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Axioms 2017, 6(4), 33; doi:10.3390/axioms6040033
Universal Enveloping Commutative Rota–Baxter Algebras of Pre- and Post-Commutative Algebras
Laboratory of Rings Theory, Sobolev Institute of Mathematics of the SB RAS, Acad. Koptyug ave., 4, 630090 Novosibirsk, Russia
Department of Mathematics and Mechanics, Novosibirsk State University, Pirogova str., 2, 630090 Novosibirsk, Russia
Received: 2 October 2017 / Accepted: 4 December 2017 / Published: 7 December 2017
Universal enveloping commutative Rota–Baxter algebras of pre- and post-commutative algebras are constructed. The pair of varieties (RBλCom, postCom) is proved to be a Poincaré–Birkhoff–Witt-pair (PBW)-pair and the pair (RBCom, preCom) is proven not to be.
Keywords:Rota–Baxter algebra; universal enveloping algebra; PBW-pair of varieties; Zinbiel algebra; dendriform algebra; pre-commutative algebra; post-commutative algebra
Any linear operator R defined on an algebra A over the field F is called a Rota–Baxter operator (RB-operator) of weight if it satisfies the relation
An algebra with a given RB-operator acting on it is called a Rota–Baxter algebra (RB-algebra).
G. Baxter defined commutative RB-algebras in 1960 ; the relation of Equation (1) with is simply a generalization of the integrated by parts formula. J.-C. Rota, P. Cartier and others studied [2,3,4] combinatorial properties of RB-operators and RB-algebras. In the 1980s, the deep connection between Lie RB-algebras and the classical Yang–Baxter equation (CYBE) was found [5,6]. In 2000, M. Aguiar showed  that a solution of the associative Yang–Baxter equation (AYBE)  gives rise to a structure of the associative RB-algebra. There are many applications of RB-operators in mathematical physics, combinatorics, number theory, and operads [9,10,11,12].
There exist many constructions of free commutative RB-algebras, including those given by J.-C. Rota, P. Cartier, and L. Guo jointly with W. Keigher [2,4,13]. We remark that the last  could be used to define a structure of a Hopf algebra on a free commutative RB-algebra . In 2008, K. Ebrahimi-Fard and L. Guo obtained free associative RB-algebra . Different linear bases of free Lie RB-algebras were recently found [16,17,18].
In 1995 , J.-L. Loday introduced algebras that satisfy the following identity:
We call such algebras “pre-commutative algebras” because they play an analogous role for “pre-algebras” that commutative algebras play for ordinary algebras. In literature, they are also known as dual Leibniz algebras (by Koszul duality) or Zinbiel algebras (the word “Leibniz” written in the inverse order). Regarding pre-commutative algebras, see, for example, [20,21,22].
In 1999 , J.-L. Loday introduced dendriform algebras (we call these “pre-associative algebras”). A linear space endowed with two bilinear products is called a pre-associative algebra if the following identities are satisfied:
Given a pre-associative algebra A, if we have for any , then A is a pre-commutative algebra because of the product . The same space A under the product is a pre-Lie algebra [24,25,26]; that is, it satisfies the identity .
In [27,28,29], post-associative, -commutative, and -Lie algebras were introduced. All of these have an additional product and satisfy certain identities.
The common definitions of the varieties of pre- and post--algebras for a variety can be found in [30,31].
In 2000 , M. Aguiar noticed that any commutative algebra with given a RB-operator R of zero weight is a pre-commutative algebra with the operation defined by . In 2007 , J.-L. Loday stated that a commutative algebra with a RB-operator of weight 1 is a post-commutative algebra under the operations and , where .
In 2013 , this connection between RB-algebras and pre- and post-algebras was generalized to any variety. In 2013 , it was proved that any pre--algebra (post--algebra) injectively embeds into its universal enveloping RB-algebra of variety and weight equal (not equal) to zero.
On the basis of the last result, we have a problem: to construct the universal enveloping RB-algebra of a variety for pre- and post--algebras. Another related problem is the following: whether the pairs of varieties and , , are Poincaré–Birkhoff–Witt (PBW)-pairs . Here, by (), we mean the variety of RB-algebras of variety and weight equal (not equal) to zero.
The objectives stated above appear in the associative case in the comments of chapter V of the monograph on RB-algebras by L. Guo  and were solved by the author in . See also a brief history of the subject for the associative case in .
The current work is devoted to the solution of the stated problems in the commutative case. Although the main method of the solution in associative and commutative cases is similar, the technical tools and the constructions are rather different, and it is hard to derive any of them from another.
In Section 2, we show the connections between RB-algebras and classical, modified and associative versions of the Yang–Baxter equation; we also give preliminaries on pre- and post-commutative algebras and PBW-pairs of varieties. Universal enveloping RB-algebras of pre-commutative (Section 3) and post-commutative (Section 4) types are constructed. As corollaries, we state that the pair of varieties is a PBW-pair and that is not.
Throughout the paper, by variety, we mean a class of algebraic structures of the same signature that is closed under the taking of homomorphic images, subalgebras and direct products . Birkhoff’s theorem  states that a variety of algebras can be equivalently defined as the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the class of all commutative RB-algebras forms the variety. On the other hand, the class of all commutative RB-algebras could be considered as the category with algebras as objects and homomorphisms as arrows. Thus, M. Aguiar  found the functor from the category of commutative RB-algebras of weight () to the category of pre-commutative (post-commutative) algebras. Within the paper, we are interested in its left adjoint functor. Finally, a variety can be considered as an operad , a particular case of the notion of multicategory. From this point of view, identities and varieties of pre- and post-algebras were investigated in [11,30].
We comment on terminology used in the paper. The history of investigations into defining identities for varieties of pre- and post-algebras can be roughly divided into two periods: the first came before the principal work of C. Bai et al.  appeared on arXiv in 2011 (see also ; for dialgebras and Koszul duals for pre-algebras, the breakthrough was made in 2008 ); the second period has been going on since this work. During the first period, the names of the varieties of pre- and post-algebras were in some sense chaotic (pre-Lie algebras, dendriform algebras, Zinbiel algebras, post-Lie algebras, tridendriform algebras, and commutative tridendriform algebras). After the common operadic foundation for all of these varieties appeared , it was logical to try to unify all of these objects of the same “world”. In light of the names of pre-Lie  and pre-Poisson  algebras given by M. Gerstenhaber and M. Aguiar respectively, the names of pre-Jordan, pre-alternative, pre-associative, and pre-Malcev algebras [39,40,41,42] appeared. Similar occurred for post-algebras after the name of post-Lie algebras was given by B. Vallette in ; for example, see . This is why we prefer to name dendriform algebras as pre-associative algebras and Zinbiel algebras as pre-commutative algebras (analogously for the varieties of post-algebras).
We consider some well-known examples of RB-operators (see, e.g., ):
Given an algebra A of continuous functions on , an integration operator is a RB-operator on A of zero weight.
Given an invertible derivation d on an algebra A, is a RB-operator on A of zero weight.
Let be an infinite sum of the field F. An operator R defined as is a RB-operator on A of weight .
Further, if nothing else is specified, by RB-operator, we mean a RB-operator of zero weight.
2.2. Yang–Baxter Equation
We let be a semisimple finite-dimensional Lie algebra over . For , we introduce the classical Yang—Baxter equation (CYBE) given by A.A. Belavin and V.G. Drinfel’d  aswhereare elements from .
A tensor is called skew-symmetric if . In [5,6], it was shown that given a skew-symmetric solution of the CYBE on , a linear map defined as is a RB-operator of zero weight on . Here denotes the Killing form on .
Up to conjugation and a scalar multiple unique skew-symmetric solution of the CYBE on is . The corresponding RB-operator is the following: , , .
We let A be an associative algebra; a tensor is a solution of the AYBE [8,45,46] ifwhere the definition of is the same as for CYBE.
(). Let be a solution of the AYBE on an associative algebra A. A linear map defined as is a RB-operator on A of zero weight.
(). Up to conjugation, transpose and a scalar multiple all nonzero solutions of the AYBE on are , , , and .
In 1983 , M.A. Semenov-Tyan-Shansky introduced the modified Yang–Baxter equation (MYBE). Letting L be a Lie algebra and R be a linear map on L, then the MYBE is
It is easy to check that R is a solution of the MYBE if and only if is a RB-operator on L of weight . Thus, there is one-to-one correspondence between the set of solutions of the MYBE and RB-operators of weight .
2.3. PBW-Pair of Varieties
In , the notion of a PBW-pair that generalizes the relation between associative and Lie algebras given by the famous Poincaré–Birkhoff–Witt theorem was introduced.
We let , be varieties of algebras and be such a functor that maps to , preserving A as a vector space but changing the operations on A. The universal enveloping algebra is an image of A of the left adjoint functor to . Defining on a natural ascending filtration, we obtain the associated graded algebra .
A pair of varieties with the functor is called a PBW-pair if and are isomorphic as elements of . Here denotes the vector space A with trivial multiplication operations.
2.4. Free Commutative RB-Algebra
We consider one of the possible constructions of free commutative RB-algebras from , which is based on shuffle algebra.
We let A be a commutative algebra over the field F. We denote by the vector space . We define the bilinear operation ⋄ on as follows: for and equals the sum of all tensors of the length whose tensor factors are exactly and , , ; moreover, the initial orders of and are preserved.
We give an explicit definition of the product by induction on . If at least one of the numbers m or n equals 0, for example, , then we define as a multiplication of the scalar on the tensor a. For , we define . For ,
Calculating , we have summands:
We consider a tensor product of algebras and . We define a linear map R on as follows:
In , it was proved that the RB-subalgebra of generated by is isomorphic to , a free commutative RB-algebra generated by A. A free commutative RB-algebra generated by a set X could be constructed as . We denote the free commutative RB-algebra of weight generated by a set X as . In short, an algebra is denoted as .
In , quasi-shuffle algebra was used for the construction of a free commutative RB-algebra of nonzero weight.
In , P. Cartier proved that a linear basis T of could be constructed by induction:
- all monomials from lie in T;
- if , then for a monomial w from .
We use the linear basis of of P. Cartier  and refer to it as a standard basis. Given a word u from the standard basis, the number of appearances of the symbol R in the notation of u is called the R-degree of the word u, with denotation . We also define the degree of u as follows:
2.5. Pre-Commutative Algebra
A pre-commutative algebra is an algebra whose product satisfies the identity
2.6. Post-Commutative Algebra
A post-commutative algebra is a linear space endowed with two bilinear products and ⊥ such that ⊥ is associative and commutative and the following identities are fulfilled:
A free post-commutative algebra can be constructed with the help of quasi-shuffle algebra .
2.7. Embedding of Loday Algebras into RB-Algebras
The common definition of the varieties of pre- and post--algebra for a variety can be found in [30,31].
Given a commutative algebra B with a RB-operator R of zero weight, the space B with respect to the operationis a pre-commutative algebra.
Given a commutative algebra with a RB-operator R of weight 1, we have that is a post-commutative algebra, where the product is defined by Equation (8). The case of a RB-operator R of any nonzero weight is reduced to the case of weight 1 as follows: the map is a RB-operator of weight 1.
Given a pre-commutative algebra , a universal enveloping commutative RB-algebra U of C is a universal algebra in the class of all commutative RB-algebras of zero weight such that there exists an injective homomorphism from C to . Analogously, we define a universal enveloping commutative RB-algebra of a post-commutative algebra. The common denotation of a universal enveloping algebra is the following: .
- Any pre--algebra can be embedded into its universal enveloping RB-algebra of the variety and zero weight.
- Any post--algebra can be embedded into its universal enveloping RB-algebra of the variety and nonzero weight.
Following Theorem 1, we have the natural question: what does a linear basis of a universal enveloping RB-algebra of a pre- or post--algebra look like for a variety ? In the case of associative pre- and post-algebras, the question appears in  and was solved in . The current article is devoted to answering the question in the commutative case.
We apply the following method to construct a universal enveloping RB-algebra. Let X be a linear basis of a pre-commutative algebra C. We define a basis of the universal enveloping as a certain subset E of the standard basis of , closed under the action of a RB-operator. By induction, we define a commutative product ∗ on the linear span of E and prove its associativity. Finally, we prove the universal property of the algebra .
In the case of post-commutative algebras, as was mentioned above, we consider a universal enveloping commutative RB-algebra of weight 1.
3. Universal Enveloping Rota–Baxter Algebra of Pre-Commutative Algebra
Within this section, we construct a universal enveloping RB-algebra of an arbitrary pre-commutative algebra .
We let B be a linear basis of C. We consider the algebra , where I is the ideal in generated by the set . Here the expressions and , , equal the results of the products in C. As B is the linear basis of C, the expressions are linear combinations of elements of B. We denote by · the product in A.
We denote by the set of all monomials in . As a result of Gröbner theory, there exists a set such that its image is a linear basis of A; moreover, for any decomposition of an element into a concatenation of nonempty , we have . Indeed, the Buchberger Theorem  (Theorem 4) states that there exists such a set that , , is a linear basis of the quotient algebra and is closed under the taking of subwords. Roughly speaking, is chosen as the subset of of all monomials that are not divided by some “forbidden” monomials (for the last form, the set is called the Gröbner basis of I). If a monomial satisfies this property, then all its nonempty subwords also satisfy the property.
An element of the standard basis is called good if v is neither of the view nor for all and some x. For example, if the elements , , , and for some and are the elements of the standard basis, then all of them are good. The answer to the question of whether the elements listed above lie in the standard basis depends on the choice of and on the product in the pre-commutative algebra C. The product directly influences the set ; thus we cannot yet say whether , , or lie in .
We define by induction -words (E-words), a subset of the standard basis of :
- elements of are E-words of type 1;
- given an E-word u, we define as an E-word of type 2;
- given an E-word x, we define for as an E-word of type 3 if is good.
Given an E-word x and , the element of the standard basis is an E-word if
- is good (otherwise is not an E-word);
- the length of is greater than 1 (otherwise is not a good E-word).
The set of all E-words forms a linear basis of universal enveloping commutative RB-algebras of C.
Let D denote a linear span of all E-words. One can define such a bilinear commutative operation as ∗ on the space D that has the following properties (labels k–l below denote the types of factors in the product ; i.e., v is an E-word of type k and u is an E-word of type l):
1–1: Given , we have
1–2: Given , , and , an E-word of type 2, we have(If , by , we mean .)
1–3: Given , an E-word of type 3, , we have
2–2: Given E-words and of type 2, we have
2–3: Given an E-word of type 2 and E-word , , , , of type 3, we have
3–3: Given E-words , and , , of type 3, we have
We define the operation ∗ with the prescribed conditions for E-words by induction on . For , we define , , which satisfies the condition 1–1.
For , we define as follows:
1–2: Given , , an E-word of type 2, , we have
1–3: Given , an E-word of type 3, , , ,
for , we define as follows:
1–2: Given , , , an E-word of type 2, we have
It is correct to write . Indeed, as is an E-word, is good. We have three variants: , , , or , . In all cases, is a sum of E-words of the form different from for .
We prove that the definition of the product by Equation (13) is correct. By the reasons stated above, is a sum of good E-words of type 2. Thus, it is remains to check that is a sum of good E-words of type 2, provided that and also are. In fact, it is enough to verify that is a sum of good E-words of type 2. Considering three variants of s, we state the last fact.
We set the product in the cases 2–1, 3–1, and 3–2 equal to . ☐
The space D with the operations ∗ and R is a RB-algebra.
This follows from Equation (12). ☐
The relation holds in D for every .
This follows from Lemma 1, the first case of Equation (10). ☐
The operation ∗ on D is commutative and associative.
The commutativity follows from Lemma 1.
Given E-words , we prove associativity:by inductions on two parameters: first, on the summary R-degree r of the triple , and second, on the summary degree d of .
For , we have , , as the product · is associative in the algebra A.
Let and suppose that associativity for all triples of E-words with the smaller total R-degree is proven. Consider .
2–1–1: Let , , , .
- Let , . Recall that for . Define a map on monomials as follows:We check that . By Equation (6) and the definition of I,Thus, the map ⊢ can be considered as a map , and it coincides on with the map . Finally,
- The case , , can be derived from (a).
- If x is good, then associativity follows from Lemma 1.
Let , . We have , as we deal with the product in A.
Let , . On the one hand,
On the other hand,
Associativity follows directly.
2–2–1: Let , , . If y is good, then associativity follows from Equations (10) and (12).
- Let , .
2–1–2: Associativity follows from case 2–2–1.
2–2–2: By induction, we have
Let . The proof for the cases 2–1–1, 1–2–1, 2–2–1, 2–1–2, and 2–2–2 is analogous to that stated above.
3–2–1: Let , , , . If , then by induction on d,
Therefore, by induction on d.
Let , . We calculate
Applying induction on d for the first summand of the RHS of Equation (21), we rewrite
Analogously, the RHS of Equation (20) equals
Finally, we have
If is good, then
3–1–2: The proof is analogous to that in case 2–1–2.
2–3–1: Associativity follows from cases 2–2–1 and 2–1–1.
3–3–2: Let , , and . If z is not good, then the proof follows from the procedure in cases 1–1–2 and 2–2–2. Supposing z is good, on the one hand,
On the other hand, applying induction on d, we derive
Thus, associativity follows from case 2–2–2.
3–2–2, 3–2–3 and 2–3–2: Associativity follows by the same inductive reasons as in case 3–3–2.
Proof of Theorem 2.
We prove that the algebra is the universal enveloping algebra of the pre-commutative algebra C; that is, it is isomorphic to the algebra
By construction, the algebra D is generated by B. Thus, D is a homomorphic image of .
As the equalityholds on , the RB-algebra can be considered as a homomorphic image of . It is well-known  that algebras and coincide. Thus,where J is the RB-ideal in generated by the set , and as above, , , and . Thus, is a homomorphic image of .
The basis S of  can be constructed by induction: first, ; second, if , then for , .
We define as a homomorphism from to mapping b to for all . We identify the algebra D with its image under .
We show by Equation (8) that the complement of the set of all E-words in the standard basis of is linearly expressed by E-words. Consequently, is a homomorphic image of D, and thus and D are isomorphic. Applying the inductions on the R-degree and the degree of basis words in , the relationswe prove that the elements of are linearly expressed by E-words. ☐
Given a pre-commutative algebra C, denotes a linear span of all E-words of zero R-degree in .
Let an algebra A be a direct sum of n copies of the field F. Consider A as a pre-commutative algebra under the operations and . From , we conclude that .
Let B be an n-dimensional vector space over the field F with trivial operations . Thus, .
The pair of varieties is not a PBW-pair.
The universal enveloping commutative RB-algebra of finite-dimensional pre-commutative algebras A and B (Examples 8 and 9) of the same dimension are not isomorphic; otherwise the spaces and are isomorphic as vector spaces. However, we have
Thus, the structure of universal enveloping commutative RB-algebra of a pre-commutative algebra C depends on the operation in C. ☐
4. Universal Enveloping Rota–Baxter Algebra of Post-Commutative Algebra
In this section, we construct a universal enveloping RB-algebra of an arbitrary post-commutative algebra . We let B be a linear basis of C.
We define by induction -words (-words), a subset of the standard basis of :
- elements of B are -words of type 1;
- given a -word u, we define as a -word of type 2;
- given a -word u, we define for as a -word of type 3.
For , the elements , , , and of the standard basis of are not -words.
For , the elements , , , and are -words.
The set of all -words forms a linear basis of universal enveloping commutative RB-algebras of C.
Let D denote a linear span of all -words. One can define such a bilinear commutative operation as ∗ on the space D that has the following properties (labels k–l below denote the types of factors in the product ; i.e., v is a -word of type k and u is a -word of type l):
1–1: Given , , , we have
1–2: Given , a -word of type 2, we have
1–3: Given , a -word of type 3, , we have
2–2: Given -words , of type 2, we have
2–3: Given a -word of type 3 and -word of type 2, , we have
3–3: Given -words , , , of type 3, we have
The proof is analogous to the proof of Lemma 1. ☐
The algebra D is an enveloping commutative RB-algebra of weight 1 for C.
In general, the proof is analogous to the proof of Lemmas 2–4. ☐
The key moment is to prove an associativity. We prove that for a triple of types k–l–m. We proceed by inductions on and on . We consider only the cases for which the proofs are technically different from the proof of Lemma 4.
2–1–1: Consider the case , , for . Then by Equation (7),
Proof of Theorem 3.
The proof is analogous to the proof of Theorem 2. ☐
The pair of varieties is a PBW-pair.
It is well known  that one can define a Hopf algebra structure on a free commutative RB-algebra.
Does there exist a Hopf algebra structure on the universal enveloping commutative RB-algebra of a pre- and post-commutative algebra?
The author expresses his gratitude to the anonymous referee for his helpful comments.
Conflicts of Interest
The author declares no conflict of interest.
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