From Rota-Baxter Algebras to Pre-Lie Algebras
|
|
- Bridget Dorsey
- 5 years ago
- Views:
Transcription
1 From Rota-Baxter lgebras to Pre-Lie lgebras Huihui n a Chengming Bai b a. Department of Mathematics & LPMC, Nankai University, Tianjin 37, P.R. China arxiv:7.39v [math-ph] 9 Nov 27 b. Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 37, P.R. China bstract Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-lie algebras from associative algebras. In this paper, we give all Rota-Baxter operators of weight on complex associative algebras in dimension 3 and their corresponding pre-lie algebras. Key Words: Rota-Baxter algebras, Lie algebras, pre-lie algebras Mathematics Subject Classification (2): 7B, R address: anhhmail.nankai.edu.cn
2 Introduction Rota-Baxter algebraisanassociative algebraover afieldfwithalinearoperatorr : satisfying the Rota-Baxter relation: R(x)R(y)+λR(xy) = R(R(x)y +xr(y)), x,y. (.) Here λ F is a fixed element which is called the weight. Obviously that for any λ, R λ R can reduce the Rota-Baxter operator R of weight λ to be of weight λ =. Rota-Baxter relation (.) first occurred in the work of G. Baxter in 96 to solve an analytic problem ([Bax]), based on a paper written by F. Spitzer ([Sp]) in 956. In fact, the Rota-Baxter relation (.) generalizes the integration-by-parts formula. G.-C. Rota ([R-R4]), F.V. tkinson ([t]) and P. Cartier ([Ca]) contributed important results. In particular, it was G.-C. Rota who realized its importance in combinatorics and other fields in mathematics ([R-R2]). Since then, it has been related to many topics in mathematics and mathematical physics. For example, Rota-Baxter algebras appeared in connection with the work of. Connes and D. Kreimer on renormalization theory in perturbative quantum field theory ([CK2-3]), see[fg] for more details. It is also related to J.-L. Loday s dendriform algebras ([Lo], [LR]), as well as to M. guiar s associative analogue of the classical Yang-Baxter equation ([g-3]). However, it is difficult to construct examples of Rota-Baxter algebras. Basically there are two ways to construct Rota-Baxter algebras. One way is to use the free Rota-Baxter algebras which in some senseare the biggest examples. Thereare a lot of references on the study of free Rota-Baxter algebras ([Ca], [R], [EG2], [GK-2] and the references therein). The other way is to get concrete examples in low dimensions, which is the main content of this paper. lthough there has already existed certain works on (finite-dimensional) Rota-Baxter algebras, e.g. [Deb], [Der], [Mi-2], [N], to our knowledge, there has been no classification in low dimensions yet. We will give all Rota-Baxter algebras in dimension 3. Though our study depends on direct computation through example one by one, these examples will be regarded as a guide for further development. n application of Rota-Baxter (associative) algebras is to get some new algebraic structures. We mainly mention two classes of algebraic structures related to Rota-Baxter algebras in this paper. One class of algebras are the Loday s dendriform algebras ([Lo], [LR]). Dendriform algebras are equipped with an associative product which can be written as a linear combination of nonassociative compositions. These notions are motivated by the natural link between asso- 2
3 ciative algebras and Lie algebras. By the work of M. guiar, P. Leroux and K. Ebrahimi-Fard ([g], [E-2], [Le-2]) the close relation of these new types of algebras to Rota-Baxter algebras as well as Nijenhuis algebras and differential algebras was established. The other class of algebras are the pre-lie algebras (or have other names such as leftsymmetric algebras, Vinberg algebras and so on). Pre-Lie algebras are a class of nonassociative algebras coming from the study of convex homogeneous cones, affine manifolds and deformations of associative algebras ([u], [G], [Ki], [Me], [V]). s it was pointed out in [CL], the pre-lie algebra deserves more attention than it has been given. It has also appeared in many fields in mathematics and mathematical physics, such as complex and symplectic structures on Lie groups and Lie algebras ([S], [Ch], [Sh]), integrable systems ([SS]), classical and quantum Yang-Baxter equations ([Bo], [ES], [GS], [Ku-2]), Poisson brackets and infinite-dimensional Lie algebras ([BN], [GD], [Z]), vertex algebras ([BK]), quantum field theory ([CK]) and operads ([CL]). In particular, an important role has been played by pre-lie algebras in mathematical physics, especially the work of Connes-Kreimer on pre-lie algebra structure on Feynman diagrams by the insertion-elimination operations (see [CK4] for a detailed interpretation). The same can be said of Rota-Baxter algebras. The connection of these two roles is still not clear, which might be clarified by careful study on the relation between Rota-Baxter algebras and pre-lie algebras, as we try to do in this paper. Since there is no suitable (matrix) representation theory of pre-lie algebras due to their nonassociativity, it is natural to consider how to construct them from some algebraic structures which we have known. This is the realization theory. We have already obtained some experience. For example, a commutative associative algebra (, ) and its derivation D can define a Novikov algebra (, ) (which is a pre-lie algebra with commutative right multiplication operators) by ([GD], [BM-2]): x y = x Dy, x,y. (.2) n analogue of the above construction in the version of Lie algebras is related to the classical Yang-Baxter equation. In fact, a Lie algebra (G,[, ]) and a linear map R : G G satisfying [R(x),R(y)] = R([R(x),y]+[x,R(y)]), x, y G (.3) can define a pre-lie algebra (G, ) by ([BM3], [GS], [Ku3], [Me]) x y = [R(x),y], x, y G. (.4) 3
4 Equation (.3) is just the operator form of classical Yang-Baxter equation on a Lie algebra which was given by M.. Semenov-Tyan-Shanskii in [Se]. Obviously it also can be regarded as a Rota-Baxter operator of weight zero on the Lie algebra G. In fact, as it was mentioned in [EGK] and [EG2], the Rota-Baxter relation (.) on associative algebras can be naturally extended to be on Lie algebras. It is natural to consider the construction of pre-lie algebras from (noncommutative) associative algebras. The answer is the construction from Rota-Baxter algebras. Let (, ) be an associative algebra and R be a Rota-Baxter operator. If the weight λ =, then from equations (.3) and (.4), it is obvious that the product x y = R(x) y y R(x), x,y (.5) defines a pre-lie algebra. When the weight λ =, we can see that the product x y = R(x) y y R(x) x y, x,y (.6) defines a pre-lie algebra (see Corollary 2.7). In fact, there are two approaches to both equations (.5) and (.6). One approach is from the relation between pre-lie algebras and the operator form of the (modified) classical Yang-Baxter equation given by I.Z. Golubchik and V.V. Sokolov in [GS]. The other approach is from the relation between dendriform dialgebras and Rota-Baxter algebras and pre-lie algebras given by M. guiar and K. Ebrahimi-Fard ([g], [E-E2]). It is also natural to consider which kind of pre-lie algebras can be obtained from Rota-Baxter algebras. Note that for a commutative associative algebra, the inverse of an invertible derivation is just a Rota-Baxter operator of weight zero. So we would like to point out that in the above three algebraic constructions (commutative associative algebras, Lie algebras and associative algebras) of pre-lie algebras, the corresponding linear transformations (derivations, operators satisfying classical Yang-Baxter equation and Rota-Baxter operators) have more or less relations to Rota-Baxter operators. We have given a detailed study of Rota-Baxter operators on pre-lie algebras of weight zero in [LHB]. more remarkable property is that for any such Rota-Baxter pre-lie algebra, equation (.5) can also define a pre-lie algebra which is called the double of the former ([LHB]). Therefore, any pre-lie algebra with its Rota-Baxter operator (of weight zero) and its doubles can construct a close category. We would like to point out that there is another different double 4
5 construction of Rota-Baxter algebras defined by Ebrahimi-Fard in [EGK], that is, for any Rota- Baxter algebra (,R), there is a new Rota-Baxter algebra ( R,R) which is called the double of (,R) in [EGK], where the product in R is given by x R y = R(x)y +xr(y) xy, a,b. (.7) Moreover, all Rota-Baxter operators of weight zero on associative algebras in dimension 3 were given in [LHB] too. In this paper, we study the Rota-Baxter operators of weight λ = on associative algebras. It is easy to see that this Rota-Baxter operator is still a Rota-Baxter operator on the induced pre-lie algebra given by equation (.6) ([EGP]). The paper is organized as follows. In section 2, we give some fundamental results and examples on Rota-Baxter algebras and pre-lie algebras. In section 3, we give all Rota-Baxter algebras on 2-dimensional complex pre-lie algebras, and in the associative cases, we give their corresponding pre-lie algebras. In section 4, we give all Rota-Baxter algebras on 3-dimensional complex associative algebras and their corresponding pre-lie algebras. In section 5, we give some discussion and conclusions. Throughout this paper, the Rota-Baxter operator is of weight λ = and all algebras are of finite dimension and over the complex field C, unless otherwise stated. stands for an associative algebra with a basis and nonzero products at each side of. 2 Preliminaries and some examples Let be an associative algebra. For any x,y, the commutator [x,y] = xy yx defines a Lie algebra. We denote the set of all Rota-Baxter operators on of weight λ = by RB(). Then the following conclusion is obvious (cf. [E], [EGP], [EG], etc.). Lemma 2. Let (, ) be an associative algebra. () linear operator R RB() if and only if R RB(), where is the identity map. In particular,, RB(). (2) Let (, ) be an algebra given by x y = R(x) y +x R(y) x y, x,y. (2.) Then (, ) is an associative algebra and R is still a Rota-Baxter operator of weight on (, ). (3) If R RB(), then B = 2R satisfies [B(x),B(y)]+[x,y] = B([B(x),y]+[x,B(y)]), x,y. (2.2) 5
6 (4) Let denote the algebra defined by a product (x,y) x y on which satisfies x y = y x for any x,y, then is still an associative algebra and RB() = RB( ). (5) If R RB() and R 2 = R, then for any α F, N α = (+α)r α satisfies the following Nijenhuis relation ([CGM], [Le-2]) N α (x)n α (y)+n 2 α (xy) = N α(n α (x)y +xn α (y)), x,y. (2.3) Remark 2.2 In [Se], equation (2.) is called the operator form of the modified classical Yang-Baxter equation on a Lie algebra. In general, it is not easy to obtain RB() for an arbitrary associative algebra. We give some examples in certain special cases as follows. Example 2.3 Let be a commutative associative algebra which is the direct sum of fields. That is, there is a basis {e,,e n } of satisfying e i e j = δ ij e j. Then by Rota-Baxter relation (.), R = n r ik e k RB() if and only if k= r lk r kl =, l k, and r ii =,r il = or,l i; or r ii =,r il = or,l i. In particular, a special case was given in [E] as (for any s n) R(e i ) = s e l, i s; R(e s+ ) =, R(e i ) = l=i i l=s+ e l, s+2 i n, that is, r ii =, r ij =, r ji =, i < j s,; r kk =, r kl =, r lk =,s+ l < k n. and r mn = in the other cases. We also list RB() for n 3 in the next two sections. Example 2.4 Let be an associative algebra in dimension n 2 satisfying the condition that for any two vectors x,y, the product x y is still in the subspace spanned by x,y. From [Bai], for any fixed n 2, there are three kinds of such (non-isomorphic) algebras. Let {e,,e n } be a basis of, then must be isomorphic to one of the following three algebras: (I) e i e j =, i,j =,,n; (II) e e i = e i,e j e i =, i =,,n, j = 2,,n (III) e i e = e i,e i e j =, i =,,n, j = 2,,n 6
7 It is obvious that RB(I) = gl(n) (all n n matrices). Notice that type (III) is just type (II) given in Lemma 2.. Hence RB(II) = RB(III). Moreover, we can prove that any operator R RB(II) if and only if R 2 = R. In fact, let R(e i ) = n r ik e k, then by the Rota-Baxter relation (.), we only need to check the following k= equations (other equations hold naturally): R(e )R(e i )+R(e i ) = R(e R(e i )+R(e )e i ), i =,,n. For any i, the left hand side is r R(e i ) + R(e i ) and the right hand side is R 2 (e i ) + r R(e i ). Therefore, R RB(II) if and only if R 2 = R. Furthermore, by conclusion (5) in Lemma 2., we know that any Rota-Baxter operator R on the pre-lie algebra of type (II) or type (III) can induce an operator N α = ( + α)r α satisfying the Nijenhuis relation (2.3) for any α C. On the other hand, Definition 2.5 Let be a vector space over a filed F with a bilinear product (x,y) xy. is called a pre-lie algebra if for any x,y,z, (xy)z x(yz) = (yx)z y(xz). (2.4) It is obvious that all associative algebras are pre-lie algebras. For a pre-lie algebra, the commutator [x,y] = xy yx, (2.5) defines a Lie algebra G = G(), which is called the sub-adjacent Lie algebra of. Proposition 2.6 ([GS]) Let(, ) beanassociative algebra. Ifalinear operator R : satisfies the modified Yang-Baxter equation (2.2), then the new product on given by x y = x y +y x+[r(x),y], x,y (2.6) defines a pre-lie algebra. By Proposition 2.6 and the conclusion (3) in Lemma 2., we can get the following conclusion. Corollary 2.7 Let be an associative algebra and R be a Rota-Baxter operator of weight. Then the product give by equation (.6), that is, x y = R(x) y y R(x) x y, x,y (2.7) defines a pre-lie algebra. 7
8 Definition 2. ([Lo]) Let be a vector space over a filed F with two bilinear products denoted by and. (,, ) is called a dendriform dialgebra if for any x,y,z, (x y) z = x (y z), (x y) z = x (y z), x (y z) = (x y) z, (2.) where x y = x y +x y. Proposition 2.9 ([g],[lo]) Let (,, ) be a dendriform dialgebra. Then the product given by x y = x y +x y, x,y, (2.9) defines an associative algebra ([Lo]) and the product given by x y = x y y x, x,y, (2.) defines a pre-lie algebra ([g]). (, ) and (, ) have the same sub-adjacent Lie algebra. Therefore, Corollary 2.7 (and equation (.5) and conclusion (2) in Lemma 2.) can also be obtained from the following conclusion (by a normalization of constant if necessary). Proposition 2.([g], [E]) Let(, ) be an associative algebra and R be a Rota-Baxter operator of weight λ, then there is a dendriform dialgebra (,, ) defined by x y = x R(y) λx y, x y = R(x) y, x,y. (2.) It is obvious that for a commutative associative algebra (, ) and any R RB(), the pre-lie algebra (, ) given by equation (2.7) is still (, ) itself. It is also obvious that for an associative algebra (, ), the pre-lie algebra (, ) given by equation (2.7) when R = is just (, ) itself and when R = is (, ) given in Lemma 2.. Moreover, we can get a more general conclusion: Let (, ) be an associative algebra and R RB(). Let (, ) be the associative algebra given in Lemma 2.. Then the pre-lie algebra given by equation (2.7) through (, R) is just the one given by equation (2.7) through (, R). Example 2. Let (, ) be the associative algebra of type (II) given in Example 2.4. Then the pre-lie algebra (, ) given by equation (2.7) satisfies n e e = e r k e k, n e e j = (r )e j, e j e = r jk e k, e j e l = r j e l, j,l = 2,,n, k=2 k=2
9 wherer(e i ) = n r ik e k andr 2 = R. Itisinterestingthatforn = 2,3, theabovepre-liealgebras k= are associative (see the next two sections). However, it is not easy to get their classification in higher dimensions and we have not known whether they are still associative. Corollary 2.2 ([EGP]) Let (, ) be an associative algebra and R RB(), then R is still a Rota-Baxter operator of weight λ = on the pre-lie algebra (, ) given by equation (2.7). Example 2.3 Let be the 2-dimensional associative algebra of type (II) in Example 2.4, then it is easy to see that the operator R given by R(e ) = e,r(e 2 ) = ae (for any a ) is a Rota-Baxter operator of (also see the next section). The pre-lie algebra obtained by equation (2.7) is given by e e = e, e e 2 = e 2 e =, e 2 e 2 = ae 2. It is a commutative associative algebra which is isomorphic to a simple form < e,e 2 e e = e,e 2 e 2 = e 2 > (it is just the algebra given in Example 2.3 in the case n = 2) by a linear transformation e e, e 2 a e 2. Note that R is a Rota-Baxter operator of (, ) under the same basis {e,e 2 } and the form R does not satisfy the conditions given in Example 2.3. In fact, under the new basis {e,e 2 }, R corresponds to the new form R given by R (e ) = e,r (e 2 ) = e which is consistent with the conclusion in Example 2.3. This is an example that the matrix presentations of Rota-Baxter operators depend on the choice of the bases. Moreover, there is a related discussion in section 5. 3 Rota-Baxter operators on 2-dimensional associative algebras and pre-lie algebras Let (, ) be an associative algebra or a pre-lie algebra and {e,e 2,,e n } be a basis of. Let R be a Rota-Baxter operator of weight on. Set n R(e i ) = r ij e j, e i e j = j= n Cije k k. (3.) Then r ij satisfies the following equations: n (Ckl m r ikr jl +Cij k r km Ckj l r ikr lm Cil k r jlr km ) =, i,j =,2,,n. (3.2) k,l,m= We know that there are two -dimensional associative algebras (D) =< e e e = > and (D) =< e e e = e >. It is easy to see that RB(D) = C and RB(D) = {R R(e ) = or R(e ) = e }. 9 k=i
10 We have known the classification of 2-dimensional complex pre-lie algebras ([Bu]), which includes the classification of 2-dimensional complex associative algebras. The following results can be obtained by direct computation. Proposition 3. The Rota-Baxter operators on 2-dimensional commutative associative algebras are given in the following table (any parameter belongs to the complex field C, unless otherwise stated). ssociative algebra Rota-Baxter «operators «RB() ««() e e = e,e 2e 2 = e 2,,,, ««««,,,, ««««,,, ««««(2) e 2e 2 = e 2,e e 2 = e 2e = e,,, ««(3) e e = e, r 22 «r 22 r r 2 (4) e ie j = r 2 r 22 «,r 2 r r 2 (5) e e = e 2 r 2 2r There are two non-commutative associative algebras in dimension 2 (B) =< e e 2 e = e,e 2 e 2 = e 2 > and (B2) =< e e e 2 = e,e 2 e 2 = e 2 >. Both of them belong to the algebras given in Example 2.4 in the case n = 2, so any Rota-Baxter operator R satisfies R 2 = R. Furthermore, we can know that (since many of their corresponding pre-lie algebras are isomorphic under a basis transformation, we give a classification of these pre-lie algebras in the sense of isomorphism, that is, the corresponding pre-lie algebras are isomorphic to some pre-lie algebras with simpler presentations) RB(B) = { ( ( ( r 2 ( r 2 ) = (B) ) = (B2) ) = (2) ) = (3) ( r r 2 r 2 r ) r, 2 r 2 r +r 2 r 2 = = ()} We also have RB(B2) = RB(B) and the corresponding pre-lie algebras are given by the conclusion before Example 2..
11 Corollary 3.2 ny 2-dimensional pre-lie algebra obtained by equation (2.7) from a Rota- Baxter (associative) algebra is associative. Corollary 3.3 Only the non-nilpotent commutative associative algebras (they are (), (2), (3)) can be obtained from 2-dimensional non-commutative associative Rota-Baxter algebras by equation (2.7). t the end of this section, we give the following conclusion by direct computation. Proposition 3.4 The Rota-Baxter operators on 2-dimensional (nonassociative) pre-lie algebras are given in the following table. Pre-Lie algebra Rota-Baxter «operators «RB() (B3) e 2e = e,e 2e 2 = e e 2 «(B4) e 2e = e,e 2e 2 = ke 2,k k : «««««r 2 r 22 r 2 k = : ««r 2 r 22 «r 2 «(B5) e e 2 = le,e 2e = (l )e,e 2e 2 = e +le 2,l l = : ««l : l ««(B6) e e = 2e,e e 2 = e 2,e 2e 2 = e ««l Since (B6) is the unique simple pre-lie algebra (without any ideals besides zero and itself) in dimension 2 ([Bu]), we have Corollary 3.5 There is no non-trivial Rota-Baxter operator on the 2-dimensional simple pre-lie algebra, that is, only, are the Rota-Baxter operators. 4 Rota-Baxter operators on 3-dimensional associative algebras and their corresponding pre-lie algebras It is easy to get the classification of 3-dimensional complex associative algebras (for example, see [LHB]). Then by direct computation, we have the following results. Proposition 4. The Rota-Baxter operators on 3-dimensional commutative associative algebras are given in the following table.
12 ssociative algebra Rota-Baxter operators RB() r r 2 r 3 (C) e ie j = r 2 r 22 r 23 r 3 r 32 r 33 r (C2) e 3e 3 = e r 2 r 22 r 3 r 32 r ± p r 2 r j r e2e 2 = e (C3) r 2 r r e 3e 3 = e =, r 3 r r r r 2 r 22 r 23, 23, r 22 = r ± p r r 3 r 23 r 2 r r r r 23, r 2 r 22 r 23 r + r 3 r 23 r + 22 = r + p r 2 r r r 22 = r p r 2 r r2 23 r r 23, r 2 r + 22 r 23 r + r 3 r 23 r 22 = r + p r 2 r r r 22 = r p r 2 r r2 23 j r 22 r 33 e2e 3 = e 3e 2 = e r 22 +r 33 r (C4) 33 = r 22 ± p r22 r e 3e 3 = e 2 r 22, 2 r22 2 r 2 = 2r 32r 33 ( r 33 ) r 3 r 32 r ( 2r )(r 22 +r 33 ) < e e = e r r (C5) e 2e 2 = e 2 r 22 r 22 : e 3e 3 = e 3 r 33 2r 33 2r 33 r 33 r r r 22 r 22 2r 33 r 33 2r 33 r 33 j e2e (C6) 2 = e 2 e 3e 3 = e 3 r 2r 22 r 22 r 33 r 2r r 22 r 33 r 2r 22 r 22 2r 33 2r 33 r 33 r 2r r 22 2r 33 2r 33 r 33 r 2r 2r r 2r 2r r 22 r 22 r 33 2r 33 r 33 r r 2r 22 r 22 2r 22 2r 22 r 22 2r 22 r 33 2r 33 r 33 r r 2r 2r r 22 2r 22 r 22 2r 22 r 33 r 33 r 2r r 2r r 22 2r 22 r 22 2r 22 r 33 r 33 r =,,r 22 =,,r 33 =, r r 22,r 22 =,,r 32 =, r 32 r r 22 r 32 r r 22 2r 22 r 33,r 22 =,,r 32 =,,r 22 =,,r 33 =, 2
13 ssociative algebra Rota-Baxter operators RB() < e e 3 = e 3e = e r (C7) e 2e 2 = e 2 r 22,r =,,r 22 =,,r 32 =, : e 3e 3 = e 3 r 32 r r 22,r =,,r 22 =,,r 32 =, r 32 r r =,,r 33 =, r 33 r,r =,,r 33 =, r 33 r r 2 (C) e 3e 3 = e 3 r 2 r 22,r 33 =, r 33 j r r 2 ee 3 = e 3e = e (C9) r 22 r e 3e 3 = e 2,r =, 3 r r r 22,r =,,r 33 =, r 33 r r 2 r 22,r =,,r 2 r < e e 3 = e 3e = e r r 2 r 2 (C) e 2e 3 = e 3e 2 = e 2 r 2 r, r 33 =, : e 3e 3 = e 3 r 33 r r 2 r 2r 2 = r r,r 33 =,,r =, r 33 r 2,r 33 =, r 33 r 2,r 33 =, r 33 j ee = e 2 (C) r22 ± p r22 2 r22 r2 r e 3e 3 = e 22,r 33 =, 3 r 33 e e = e 2 >< r e e 3 = e 3e = e (C2) r,r e 2e 3 = e 3e 2 = e 33 =,,r =, 2 >: r 33 e 3e 3 = e 3 Proposition 4.2 The Rota-Baxter operators on 3-dimensional non-commutative associative algebras and their corresponding pre-lie algebras given by equation (2.7) (in the sense of isomorphism) are given in the following table. ssociative algebra Rota-Baxter operators RB() Pre-Lie algebra j e e (T) 2 = r r 2 r 3 2 e3 e 2 e = r 2 r r 23, r 2 e3 2 r +r 2r 2 = (T)(C3) r 33 r r 2 r 3 (T),(T2), r 2 r 22 r 23, r +r 22 (T3) λ,λ r r 22 r 2 r 2 r +r 22 3
14 ssociative algebra Rota-Baxter operators RB() Pre-Lie algebra r 3 (T2) e 2 e = e 3 r 23 (C) r 33 r 3 r 23 (T) r 33 r r 3 r 22 r 23 r +r 22 (T2), (T3) λ,λ r r 22 r +r 22 r 3 r 2 r 3 r 2 r 23 (r 2 ), r 23 (r 2 ) (C2) < (T3) λ : e e = e 3 e e 2 = e 3 e 2 e 2 = λe 3 λ j e3 e 2 = e 2 (T4) e 3 e 3 = e 3 r 3 r 2 r 23 (r 2 ), r 2 r 3 r 23 (r 2 ) (C3) r 2 r 3 r 22 r 23 r 2,r 22 (T2) r 2 r 3 r 22 r 23 r 2,r 22 (T2), (T3) λ,λ r r 3 r 2 r 23 r,r 2 (T2), (T3) λ,λ r r 3 r 2 r 23 r,r 2 (T2) r r 2 r 3 r 2 r 22 r 23 r 3 r 32 r 33 (T3) λ, λ r 2 +r (r 2 2r 33 )+(λr2 2 +r 33 r 2 r 33 ) = ; r r 2 +r r 22 +λr 2 r 22 +r 33 ( r λr 2 r 2 r 22 ) = ; r 2 r +r 2 r 2 +λr 22 r 2 r 2 r 33 λr 2 r 33 = ; r 2 2 +r 2(r 22 r 33 )+(λr22 2 +λr 33 2λr 22 r 33 ) = r r 2 r 2 r r 2 r 2 r r r 2 r 2r 32 r 32 r r 2 r 2r 32 r 32 r (T5) e e 3 = e 2 >< e (N) 3 e = e 2 e 3 e 2 = e 2 >: e 3 e 3 = e 3 (T4) (C9) (C) (T5) 4
15 ssociative algebra Rota-Baxter operators RB() r r 22 r 23, r 32 r 22 r 23 r 2 22 r 22 +r 23r 32 = Pre-Lie algebra j e2 e 3 = e 2 (T5) e 3 e 3 = e 3 RB(T4) The same as in (T4) (=(T4) ) through Corollary 2.2 < e e = e (T6) e 3 e 2 = e 2 : e 3 e 3 = e 3 r,r =,,r 3 =, r 3 r,r =,,r 3 =, r 3 r,r =,,r 3 =, r 3 r 32 r,r =,,r 3 =, r 3 r 32 r r 22 r 23 r 32 r 22 r r 23 r 22 r 23 r 22 r 32 r 22 r r 23 r 22 r 23 r 22 r 32 r 22 r 2 r 2 r 2 r 2 r 2 r 2 r 2 r 2 r 23, r =, r22 2 r 22 +r 23r 32 = r 23, r =, r22 2 r 22 +r 23r 32 =, r 23 r =, r 2 22 r 22 +r 23r 32 = (C6) (T6) (T7) (C7) (C6) (C5) (C5) (C5) e e = e +2e 3 >< e e 3 = e 3 (N2) e 3 e = e 3 >: e 3 e 2 = e 2 e 3e 3 = e 3 e e = e >< e e 3 = e 3 (N3) e 3 e = e 3 >: e 3 e 2 = e 2 e 3e 3 = e 3 (T9) e e = e e e 2 = e 2 >< e 2 e = e 2 (N4) e e 3 = e 3 e 3 e = e 3 >: e 3 e 2 = e 2 e 3e 3 = e 3 e e = e >< e e 2 = e 2 (N5) e 2 e = e 2 >: e 3 e 2 = e 2 e 3e 3 = e 3 e e = e >< e e 2 = e 2 (N6) e 2 e = e 2 >: e 3 e 2 = e 2 e 3e 3 = e 3 5
16 ssociative algebra Rota-Baxter operators RB() Pre-Lie algebra r 2 (T5) r 2 r 2 < e e = e (N7) e 3 e 2 = e 2 : r 2 e 3e 3 = e 3 < e e = e (T7) e 2 e 3 = e 2 RB(T6) The same as in (T6) : e 3 e 3 = e 3 (=(T6) ) through Corollary 2.2 e e 3 = e >< r e (T) 3 e = e,r =, (T) e 3 e 2 = e 2 >: e 3 e 3 = e 3 e r e 3 = e >< e 3 e = e,r =, (N) e 2 e 3 = e 2 >: e 3e 3 = e 3 r,r =, (C) r 32 r,r =, (C9) r 32 r r 23 r 22 r 23, r =, (C7) r 32 r 22 r22 2 r 22 +r 23r 32 = r 2,r 2 (C) r 2r 32 r 32 r 2,r 2 (C9) r 2r 32 r 32 e e = e e e 2 = e 2 >< e e 3 = e 3 (T9) e 2 e = e 2 e 3 e = e 3 >: e 3 e 2 = e 2 e 3 e 3 = e 3 r 2 r 2 r 2,r 2, r 32,r 3 =, r 3,r 3 =, r 3 r 32 >< (N9) >: (T) (T9) (C7) (T9) (C7) e e 3 = e +e 2 e 2 e 3 = e 2 e 3 e = e +e 2 e 3 e 3 = e 3 6
17 ssociative algebra Rota-Baxter operators RB(),r 3 =, r 3 r 32 r 32,r 3 =, r 3 r r 22 r 23, r 32 r 22 r 23 r 22 r 23 r 22 r 32 r 22 r 23 r 22 r 23 r 22 r 32 r 22 r r 22 r r 22 2 r 22 r 22, r 23 r =, r22 2 r 22 +r 23r 32 =, r 23 r 2 22 r 22 +r 23r 32 = r 23 r 2 22 r 22 +r 23r 32 =,r 22 =,,r =,,r 22 =,,r =,,r 22 =, r r 2 r r 2 r 2 r r 2 r 2 r r 2 2 r 2 2 r 2 r 2 r 2 r 2 r 2,r 22 =,,r =,,r 2,r =,,r 2,r =,,r 2,r =,,r 2,r 2,r 2,r 2,r 2 Pre-Lie algebra (C7) (C7) (T9) (C5) (C5) (C5) (N5) (T7) (N5) (T7) (N5) (N5) (T7) (T7) (N5) (N5) (T7) (T7) 7
18 ssociative Rota-Baxter operators RB() algebra < e 3 e = e (T) e 3 e 2 = e 2 {R R 2 = R} : e 3 e 3 = e 3 < e e 3 = e (T) e 2 e 3 = e 2 {R R 2 = R} : e 3 e 3 = e 3 < e 3 e = e (T2) e 2 e 3 = e 2, : e 3 e 3 = e 3 r 3 r 3 r 32 r 32 r 32 r 3 r 32 r 3 r 2,r 2 r 2r 32 r 32 r 2,r 2 r 2r 32 r 32 r r 22 r 23 r 32 r 22 r r 2 r 22 r 23 r 2 ( r r 22 ) r 23 r 32 r 22 r r 3 r 22 r 3 r r r 2 r 3 r 22 r r 2 ( r r 22 ) 3 r 3 r r 2,r 2 r 3 r 3r 2 r 2,r 2 r 3 r 3r 2 r 23 r =, r 2 22 r 22 +r 23r 32 =, r 23 r 2 r =, r22 2 r 22 +r 23r 32 = r 3, r 22 =, r 2 r +r 3r 3 =, r 3 r 2 r 22 =, r 2 r +r 3r 3 = Pre-Lie algebra (T6),(T),(T9), (T),(T),(C) (T6),(T),(T9), (T),(T),(C) (T2) (N) (C9) (N) (T4) (C9) (T4) (N) (N9) e e = e >< e e 3 = e 3 (N) e 3 e = e 3 >: e 3 e 2 = e 2 e 3 e 3 = e 3 (N6) (N6), (N) (N6), (N) (N6), (N) (N9) (N)
19 Corollary 4.4 The algebras of type (N)-(N) are the only nonassociative pre-lie algebras obtained from 3-dimensional Rota-Baxter algebras. Corollary 4.5 The sub-adjacent Lie algebras of the nonassociative pre-lie algebras obtained from 3-dimensional Rota-Baxter algebras are unique up to isomorphism: < e,e 2,e 3 [e 2,e 3 ] = e 2 >. It is the direct sum of the 2-dimensional non-abelian Lie algebra and -dimensional center. Corollary 4.6 Besides the algebras of type (C4), (C) and (C2), the 3-dimensional commutative associative algebras can be obtained from noncommutative associative Rota-Baxter algebras by equation (2.7). 5 Discussion and conclusions From the study in the previous sections, we give the following discussion and conclusions. () We have given all the Rota-Baxter operators of weight on complex associative algebras in dimension 3. They can help us to construct pre-lie algebras. We would like to point out that such constructions have some constraints. For example, all the pre-lie algebras obtained from 2-dimensional Rota-Baxter algebras are associative and the sub-adjacent Lie algebras of the nonassociative pre-lie algebras obtained from 3-dimensional Rota-Baxter algebras are unique up to isomorphism. (2) By conclusion (3) in Lemma 2., the Rota-Baxter operators that we obtained in this paper can help us to get the examples of operators satisfying (the operator form of) the modified classical Yang-Baxter equation in the sub-adjacent Lie algebras of these associative algebras. (3) It is hard and less practicable to extend our study to be in higher dimensions since the Rota-Baxter relation involves the nonlinear quadratic equations (3.2). Moreover, for a Rota- Baxter algebra, both the set RB() and the corresponding pre-lie algebras obtained from rely on the choice of a basis of and its corresponding structural constants (see Example 2.3). So it might be enough to search some interesting examples (not necessary to get the whole set RB()) in higher dimensions, even in infinite dimension ([E]). In this sense, our study can be a good guide (like Examples ). (4) The construction in Corollary 2.7 cannot be extended to the nonassociative pre-lie algebras, that is, we cannot obtain pre-lie algebras from a nonassociative Rota-Baxter pre-lie algebra by equation (2.7). However, if the induced pre-lie algebra (, ) = (, ) from a Rota- 9
20 Baxter (associative) algebra (,,R) is still associative, then (,,R) is still a Rota-Baxter algebra which can induce a new pre-lie algebra (, 2 ) with R being a Rota-Baxter operator (it is also a double construction, see [EGK] and [LHB]). Therefore, we can get a series of Rota- Baxter (associative) algebras (, n,r) for any n N or there exists some N N such that (, n,r) is a Rota-Baxter associative algebra for any n < N and (, N,R) is a nonassociative Rota-Baxter pre-lie algebra. (5) We have also given the Rota-Baxter operators of weight on 2-dimensional complex pre- Lie algebras. It is interesting to consider certain geometric structures related to these examples and the possible application in physics. cknowledgements The authors thank Professor Li Guo and the referees valuable suggestion. In particular, the authors are grateful of being told the close relations between dendriform dialgebras and pre-lie algebras. This work was supported in part by the National Natural Science Foundation of China (579, 62), NKBRPC (26CB595) and Program for New Century Excellent Talents in University. References [S ]. ndrada, S. Salamon, Complex product structures on Lie algebras, Forum Math. 7 (25) [g ] M. guiar, Pre-Poisson algebras, Lett. Math. Phys. 54 (2) [g2 ] M. guiar, Infinitesimal Hopf algebras, Contemporary Mathematics 267, mer. Math. Soc., (2) -29. [g3 ] M. guiar, On the associative analog of Lie bialgebras, J. lgebra 244 (2), no. 2, [t ] F.V. tkinson, Some aspects of Baxter s functional equation, J. Math. nal. ppl. 7(963) -3. [u ] uslander, L. Simply transitive groups of affine motions. mer. J. Math. 99 (977), no. 4, [Bai ] C.M. Bai, Left-symmetric algebras from linear functions, J. lgebra 2 (24), no. 2, [BM ] C.M. Bai, D.J. Meng, On the realization of transitive Novikov algebras, J. Phys. : Math. Gen. 34 (2), no. 6,
21 [BM2 ] C.M. Bai, D.J. Meng, The realizations of non-transitive Novikov algebras, J. Phys. : Math. Gen. 34 (2), no.33, [BM3 ] C.M. Bai, D.J. Meng, Lie algebraic approach to Novikov algebras, J. Geom. Phys. 45 (23), no. -2, [BK ] B. Bakalov, V. Kac, Field algebras, Int. Math. Res. Not. (23), no. 3, [BN ].. Balinskii, S.P. Novikov, Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras, Soviet Math. Dokl. 32 (95) [Bax ] G. Baxter, n analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. (96) [Bo ] M. Bordemann, Generalized Lax pairs, the modified classical Yang-Baxter equation, and affine geometry of Lie groups, Comm. Math. Phys. 35 (99), no., [Bu ] D. Burde, Simple left-symmetric algebras with solvable Lie algebra, Manuscipta Math. 95 (99), no. 3, [CGM ]J. Carinena, J. Grabowski, G. Marmo, Quantum bi-hamiltonian systems, Int. J. Modern Phys. 5 (2), no. 3, [Ca ] P. Cartier, On the structure of free Baxter algebras, dvances in Math. 9 (972) [CL ] F. Chapoton, M. Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not. (2), no., [Ch ] B.Y. Chu, Symplectic homogeneous spaces, Trans. mer. Math. Soc. 97 (974) [CK ]. Connes, D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 99 (99), no., [CK2 ]. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann- Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 2 (2), no., [CK3 ]. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann- Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group, Comm. Math. Phys. 26 (2), no., [CK4 ]. Connes, D. Kreimer, Insertion and Elimination: the doubly infinite Lie algebra of Feynman graphs, nnales Henri Poincare, 3 (22) no. 3, [Deb ] S.Luiz, de Braganca, Finite dimensional Baxter algebras, Studies in pplied Math. 54 (975), no., [Der ] N.. Derzko, Mappings satisfying Baxter s identity in the algebra of matrices, J. Math. nal. ppl. 42 (973) -9. [E ] K. Ebrahimi-Fard, Loday-type algebras and the Rota-Baxter relation, Lett. Math. Phys. 6 (22), no. 2,
22 [E2 ] K. Ebrahimi-Fard, On the associative Nijenhuis relation, Elect. J. Comb., (24), no., Research Paper 3. [EGP ] K. Ebrahimi-Fard, J.M. Gracia-Bondia, F. Patras, Rota-Baxter algebras and new combinatorial identities, arxiv:math.co/73. [EG ] K. Ebrahimi-Fard, L. Guo, Rota-Baxter algebras and dendriform algebras, arxiv:math/ [EG2 ] K. Ebrahimi-Fard, L. Guo, On free Rota-Baxter algebras, arxiv:math.r/5266. [EGK] K. Ebrahimi-Fard, L. Guo, D. Kreimer, Integrable renormalization : The ladder case, J. Math. Phys, 45 (24), No., [ES ] P. Etingof,. Soloviev, Quantization of geometric classical r-matrix, Math. Res. Lett. 6 (999), no. 2, [FG ] H. Figueroa, J.M. Gracia-Bondia, Combinatorial Hopf algebras in qunantum field theory I, Rev. Math. Phys. 7 (25), no., [GD ] I.M. Gel fand, I. Ja. Dorfman, Hamiltonian operators and algebraic structures associated with them, Funktsional. nal. i Prilozhen. 3 (979), no. 4, 3-3, 96. [G ] M. Gerstenhaber, The cohomology structure of an associative ring, nn. of Math (2). 7 (963) [GS ] I.Z. Golubchik, V.V. Sokolov, Generalized operator Yang-Baxter equations, integrable ODEs and nonassociative algebras, J. Nonlinear Math. Phys., 7 (2), no.2, [GK ] L. Guo, W. Keigher, Baxter algebras and shuffle products, dv. Math. 5 (2), no., [GK2 ] L. Guo, W. Keigher, On free Baxter algebras: completions and the internal construction, dv. Math. 5 (2), no., -27. [Ki ] H. Kim, Complete left-invariant affine structures on nilpotent Lie groups, J. Differential Geometry 24 (96), no. 3, [Ku ] B.. Kupershmidt, Non-abelian phase spaces, J. Phys. : Math. Gen. 27 (994), no., [Ku2 ] B.. Kupershmidt, On the nature of the Virasoro algebra, J. Nonlinear Math. Phy., 6 (999), no. 2, [Ku3 ] B.. Kupershmidt, What a classical r-matrix really is, J. Nonlinear Math. Phy., 6 (999), no. 4, [Le ] P. Leroux, Construction of Nijenhuis operators and dendriform trialgebras, arxiv:math.q/ [Le2 ] P. Leroux, Ennea-algebras, J. lgebra 2 (24), no.,
23 [LHB ] X.X. Li, D.P. Hou, C.M. Bai, Rota-Baxter operators on pre-lie algebras, J. Nonlinear Math. Phy., 4 (27), no. 2, [Lo ] J.-L. Loday, Dialgebras, in Dialgebras and related operads, Lecture Notes in Math. 763 (22) [LR ] J.-L. Loday, M. Ronco, Trialgebras and families of polytopes, in Homotopy Theory: Relations with lgebraic Geometry, Group Cohomology, and lgebraic K-theory, Comtep. Math. 346 (24) [Me ]. Medina, Flat left-invariant connections adapted to the automorphism structure of a Lie group, J. Differential Geometry 6 (9), no. 3, [Mi ] J.B. Miller, Some properties of Baxter operators, cta Math. cad. Sci. Hungar. 7 (966) [Mi2 ] J.B. Miller, Baxter operators and endomorphisms on Banach algebras, J. Math. nal. ppl. 25 (969) [N ] Nguyen-Huu-Bong, Some apparent connection between Baxter and averaging operators, J. Math. nal. ppl. 56 (976), no. 2, [R ] G.-C. Rota, Baxter algebras and combinatorial identities I, Bull. mer. Math. Soc. 75 (969) [R2 ] G.-C. Rota, Baxter algebras and combinatorial identities II, Bull. mer. Math. Soc. 75 (969) [R3 ] G.-C. Rota, Baxter operators, an introduction, In: Gian-Carlo Rota on Combinatorics, Introductory papers and commentaries, Joseph P.S. Kung, Editor, Birkhäuser, Boston, 995. [R4 ] G.-C. Rota, Ten mathematics problems I will never solve, Mitt. Dtsch. Math,-Ver. (99), no. 2, [Se ] M.. Semenov-Tyan-Shanskii, What is a classical R-matrix? Funct. nal. ppl. 7 (93) [Sh ] H. Shima, Homogeneous Hessian manifolds, nn. Inst. Fourier (Grenoble) 3 (9), no. 3, 9-2. [Sp ] F. Spitzer, combinatorial lemma and its application to probability theory, Trans. mer. Math. Soc. 2 (956) [SS ] S.I. Svinolupov, V.V. Sokolov, Vector-matrix generalizations of classical integrable equations, Theoret. and Math. Phys. (994), no. 2, [V ] E.B. Vinberg, The theory of homogeneous convex cones, Transl. of Moscow Math. Soc. No. 2 (963) [Z ] E.I. Zel manov, On a class of local translation invariant Lie algebras, Soviet Math. Dokl. 35 (97)
Rota-Baxter operators on pre-lie algebras
Journal of Nonlinear Mathematical Physics Volume 14, Number (007), 69 89 Article Rota-Baxter operators on pre-lie algebras Xiuxian LI a,b, Dongping HOU a and Chengming BAI c a Department of Mathematics
More informationA brief introduction to pre-lie algebras
Chern Institute of Mathematics, Nankai University Vasteras, November 16, 2016 Outline What is a pre-lie algebra? Examples of pre-lie algebras Pre-Lie algebras and classical Yang-Baxter equation Pre-Lie
More informationLoday-type Algebras and the Rota-Baxter Relation
Loday-type Algebras and the Rota-Baxter Relation arxiv:math-ph/0207043v2 26 Aug 2002 K. Ebrahimi-Fard 1 Physikalisches Institut der Universität Bonn Nußallee 12, D 53115 Bonn, Germany Abstract In this
More informationarxiv: v1 [math-ph] 5 May 2015
FERMIONIC NOVIKOV ALGEBRAS ADMITTING INVARIANT NON-DEGENERATE SYMMETRIC BILINEAR FORMS ARE NOVIKOV ALGEBRAS ZHIQI CHEN AND MING DING arxiv:155967v1 [math-ph] 5 May 215 Abstract This paper is to prove that
More informationClassical Yang-Baxter Equation and Its Extensions
(Joint work with Li Guo, Xiang Ni) Chern Institute of Mathematics, Nankai University Beijing, October 29, 2010 Outline 1 What is classical Yang-Baxter equation (CYBE)? 2 Extensions of CYBE: Lie algebras
More informationCLASSICAL R-MATRICES AND NOVIKOV ALGEBRAS
CLASSICAL R-MATRICES AND NOVIKOV ALGEBRAS DIETRICH BURDE Abstract. We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting
More informationCLASSIFICATION OF NOVIKOV ALGEBRAS
CLASSIFICATION OF NOVIKOV ALGEBRAS DIETRICH BURDE AND WILLEM DE GRAAF Abstract. We describe a method for classifying the Novikov algebras with a given associated Lie algebra. Subsequently we give the classification
More informationarxiv: v2 [math.ra] 17 Mar 2010
O-OPERTORS ON SSOCITIVE LGEBRS ND DENDRIFORM LGEBRS CHENGMING BI, LI GUO, ND XING NI arxiv:1003.2432v2 [math.r] 17 Mar 2010 bstract. We generalize the well-known construction of dendriform dialgebras and
More informationResearch Article Rota-Baxter Operators on 3-Dimensional Lie Algebras and the Classical R-Matrices
Hindawi Advances in Mathematical Physics Volume 07, Article ID 680, 7 pages https://doi.org/0.55/07/680 Research Article Rota-Baxter Operators on 3-Dimensional Lie Algebras and the Classical R-Matrices
More informationarxiv:math/ v3 [math.ra] 31 May 2007
ROTA-BAXTER ALGEBRAS AND DENDRIFORM ALGEBRAS KURUSCH EBRAHIMI-FARD AND LI GUO arxiv:math/0503647v3 [math.ra] 31 May 2007 Abstract. In this paper we study the adjoint functors between the category of Rota-
More informationPREPOISSON ALGEBRAS MARCELO AGUIAR
PREPOISSON ALGEBRAS MARCELO AGUIAR Abstract. A definition of prepoisson algebras is proposed, combining structures of prelie and zinbiel algebra on the same vector space. It is shown that a prepoisson
More informationarxiv: v2 [math.ra] 23 Oct 2015
SPLITTING OF OPERATIONS FOR ALTERNATIVE AND MALCEV STRUCTURES arxiv:1312.5710v2 [math.ra] 23 Oct 2015 SARA MADARIAGA Abstract. In this paper we define pre-malcev algebras and alternative quadrialgebras
More informationHopf algebras in renormalisation for Encyclopædia of Mathematics
Hopf algebras in renormalisation for Encyclopædia of Mathematics Dominique MANCHON 1 1 Renormalisation in physics Systems in interaction are most common in physics. When parameters (such as mass, electric
More informationOn the Associative Nijenhuis Relation
On the Associative Nijenhuis Relation Kurusch Ebrahimi-Fard Institut Henri Poincaré 11, rue Pierre et Marie Curie F-75231 Paris Cedex 05 FRANCE kurusch@ihes.fr and Universität Bonn - Physikalisches Institut
More informationarxiv: v1 [math.ra] 11 Apr 2016
arxiv:1604.02950v1 [math.ra] 11 Apr 2016 Rota-Baxter coalgebras and Rota-Baxter bialgebras Tianshui Ma and Linlin Liu 1 Department of Mathematics, School of Mathematics and Information Science, Henan Normal
More information3-Lie algebras and triangular matrices
Mathematica Aeterna, Vol. 4, 2014, no. 3, 239-244 3-Lie algebras and triangular matrices BAI Ruipu College of Mathematics and Computer Science, Hebei University, Baoding, 071002, China email: bairuipu@hbu.cn
More informationarxiv: v1 [math.ct] 2 Feb 2013
THE CATEGORY AND OPERAD OF MATCHING DIALGEBRAS YONG ZHANG, CHENGMING BAI, AND LI GUO arxiv:1302.0399v1 [math.ct] 2 Feb 2013 Abstract. This paper gives a systematic study of matching dialgebras corresponding
More informationarxiv: v1 [math.qa] 30 Dec 2018
DERIVED IDENTITIES OF DIFFERENTIAL ALGEBRAS P. S. KOLESNIKOV arxiv:1812.11516v1 [math.qa] 30 Dec 2018 Abstract. Suppose A is a not necessarily associative algebra with a derivation d. Then A may be considered
More informationON BIALGEBRAS AND HOPF ALGEBRAS OF ORIENTED GRAPHS
Confluentes Mathematici, Vol. 4, No. 1 (2012) 1240003 (10 pages) c World Scientific Publishing Company DOI: 10.1142/S1793744212400038 ON BIALGEBRAS AND HOPF ALGEBRAS OF ORIENTED GRAPHS DOMINIQUE MANCHON
More informationarxiv: v1 [math.ra] 19 Dec 2013
SPLITTING OF OPERATIONS FOR ALTERNATIVE AND MALCEV STRUCTURES arxiv:1312.5710v1 [math.ra] 19 Dec 2013 SARA MADARIAGA Abstract. In this paper we define pre-malcev algebras and alternative quadrialgebras
More informationSome constructions and applications of Rota-Baxter algebras I
Some constructions and applications of Rota-Baxter algebras I Li Guo Rutgers University at Newark joint work with Kurusch Ebrahimi-Fard and William Keigher 1. Basics of Rota-Baxter algebras Definition
More informationA Note on Poisson Symmetric Spaces
1 1 A Note on Poisson Symmetric Spaces Rui L. Fernandes Abstract We introduce the notion of a Poisson symmetric space and the associated infinitesimal object, a symmetric Lie bialgebra. They generalize
More informationBirkhoff type decompositions and the Baker Campbell Hausdorff recursion
Birkhoff type decompositions and the Baker Campbell Hausdorff recursion Kurusch Ebrahimi-Fard Institut des Hautes Études Scientifiques e-mail: kurusch@ihes.fr Vanderbilt, NCGOA May 14, 2006 D. Kreimer,
More informationarxiv: v1 [math.ra] 10 May 2013
ON FINITE-DIMENSIONAL SUBALGEBRAS OF DERIVATION ALGEBRAS arxiv:1305.2285v1 [math.ra] 10 May 2013 Abstract. Let K be a field, R be an associative and commutative K-algebra and L be a Lie algebra over K.
More informationA REMARK ON SIMPLICITY OF VERTEX ALGEBRAS AND LIE CONFORMAL ALGEBRAS
A REMARK ON SIMPLICITY OF VERTEX ALGEBRAS AND LIE CONFORMAL ALGEBRAS ALESSANDRO D ANDREA Ad Olivia, che mi ha insegnato a salutare il Sole ABSTRACT. I give a short proof of the following algebraic statement:
More informationOn Central Extensions of Associative Dialgebras
Journal of Physics: Conference Series PAPER OPEN ACCESS On Central Extensions of Associative Dialgebras To cite this article: Isamiddin S. Rakhimov 2016 J. Phys.: Conf. Ser. 697 012009 View the article
More informationLeft-symmetric algebra structures on the W -algebra W (2, 2)
Left-symmetric algebra structures on the W -algebra W (2, 2) Hongjia Chen 1 Junbo Li 2 1 School of Mathematical Sciences, University of Science Technology of China, Hefei 230026, Anhui, P. R. China hjchen@mail.ustc.edu.cn
More informationCYCLIC HOMOLOGY AND THE BEILINSON-MANIN-SCHECHTMAN CENTRAL EXTENSION. Ezra Getzler Harvard University, Cambridge MA 02138
CYCLIC HOMOLOGY AND THE BEILINSON-MANIN-SCHECHTMAN CENTRAL EXTENSION. Ezra Getzler Harvard University, Cambridge MA 02138 Abstract. We construct central extensions of the Lie algebra of differential operators
More informationPOISSON BRACKETS OF HYDRODYNAMIC TYPE, FROBENIUS ALGEBRAS AND LIE ALGEBRAS
POISSON BRACKETS OF HYDRODYNAMIC TYPE, FROBENIUS ALGEBRAS AND LIE ALGEBRAS A. A. BALINSKIĬ AND S. P. NOVIKOV 1. Poisson bracets of hydrodynamic type, (1) {u i (x), u j (y)} = g ij (u(x))δ (x y) + u xb
More informationEmbedding dendriform dialgebra into its universal enveloping Rota-Baxter algebra
Embedding dendriform dialgebra into its universal enveloping Rota-Baxter algebra arxiv:1005.2717v1 [math.ra] 16 May 2010 Yuqun Chen and Qiuhui Mo School of Mathematical Sciences, South China Normal University
More informationON THE MATRIX EQUATION XA AX = X P
ON THE MATRIX EQUATION XA AX = X P DIETRICH BURDE Abstract We study the matrix equation XA AX = X p in M n (K) for 1 < p < n It is shown that every matrix solution X is nilpotent and that the generalized
More informationLECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
More informationCentroids Of Dendriform Algebras
Volume 9 No. 5 208, 427-435 ISSN 34-3395 (on-line version) url http//www.acadpubl.eu/hub/ http//www.acadpubl.eu/hub/ Centroids Of Dendriform Algebras Mohammed Abdul Daim Zoba College of Engineering\ Al-Musayab,
More informationQuantum Co-Adjoint Orbits of the Group of Affine Transformations of the Complex Line 1
Beiträge zur lgebra und Geometrie Contributions to lgebra and Geometry Volume 4 (001, No., 419-430. Quantum Co-djoint Orbits of the Group of ffine Transformations of the Complex Line 1 Do Ngoc Diep Nguyen
More informationarxiv: v1 [math.ra] 6 Oct 2017
Rota Baxter operators of zero weight on simple Jordan algebra of Clifford type V. Gubarev arxiv:1710.02377v1 [math.ra] 6 Oct 2017 Abstract It is proved that any Rota Baxter operator of zero weight on Jordan
More informationINCIDENCE CATEGORIES
INCIDENCE CATEGORIES MATT SZCZESNY ABSTRACT. Given a family F of posets closed under disjoint unions and the operation of taking convex subposets, we construct a category C F called the incidence category
More informationTwisted Heisenberg-Virasoro type left-symmetric algebras
Twisted Heisenberg-Virasoro type left-symmetric algebras Hongjia Chen 1 and Junbo Li 2 1 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, P. R. China
More informationLIE ALGEBRA PREDERIVATIONS AND STRONGLY NILPOTENT LIE ALGEBRAS
LIE ALGEBRA PREDERIVATIONS AND STRONGLY NILPOTENT LIE ALGEBRAS DIETRICH BURDE Abstract. We study Lie algebra prederivations. A Lie algebra admitting a non-singular prederivation is nilpotent. We classify
More informationRota-Baxter Algebra I
1 Rota-Baxter Algebra I Li GUO Rutgers University at Newark . From differentiation to integration. d Differential operator: dx (f)(x) = lim h differential equations differential geometry differential topology
More informationCombinatorial Hopf algebras in particle physics I
Combinatorial Hopf algebras in particle physics I Erik Panzer Scribed by Iain Crump May 25 1 Combinatorial Hopf Algebras Quantum field theory (QFT) describes the interactions of elementary particles. There
More informationUNIVERSAL DERIVED EQUIVALENCES OF POSETS
UNIVERSAL DERIVED EQUIVALENCES OF POSETS SEFI LADKANI Abstract. By using only combinatorial data on two posets X and Y, we construct a set of so-called formulas. A formula produces simultaneously, for
More informationPRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES
PRE-LIE ALGEBRAS AND INCIDENCE CATEGORIES OF COLORED ROOTED TREES MATT SZCZESNY Abstract. The incidence category C F of a family F of colored posets closed under disjoint unions and the operation of taking
More informationDISCRIMINANTS, SYMMETRIZED GRAPH MONOMIALS, AND SUMS OF SQUARES
DISCRIMINANTS, SYMMETRIZED GRAPH MONOMIALS, AND SUMS OF SQUARES PER ALEXANDERSSON AND BORIS SHAPIRO Abstract. Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed
More informationUniversal Enveloping Commutative Rota Baxter Algebras of Pre- and Post-Commutative Algebras
axioms Article Universal Enveloping Commutative Rota Baxter Algebras of Pre- and Post-Commutative Algebras Vsevolod Gubarev 1,2 ID 1 Laboratory of Rings Theory, Sobolev Institute of Mathematics of the
More informationDEFORMATION OF LEIBNIZ ALGEBRA MORPHISMS
Homology, Homotopy and Applications, vol. 9(1), 2007, pp.439 450 DEFORMATION OF LEIBNIZ ALGEBRA MORPHISMS ASHIS MANDAL (communicated by Jean-Louis Loday) Abstract We study formal deformations of Leibniz
More informationCombinatorial Dyson-Schwinger equations and systems I Feynma. Feynman graphs, rooted trees and combinatorial Dyson-Schwinger equations
Combinatorial and systems I, rooted trees and combinatorial Potsdam November 2013 Combinatorial and systems I Feynma In QFT, one studies the behaviour of particles in a quantum fields. Several types of
More informationAn introduction to arithmetic groups. Lizhen Ji CMS, Zhejiang University Hangzhou , China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109
An introduction to arithmetic groups Lizhen Ji CMS, Zhejiang University Hangzhou 310027, China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109 June 27, 2006 Plan. 1. Examples of arithmetic groups
More informationDifferential Type Operators, Rewriting Systems and Gröbner-Shirshov Bases
1 Differential Type Operators, Rewriting Systems and Gröbner-Shirshov Bases Li GUO (joint work with William Sit and Ronghua Zhang) Rutgers University at Newark Motivation: Classification of Linear Operators
More informationStrongly Nil -Clean Rings
Strongly Nil -Clean Rings Abdullah HARMANCI Huanyin CHEN and A. Çiğdem ÖZCAN Abstract A -ring R is called strongly nil -clean if every element of R is the sum of a projection and a nilpotent element that
More informationA short survey on pre-lie algebras
A short survey on pre-lie algebras Dominique Manchon Abstract. We give an account of fundamental properties of pre-lie algebras, and provide several examples borrowed from various domains of Mathematics
More informationAlgebras with operators, Koszul duality, and conformal algebras
Algebras with operators, Koszul duality, and conformal algebras Pavel Kolesnikov Sobolev Institute of Mathematics Novosibirsk, Russia Third Mile High Conference on Nonassociative Mathematics Denver, Aug.
More informationFactorization of the Loop Algebras and Compatible Lie Brackets
Journal of Nonlinear Mathematical Physics Volume 12, Supplement 1 (2005), 343 350 Birthday Issue Factorization of the Loop Algebras and Compatible Lie Brackets I Z GOLUBCHIK and V V SOKOLOV Ufa Pedagogical
More informationLECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C)
LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) IVAN LOSEV Introduction We proceed to studying the representation theory of algebraic groups and Lie algebras. Algebraic groups are the groups
More information204 H. Almutari, A.G. Ahmad
International Journal of Pure and Applied Mathematics Volume 113 No. 2 2017, 203-217 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v113i2.2
More informationThe Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Noncommutative Contact Algebras Hideki Omori Yoshiaki Maeda Naoya Miyazaki Akira Yoshioka
More informationProduct Zero Derivations on Strictly Upper Triangular Matrix Lie Algebras
Journal of Mathematical Research with Applications Sept., 2013, Vol.33, No. 5, pp. 528 542 DOI:10.3770/j.issn:2095-2651.2013.05.002 Http://jmre.dlut.edu.cn Product Zero Derivations on Strictly Upper Triangular
More informationThe endomorphism semigroup of a free dimonoid of rank 1
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 3(76), 2014, Pages 30 37 ISSN 1024 7696 The endomorphism semigroup of a free dimonoid of rank 1 Yurii V.Zhuchok Abstract. We describe
More informationOn the renormalization problem of multiple zeta values
On the renormalization problem of multiple zeta values joint work with K. Ebrahimi-Fard, D. Manchon and J. Zhao Johannes Singer Universität Erlangen Paths to, from and in renormalization Universität Potsdam
More informationEXPLICIT QUANTIZATION OF THE KEPLER MANIFOLD
proceedings of the american mathematical Volume 77, Number 1, October 1979 society EXPLICIT QUANTIZATION OF THE KEPLER MANIFOLD ROBERT J. BLATTNER1 AND JOSEPH A. WOLF2 Abstract. Any representation ir of
More informationStrongly nil -clean rings
J. Algebra Comb. Discrete Appl. 4(2) 155 164 Received: 12 June 2015 Accepted: 20 February 2016 Journal of Algebra Combinatorics Discrete Structures and Applications Strongly nil -clean rings Research Article
More informationarxiv:math-ph/ v1 25 Feb 2002
FROM THE TODA LATTICE TO THE VOLTERRA LATTICE AND BACK arxiv:math-ph/0202037v1 25 Feb 2002 (1) PANTELIS A DAMIANOU AND RUI LOJA FERNANDES Abstract We discuss the relationship between the multiple Hamiltonian
More informationGroup Gradings on Finite Dimensional Lie Algebras
Algebra Colloquium 20 : 4 (2013) 573 578 Algebra Colloquium c 2013 AMSS CAS & SUZHOU UNIV Group Gradings on Finite Dimensional Lie Algebras Dušan Pagon Faculty of Natural Sciences and Mathematics, University
More informationarxiv:math/ v1 [math.qa] 22 Nov 2005
Derivation Algebras of Centerless Perfect Lie Algebras Are Complete 1 (appeared in J. Algebra, 285 (2005), 508 515.) arxiv:math/0511550v1 [math.qa] 22 Nov 2005 Yucai Su, Linsheng Zhu Department of Mathematics,
More informationNIL, NILPOTENT AND PI-ALGEBRAS
FUNCTIONAL ANALYSIS AND OPERATOR THEORY BANACH CENTER PUBLICATIONS, VOLUME 30 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1994 NIL, NILPOTENT AND PI-ALGEBRAS VLADIMÍR MÜLLER Institute
More informationIntegrable Renormalization II: the general case
Integrable enormalization II: the general case arxiv:hep-th/0403118v1 10 Mar 2004 KUUSCH EBAHIMI-FAD 1 I.H.É.S. Le Bois-Marie, 35, oute de Chartres F-91440 Bures-sur-Yvette, France and Universität Bonn
More informationRepresentations of Sp(6,R) and SU(3) carried by homogeneous polynomials
Representations of Sp(6,R) and SU(3) carried by homogeneous polynomials Govindan Rangarajan a) Department of Mathematics and Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560 012,
More informationRingel-Hall Algebras II
October 21, 2009 1 The Category The Hopf Algebra 2 3 Properties of the category A The Category The Hopf Algebra This recap is my attempt to distill the precise conditions required in the exposition by
More informationTHE CLASSICAL HOM-YANG-BAXTER EQUATION AND HOM-LIE BIALGEBRAS. Donald Yau
International Electronic Journal of Algebra Volume 17 (2015) 11-45 THE CLASSICAL HOM-YANG-BAXTER EQUATION AND HOM-LIE BIALGEBRAS Donald Yau Received: 25 January 2014 Communicated by A. Çiğdem Özcan Abstract.
More informationπ X : X Y X and π Y : X Y Y
Math 6130 Notes. Fall 2002. 6. Hausdorffness and Compactness. We would like to be able to say that all quasi-projective varieties are Hausdorff and that projective varieties are the only compact varieties.
More informationON THE CHERN CHARACTER OF A THETA-SUMMABLE FREDHOLM MODULE.
ON THE CHERN CHARACTER OF A THETA-SUMMABLE FREDHOLM MODULE. Ezra Getzler and András Szenes Department of Mathematics, Harvard University, Cambridge, Mass. 02138 USA In [3], Connes defines the notion of
More informationHypertoric varieties and hyperplane arrangements
Hypertoric varieties and hyperplane arrangements Kyoto Univ. June 16, 2018 Motivation - Study of the geometry of symplectic variety Symplectic variety (Y 0, ω) Very special but interesting even dim algebraic
More informationFLABBY STRICT DEFORMATION QUANTIZATIONS AND K-GROUPS
FLABBY STRICT DEFORMATION QUANTIZATIONS AND K-GROUPS HANFENG LI Abstract. We construct examples of flabby strict deformation quantizations not preserving K-groups. This answers a question of Rieffel negatively.
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 435 (2011) 2889 2895 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Idempotent elements
More informationThe Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases
arxiv:8.976v [math.rt] 3 Dec 8 The Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases Central European University Amadou Keita (keita amadou@student.ceu.edu December 8 Abstract The most famous
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationON SUM OF SQUARES DECOMPOSITION FOR A BIQUADRATIC MATRIX FUNCTION
Annales Univ. Sci. Budapest., Sect. Comp. 33 (2010) 273-284 ON SUM OF SQUARES DECOMPOSITION FOR A BIQUADRATIC MATRIX FUNCTION L. László (Budapest, Hungary) Dedicated to Professor Ferenc Schipp on his 70th
More informationMATH 326: RINGS AND MODULES STEFAN GILLE
MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called
More informationON GRADED MORITA EQUIVALENCES FOR AS-REGULAR ALGEBRAS KENTA UEYAMA
ON GRADED MORITA EQUIVALENCES FOR AS-REGULAR ALGEBRAS KENTA UEYAMA Abstract. One of the most active projects in noncommutative algebraic geometry is to classify AS-regular algebras. The motivation of this
More informationOn Regularity of Incline Matrices
International Journal of Algebra, Vol. 5, 2011, no. 19, 909-924 On Regularity of Incline Matrices A. R. Meenakshi and P. Shakila Banu Department of Mathematics Karpagam University Coimbatore-641 021, India
More informationCartan A n series as Drinfeld doubles
Monografías de la Real Academia de Ciencias de Zaragoza. 9: 197 05, (006). Cartan A n series as Drinfeld doubles M.A. del Olmo 1, A. Ballesteros and E. Celeghini 3 1 Departamento de Física Teórica, Universidad
More informationAn Introduction to Rota-Baxter Algebra
Surveys of Modern Mathematics Volume IV An Introduction to Rota-Baxter Algebra Li Guo International Press www.intlpress.com HIGHER EDUCATION PRESS Surveys of Modern Mathematics, Volume IV An Introduction
More informationON EXACT POISSON STRUCTURES
ON EXACT POISSON STRUCTURES YINGFEI YI AND XIANG ZHANG Abstract By studying the exactness of multi-linear vectors on an orientable smooth manifold M, we give some characterizations to exact Poisson structures
More informationLIE ALGEBRAS: LECTURE 7 11 May 2010
LIE ALGEBRAS: LECTURE 7 11 May 2010 CRYSTAL HOYT 1. sl n (F) is simple Let F be an algebraically closed field with characteristic zero. Let V be a finite dimensional vector space. Recall, that f End(V
More informationHopf subalgebras of the Hopf algebra of rooted trees coming from Dyson-Schwinger equations and Lie algebras of Fa di Bruno type
Hopf subalgebras of the Hopf algebra of rooted trees coming from Dyson-Schwinger equations and Lie algebras of Fa di Bruno type Loïc Foissy Contents 1 The Hopf algebra of rooted trees and Dyson-Schwinger
More informationarxiv: v1 [math.qa] 23 Jul 2016
Obstructions for Twist Star Products Pierre Bieliavsky arxiv:1607.06926v1 [math.qa] 23 Jul 2016 Faculté des sciences Ecole de mathématique (MATH) Institut de recherche en mathématique et physique (IRMP)
More informationAFFINE ACTIONS ON NILPOTENT LIE GROUPS
AFFINE ACTIONS ON NILPOTENT LIE GROUPS DIETRICH BURDE, KAREL DEKIMPE, AND SANDRA DESCHAMPS Abstract. To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations
More informationRANK AND PERIMETER PRESERVER OF RANK-1 MATRICES OVER MAX ALGEBRA
Discussiones Mathematicae General Algebra and Applications 23 (2003 ) 125 137 RANK AND PERIMETER PRESERVER OF RANK-1 MATRICES OVER MAX ALGEBRA Seok-Zun Song and Kyung-Tae Kang Department of Mathematics,
More informationThe real root modules for some quivers.
SS 2006 Selected Topics CMR The real root modules for some quivers Claus Michael Ringel Let Q be a finite quiver with veretx set I and let Λ = kq be its path algebra The quivers we are interested in will
More informationINTRODUCTION TO LIE ALGEBRAS. LECTURE 7.
INTRODUCTION TO LIE ALGEBRAS. LECTURE 7. 7. Killing form. Nilpotent Lie algebras 7.1. Killing form. 7.1.1. Let L be a Lie algebra over a field k and let ρ : L gl(v ) be a finite dimensional L-module. Define
More informationRealization of locally extended affine Lie algebras of type A 1
Volume 5, Number 1, July 2016, 153-162 Realization of locally extended affine Lie algebras of type A 1 G. Behboodi Abstract. Locally extended affine Lie algebras were introduced by Morita and Yoshii as
More informationTorus actions and Ricci-flat metrics
Department of Mathematics, University of Aarhus November 2016 / Trondheim To Eldar Straume on his 70th birthday DFF - 6108-00358 Delzant HyperKähler G2 http://mscand.dk https://doi.org/10.7146/math.scand.a-12294
More informationGorenstein rings through face rings of manifolds.
Gorenstein rings through face rings of manifolds. Isabella Novik Department of Mathematics, Box 354350 University of Washington, Seattle, WA 98195-4350, USA, novik@math.washington.edu Ed Swartz Department
More informationarxiv:math/ v2 [math.dg] 14 Mar 2003
Quasi-derivations and QD-algebroids arxiv:math/0301234v2 [math.dg] 14 Mar 2003 Janusz Grabowski Mathematical Institute, Polish Academy of Sciences ul. Śniadeckich 8, P.O. Box 21, 00-956 Warszawa 10, Poland
More informationBoundedly complete weak-cauchy basic sequences in Banach spaces with the PCP
Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,
More informatione j = Ad(f i ) 1 2a ij/a ii
A characterization of generalized Kac-Moody algebras. J. Algebra 174, 1073-1079 (1995). Richard E. Borcherds, D.P.M.M.S., 16 Mill Lane, Cambridge CB2 1SB, England. Generalized Kac-Moody algebras can be
More informationarxiv: v1 [math.ra] 11 Dec 2017
NARROW POSITIVELY GRADED LIE ALGEBRAS arxiv:1712.03718v1 [math.ra] 11 Dec 2017 DMITRY V. MILLIONSHCHIKOV Abstract. We classify real and complex infinite-dimensional positively graded Lie algebras g = +
More informationREAL RENORMINGS ON COMPLEX BANACH SPACES
REAL RENORMINGS ON COMPLEX BANACH SPACES F. J. GARCÍA PACHECO AND A. MIRALLES Abstract. In this paper we provide two ways of obtaining real Banach spaces that cannot come from complex spaces. In concrete
More informationProjective Quantum Spaces
DAMTP/94-81 Projective Quantum Spaces U. MEYER D.A.M.T.P., University of Cambridge um102@amtp.cam.ac.uk September 1994 arxiv:hep-th/9410039v1 6 Oct 1994 Abstract. Associated to the standard SU (n) R-matrices,
More informationLie bialgebras real Cohomology
Journal of Lie Theory Volume 7 (1997) 287 292 C 1997 Heldermann Verlag Lie bialgebras real Cohomology Miloud Benayed Communicated by K.-H. Neeb Abstract. Let g be a finite dimensional real Lie bialgebra.
More informationLinear Algebra. Preliminary Lecture Notes
Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date April 29, 23 2 Contents Motivation for the course 5 2 Euclidean n dimensional Space 7 2. Definition of n Dimensional Euclidean Space...........
More information