a bialgebra structure by a cocycle construction. Under the assumption that the generating algebra is a connected bialgebra, we further equip the free modified Rota-Baxter algebra with a Hopf algebra structure.
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Free modified Rota-Baxter algebras and Hopf algebras
Xigou Zhang
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
,
Xing Gao
School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, P. R. China
and
Li Guo
Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, USA
Abstract.
The notion of a modified Rota-Baxter algebra comes from the combination of those of a Rota-Baxter algebra and a modified Yang-Baxter equation. In this paper, we first construct free modified Rota-Baxter algebras. We then equip a free modified Rota-Baxter algebra with a bialgebra structure by a cocycle construction. Under the assumption that the generating algebra is a connected bialgebra, we further equip the free modified Rota-Baxter algebra with a Hopf algebra structure.
Key words and phrases:
Modified Rota-Baxter algebra, Rota-Baxter algebra, Hopf algebra, bracketed word, cocycle
2010 Mathematics Subject Classification:
16T99,16W99,16S10
Contents
-
2.1 The general construction of the free modified Rota-Baxter algebras
-
3 The Hopf algebra structure on free modified Rota-Baxter algebras
1. Introduction
This paper studies free objects in the category of modified Rota-Baxter algebras, a concept coming from the combination of a Rota-Baxter algebra and a modified Yang-Baxter equation. It also equips the free objects with bialgebra and Hopf algebra structures.
For a fixed constant , a Rota-Baxter operator of weight is a linear operator on an associative algebra that satisfies the Rota-Baxter equation:
[TABLE]
An associative algebra equipped with a Rota-Baxter operator is called a Rota-Baxter algebra, a notion originated from the probability study of G. Baxter [9] in 1960. Later it attracted the attention of well-known mathematicians such as Atkinson, Cartier and Rota [3, 12, 26]. After some years of dormancy, its study experienced a quite remarkable renascence since late 1990s, with many applications in mathematics and physics [2, 4, 14, 18, 20, 24, 22, 23, 25]. In particular, it appeared as one of the fundamental algebraic structures in the profound work of Connes and Kreimer on renormalization of quantum field theory [11]. See [17] for further details and references.
The concept of the classical Yang-Baxter equation arose from the study of inverse scattering theory and is also related to Schouten bracket in differential geometry. Further it can be regarded as the classical limit of the quantum Yang-Baxter equation, named after C.-Y. Yang and R. Baxter. In the 1980s, Semonov-Tian-Shansky [28] found that, under suitable conditions, the operator form of the classical Yang-Baxter equation is precisely the Rota-Baxter identity (1) (of weight 0) on a Lie algebra. As a modified form of the operator form of the classical Yang-Baxter equation, he also introduced in that paper the modified classical Yang-Baxter equation:
[TABLE]
later found applications in the study of generalized Lax pairs and affine geometry on Lie groups [5, 10, 21]. As the associative analogue of Eq. (2), the equation
[TABLE]
is called the modified associative Yang-Baxter equation, which has been applied to the study of extended -operators, associative Yang-Baxter equations, infinitesimal bialgebras and dendriform algebras [6, 7, 13].
In the spirit of the aforementioned Yang-Baxter equation to Rota-Baxter operator connection, a linear operator satisfying Eq. (3) is called a modified Rota-Baxter operator and an associative algebra equipped with a modified Rota-Baxter operator is called a modified Rota-Baxter algebra.
Integrating the notions of the Rota-Baxter algebra and modified Rota-Baxter algebra, the concept of a modified Rota-Baxter algebra with a weight was introduced in [6] as a special case of extended -operators in connection with the extended associative Yang-Baxter equation. The latter motivated their study in the Lie algebra context [8]. In [30], free commutative modified Rota-Baxter algebras were constructed by means of a modified quasi-shuffle product and modified stuffle product, in analogy to the case of free commutative Rota-Baxter algebras [12, 18].
Considering the close relationship between the modified Rota-Baxter (associative) algebras and the modified Yang-Baxter equation for Lie algebras, it is especially interesting to consider noncommutative modified Rota-Baxter algebras. This is the subject of study of this paper, focusing on the construction of the free objects and the Hopf algebra structures on the free objects. More precisely, in Section 2, we obtain an explicit construction of the free modified Rota-Baxter algebra on an algebra, by giving a natural basis of the algebra and the corresponding multiplication table. In Section 3, we further provide a bialgebra and then a Hopf algebra structure on the free modified Rota-Baxter algebra.
Notations. For the rest of this paper, unless otherwise specified, algebras are associative unitary algebras over a commutative unitary algebra .
2. Free Modified Rota-Baxter Algebras
In this section we construct free modified Rota-Baxter algebras. We give the construction in Section 2.1, leading to the main Theorem 2.6 of this section. The proof of the theorem is completed in Section 2.2.
2.1. The general construction of the free modified Rota-Baxter algebras
We begin with the general definition of modified Rota-Baxter algebras.
Definition 2.1**.**
Let be a -algebra and . A linear map is called a modified Rota-Baxter operator of weight if satisfies the operator identity
[TABLE]
Then the pair or simply is called a modified Rota-Baxter algebra of weight .
Together with the algebra homomorphisms between the algebras that preserves the linear operators, the class of modified Rota-Baxter algebras of weight forms a category. We refer the reader to [30] and the references therein for basic properties of modified Rota-Baxter algebras and focus our attention to the construction of free modified Rota-Baxter algebras. We first give the definition.
Definition 2.2**.**
Let be a -algebra. A free modified Rota-Baxter algebra on is a modified Rota-Baxter algebra together with an algebra homomorphism with the property that, for any given modified Rota-Baxter algebra and algebra homomorphism , there is a unique homomorphism of modified Rota-Baxter algebras such that .
Note that taking to be the free algebra on a set , we obtain the free modified Rota-Baxter algebra on the set . Let be a -algebra with a -basis . We first display a -basis of free modified Rota-Baxter algebras in terms of bracketed words from the alphabet set .
Remark 2.3**.**
The set is called the set of Rota-Baxter words that was applied to construct free Rota-Baxter algebras [14]. Enumeration properties and generating functions of Rota-Baxter words were obtained in [19] to which we refer the reader for further details.
Let and be two different symbols not in , called brackets, and let . Denote by the free monoid generated by .
Definition 2.4**.**
([15, 17]) Let be two subsets of . Define the alternating product of and to be
[TABLE]
Here stands for disjoint union.
For example, are elements in . But are not in .
We construct a sequence of subsets of by the following recursion on . For the initial step, we define . For the inductive step, we define
[TABLE]
For example, for , the elements and are all in , the first two are in and the first one is in .
From the definition we have . Assuming , we get
[TABLE]
Thus we can define
[TABLE]
For , we define the depth of to be
[TABLE]
Further, every has a unique standard decomposition:
[TABLE]
where , , are alternatively in or in . We call to be the breadth of , denoted by . We define the head of to be 0 (resp. 1) if is in (resp. in ). Similarly define the tail of to be 0 (resp. 1) if is in (resp. in ).
Fix a . We will equip the free -module
[TABLE]
with a multiplication . This is accomplished by defining for basis elements and then extending bilinearly. Roughly speaking, the product of and is defined to be the concatenation whenever . When , the product is defined by the product in or by the modified Rota-Baxter identity in Eq. (4).
To be precise, we use induction on the sum to define . For the initial step of , are in and so are in . Then we define
[TABLE]
Here is the product in .
For the inductive step, let be given and assume that have been defined for all with . Then consider with . First treat the case when . Then and are in or . Since , and cannot be both in . We accordingly define
[TABLE]
Here the product in the first and second case are by concatenation and in the third case is by the induction hypothesis since for the three products on the right hand side we have
[TABLE]
We next treat the case when or . Let and be the standard decompositions from Eq. (5). We then define
[TABLE]
where is defined by Eq. (7) and the rest is given by concatenation. Extending bilinearly, we obtain a binary operation
[TABLE]
This completes the definition of .
Lemma 2.5**.**
Let .
- (a)
and . 2. (b)
If , then (concatenation). 3. (c)
If , then for any ,
[TABLE]
Proof.
Items (a) and (b) follow from the definition of . The proof of Item (c) is the same as [17, Lemma 4.4.5]. ∎
We next define a linear operator
[TABLE]
In the rest of the paper, we will use the infix notation interchangeably with for any . Let
[TABLE]
be the natural injection which extends to an algebra injection
[TABLE]
Now we state our first main result, to be proved in the next subsection.
Theorem 2.6**.**
Let be a -algebra with a -basis and be given.
- (a)
The pair is an algebra. 2. (b)
The triple is a modified Rota-Baxter algebra of weight . 3. (c)
The triple ) together with the embedding is the free modified Rota-Baxter algebra of weight on the algebra .
2.2. The proof of Theorem 2.6
Proof.
(a). It is enough to verify the associativity for basis elements:
[TABLE]
We carry out the verification by induction on the sum of the depths
[TABLE]
If , then
[TABLE]
and so . In this case the product is given by the product in and so is associative.
Assume that Eq. (9) holds for for any given and consider with
[TABLE]
If , then by Lemma 2.5,
[TABLE]
A similar argument holds when . Thus we only need to verify the associativity when and . We next reduce the proof to the breadths of the words and depart to show a lemma.
Lemma 2.7**.**
If Eq. (9) holds for all and in of breadth one, then it holds for all and in .
Proof.
We use induction on the sum of breadths . The case when is the assumption of the lemma. Assume the associativity holds for for some and take with So at least one of has breadth greater than or equal to 2.
First assume that . Then we may write
[TABLE]
By Lemma 2.5, we obtain
[TABLE]
Similarly,
[TABLE]
Thus
[TABLE]
whenever
[TABLE]
which follows from the induction hypothesis. A similar proof works if
Finally if , we may write
[TABLE]
By Lemma 2.5 again, we get
[TABLE]
In the same way, we have
[TABLE]
This proves the associativity.
∎
In summary, the proof of the associativity has been reduced to the special case when are chosen so that
- (a)
with the assumption that the associativity holds when . 2. (b)
the elements have breadth one and 3. (c)
and .
By Item (b), the head and tail of each of the elements are the same. Therefore by Item (c), either all the three elements are in or they are all in . If all of are in , then as already shown, the associativity follows from the associativity in . So it remains to consider the case when are all in . Then we may write
[TABLE]
Applying Eq. (7) and bilinearity of the product , we get
[TABLE]
Similarly we obtain
[TABLE]
Now by the induction hypothesis, the -th term in the expansion of coincides with the -th term in the expansion of . Here is the permutation given by
[TABLE]
This completes the proof of Theorem 2.6 (a).
(b). The proof follows from the definition and Eq. (7).
(c). Let be a modified Rota-Baxter algebra with multiplication and let be a -algebra homomorphism. We will construct a -linear map by defining for . We achieve this by defining for , inductively on . For , define Then is satisfied. Suppose has been defined for and consider in which is, by definition,
[TABLE]
Let be in the first union component above. Then
[TABLE]
for and , . By the construction of the multiplication and the modified Rota-Baxter operator , we have
[TABLE]
Define
[TABLE]
where the right hand side is well-defined by the induction hypothesis. Similarly define if is in the other union components. For any , we have , and by the definition of in (Eq. (10)), we have
[TABLE]
So commutes with the modified Rota-Baxter operators. Combining this equation with Eq. (10) we see that if is the standard decomposition of , then
[TABLE]
Note that this is the only possible way to define in order for to be a modified Rota-Baxter algebra homomorphism extending . It remains to prove that the map defined in Eq. (10) is indeed an algebra homomorphism. For this we only need to check the multiplicity
[TABLE]
for all . For this we use induction on the sum of depths . Then . When , we have . Then Eq. (12) follows from the multiplicity of . Assume the multiplicity holds for with and take with . Let and be the standard decompositions. Since , at least one of and is in . Then by Eq. (7) we have
[TABLE]
In the first two cases, the right hand side is by the definition of . In the third case, applying Eq. (11), the induction hypothesis and the modified Rota-Baxter relation of the operator on , we have
[TABLE]
Therefore . Then
[TABLE]
as required.
This completes the proof of Theorem 2.6 ∎
3. The Hopf algebra structure on free modified Rota-Baxter algebras
In this section, starting with the assumption that is a bialgebra with its coproduct and its counit , we provide a bialgebraic and then a Hopf algebraic structure on the free modified Rota-Baxter algebras obtained in Section 2, when . It would be interesting to see how to extend this construction to other weights . For Hopf algebra structures on free Rota-Baxter algebras, see [16, 29] for Hopf algebra structures on free Rota-Baxter algebras.
3.1. The bialgebraic structure
We now build on results from previous subsections to obtain a bialgebra structure on . We first record some lemmas for a preparation.
Lemma 3.1**.**
Let be a given element of .
- (a)
The linear map is a modified Rota-Baxter operator of weight on . 2. (b)
There exists a unique modified Rota-Baxter algebra morphism such that
[TABLE]
Proof.
(a) It follows from
[TABLE]
(b) By Item (a), is a modified Rota-Baxter algebra of weight . Then the remainder follows from Theorem 2.6 (c). ∎
Note that is a modified Rota-Baxter operator on ;however is not a modified Rota-Baxter operator on . The following result constructs a modified Rota-Baxter operator on .
Lemma 3.2**.**
Let be a given element of . Define the linear map
[TABLE]
by taking
[TABLE]
Then is a modified Rota-Baxter operator of weight on .
Proof.
Let . On the one hand,
[TABLE]
On the other hand,
[TABLE]
This completes the proof. ∎
With a similar argument, we can obtain
Lemma 3.3**.**
Let be a given element of . Define the linear map
[TABLE]
by taking
[TABLE]
Then is a modified Rota-Baxter operator of weight on .
Now we are ready for our main result of this subsection. Recall is an algebra homomorphism given in Lemma 3.1. Let be the natural embedding. By Theorem 2.6 (c) and Lemma 3.2, there is a (unique) modified Rota-Baxter algebra morphism
[TABLE]
such that .
Theorem 3.4**.**
Let be a bialgebra and . Then the quintuple is a bialgebra.
Proof.
It suffices to prove the counity of and coassociativity of . For the former, denote by
[TABLE]
Then is an algebra homomorphism, since and are algebra homomorphisms. Further it is a modified Rota-Baxter algebra morphism. Indeed, for any ,
[TABLE]
By unicity in the universal property of , we have
[TABLE]
and so is a left counit. By symmetry, we can prove is also a right counit.
Moreover, both and are modified Rota-Baxter algebra morphisms from to , which is equipped with the modified Rota-Baxter operator of weight given in Lemma 3.3. As they coincide on
[TABLE]
they are equal and so is coassociative. Here is the coproduct on . Thus the quintuple is a bialgebra. ∎
Remark 3.5**.**
For any , we have
[TABLE]
In other words,
[TABLE]
which is analogue to the 1-cocycle condition in the well-known Connes-Kreimer Hopf algebra on rooted trees [11].
3.2. The Hopf algebraic structure
In this last part of the paper we show that if we start with being a connected filtered bialgebra and , then the bialgebra also has a connected filtration and hence is a Hopf algebra.
Definition 3.6**.**
A bialgebra is called filtered if it has an increasing filtration , , such that
[TABLE]
A filtered bialgebra is called connected if and .
The following result is well-known.
Lemma 3.7**.**
[27] A connected filtered bialgebra is a Hopf algebra.
Our discussion in this section will be based on the following condition.
Definition 3.8**.**
A -basis of a connected filtered bialgebra is called a filtered basis of if there is an increasing filtration such that
[TABLE]
Here is the identity of . Elements are said to have degree , denoted by .
Let be a connected filtered bialgebra with a filtered basis . Recall that constructed in Subsection 2.1 is a k-basis of the free modified Rota-Baxter algebra . We now define the degree for by induction on . For the initial step of , we get and define
[TABLE]
For the inductive step of , if , then and we define
[TABLE]
if , then write in the standard decomposition and define
[TABLE]
where each is defined either in Eq. (17) or in Eq. (18) by the induction hypothesis.
Remark 3.9**.**
For later applications, we also use the notion for .
Denote
[TABLE]
Then
[TABLE]
Now we are going to prove that is a filtered bialgebra, beginning with the compatibility of the multiplication with the filtration.
Lemma 3.10**.**
For , we have
[TABLE]
Proof.
Let and be two basis elements in . Then
[TABLE]
We now verify Eq. (22) by induction on the sum . When , then . By Eq. (21), we obtain that and so . This finishes the initial step.
Given an , assume that Eq. (22) holds for with and consider case . If or , without loss of generality, letting , then and
[TABLE]
So we may suppose . Write
[TABLE]
in their standard decompositions. Under this condition, we proceed to prove Eq. (22) by induction on the sum . When , then . If or , then by Eq. (19),
[TABLE]
It remains to check the outstanding case of
[TABLE]
where
[TABLE]
Then
[TABLE]
By the induction hypothesis on , we have
[TABLE]
which implies from Eq. (21) that
[TABLE]
Hence by Eq. (7),
[TABLE]
Assume that Eq. (22) holds for and and consider the case when and . So either or has breadth greater than or equal to , giving us three cases to consider:
Case 1. . Let , where with breadths respectively. By Eq. (19), we obtain . From Eq. (8),
[TABLE]
By the induction on , we have
[TABLE]
whence by Eq. (19),
[TABLE]
Case 2. . The proof of this case is similar to Case 1.
Case 3. and . Let and , where with breadths respectively. By Eq. (19), we obtain
[TABLE]
Thus by Eq. (8),
[TABLE]
By the induction on , we have
[TABLE]
With a similar argument to Case 1. we get
[TABLE]
This finishes the proof. ∎
For the compatibility of the coproduct with the filtration, we have
Lemma 3.11**.**
For , we have
[TABLE]
Proof.
We verify Eq. (23) by showing
Claim 3.12**.**
For any , we have
[TABLE]
where and are non-zero linear multiples of elements of with . Here we have adapted the notation in Remark 3.9.
To prove this claim we proceed by induction on . For the initial step of , we get and the result holds. Assume that Claim (24) holds for and consider for some .
In this case, we prove Claim (24) by induction on the breadth . If , we have or for some . For the former, Claim (24) holds since is given by and is a connected filtered bialgebra by our hypothesis. For the latter, applying the induction hypothesis on , we can write
[TABLE]
where , with the notion in Remark 3.9. By Eq. (16), we have
[TABLE]
By Eq. (20), it is sufficient to show that the sum of degrees of tensor factors in each summand is less than or equal to , which follows from
[TABLE]
Assume that Claim (24) holds for with and consider the case with . Let , where with . From Eq. (19), we have
[TABLE]
Write
[TABLE]
By the induction hypothesis on , we have
[TABLE]
So we have
[TABLE]
By Eq. (22),
[TABLE]
which implies from Eqs. (20), (25) and (26) that Claim 24 holds. ∎
We now arrive at our last main result.
Theorem 3.13**.**
Let be a connected filtered bialgebra with a filtered basis. Then is also a connected filtered bialgebra, and hence a Hopf algebra.
Proof.
By Lemma 3.7, we just need to prove that is a connected filtered bialgebra. This follows from Lemmas 22, 23 and Eq. (21). ∎
Acknowledgements: This work was supported by the National Natural Science Foundation of China (No. 11771190), the Fundamental Research Funds for the Central Universities (No. lzujbky-2017-162) and the Natural Science Foundation of Gansu Province (No. 17JR5RA175). The authors thank the referees for helpful suggestions.
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