An application of non-associative Composition-Diamond lemma

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1 An applcaton of non-assocatve Composton-Damond lemma arxv: v1 [math.ra] 6 Apr 2008 Yuqun Chen and Yu L School of Mathematcal Scences, South Chna Normal Unversty Guangzhou , P. R. Chna Emal: yqchen@scnu.edu.cn LYu820615@126.com Abstract: In ths paper, by usng Gröbner Shrshov bases for non-assocatve algebras nvented by A. I. Shrshov n 1962, we show I. P. Shestakov s result that any Akvs algebra can be embedded nto ts unversal envelopng algebra. Key words: non-assocatve algebra; Akvs algebra; unversal envelopng algebra; Gröbner-Shrshov bass. AMS 2000 Subject Classfcaton: 17A01, 16S15, 13P10 1 Introducton A. G. Kurosh [11] ntated to study free non-assocatve algebras over a feld provng that any subalgebra of a free non-assocatve algebra s free. Hs student, A. I. Zhukov, proved n [20] that the word problem s algorthmcally decdable n the class of nonassocatve algebras. Namely, he proved that word problem s decdable for any fntely presented non-assocatve algebra. A. I. Shrshov, also a student of Kurosh, proved n [15], 1953, that any subalgebra of a free Le algebra s free. Ths theorem s now known as the Shrshov-Wtt theorem (see, for example, [12]) for t was proved also by E. Wtt [19]. Some later, Shrshov [16] gave a drect constructon of a free (ant-) commutatve algebra and proved that any subalgebra of such an algebra s agan free (ant-) commutatve algebra. Almost ten years later, Shrshov came back to, we may say, the Kurosh programme, and publshed two papers [17] and [18]. In the former, he gave a conceptual proof that the word problem s decdable n the class of (ant-) commutatve non-assocatve algebras. Namely, he created the theory that s now known as Gröbner Shrshov bases theory for (ant-) commutatve non-assocatve algebras. In the latter, he dd the same for Le algebras (explctly) and assocatve algebras (mplctly). Ther man applcatons were the decdablty of the word problem for any one-relater Le algebra, the Frehetsatz (the Freeness theorem) for Le algebras, and the algorthm for decdablty of the word problem for any fntely presented homogeneous Le algebra. The same algorthm s vald for any fntely presented homogeneous assocatve algebra as well. Shrshov s man techncal Supported by the NNSF of Chna (No ) and the NSF of Guangdong Provnce (No ). 1

2 dscovery of [18] was the noton of composton of two Le polynomals and mplctly two assocatve polynomals. Based on t, he gave the algorthm to construct the Gröbner Shrshov bass for any deal of a free Le algebra. The same algorthm s vald n the assocatve case. Ths algorthm s n general nfnte as well as, for example, Knuth- Bendx algorthm [10]. Shrshov proved that f a Gröbner Shrshov bass of an deal s recursve, then the word problem for the quotent algebra s decdable. It follows from Shrshov s Composton-Damond lemma that t s vald for free non-assocatve, free (ant- ) commutatve, free Le and free assocatve algebras (see [17] and [18]). Explctly the assocatve case was treated n the papers by L. A. Bokut [3] and G. Bergman [2]. Independently, B. Buchberger n hs thess (1965) (see [7]) created the Gröbner bases theory for the classcal case of commutatve assocatve algebras. Also, H. Hronaka n hs famous paper [9] dd the same for (formal or convergent) nfnte seres rather than polynomals. He called hs bases as the standard bases. Ths term s used untl now as a synonym of Gröbner (n commutatve case) or Gröbner-Shrshov (n non-assocatve and non-commutatve cases) bases. There are a lot of sources of the hstory of Gröbner and Gröbner Shrshov bases theory (see, for example, [8], [4], [5], [6]). In the present paper we are dealng wth the Composton-Damond lemma for a free non-assocatve algebra, callng t as non-assocatve Composton-Damond lemma. Shrshov mentoned t n [17] that all hs results are vald for the case of free non-assocatve algebras rather than free (ant-) commutatve algebras. For completeness, we prove ths lemma n Secton 2 n ths paper. Then we apply ths lemma to the unversal envelopng algebra of an Akvs algebra gvng an another proof of I. P. Shestakov s result that any Akvs algebra s lnear (see [13]). An Akvs algebra s a vector space V over a feld k endowed wth a skew-symmetrc blnear product [x, y] and a trlnear product (x,y,z) that satsfy the dentty [[x, y], z] + [[y, z], x] + [[z, x], y] = (x, y, z) + (z, x, y) + (y, z, x) (x, z, y) (y, x, z) (z, y, x). These algebras were ntroduced n 1976 by M. A. Akvs [1] as tangent algebras of local analtc loops. For any (non-assocatve) algebra B one may obtan an Akvs algebra Ak(B) by consderng n B the usual commutator [x, y] = xy yx and assocator (x, y, z) = (xy)z x(yz). Let {e } I be a bass of an Akvs algebra A. Then the nonassocatve algebra U(A) = M({e } I e e j e j e = [e, e j ], (e e j )e k e (e j e k ) = (e, e j, e k ),, j, k I) gven by the generators and relatons s the unversal envelopng algebra of A, where [e, e j ] = m αm j e m, (e, e j, e k ) = n βn jk e n and each αj m, βn jk k. The lnearty of A means that A s a subspace of U(A) (see [13]). Remark also that any subalgebra of a free Akvs algebra s agan free (see [14]). 2 Composton-Damond lemma for non-assocatve algebras Let X = {x I} be a set, X the set of all assocatve words u n X, and X the set of all non-assocatve words (u) n X. Let k be a feld and M(X) be a k-space spanned by X. We defne the product of non-assocatve words by the followng way: (u)(v) = ((u)(v)). 2

3 Then M(X) s a free non-assocatve algebra generated by X. Let I be a lnearly ordered set. We order X by the nducton on the length ((u)(v)) of the words (u) and (v) n X : () If ((u)(v)) = 2, then (u) = x > (v) = x j ff > j. () If ((u)(v)) > 2, then (u) > (v) ff one of the followng cases holds: (a) (u) > (v). (b) If (u) = (v) and (u) = ((u 1 )(u 2 )), (v) = ((v 1 )(v 2 )), then (u 1 ) > (v 1 ) or ((u 1 ) = (v 1 ) and (u 2 ) > (v 2 )). It s easy to check that the order < on X s a monomal order n the followng sense: (a) < s a well order. (b) (u) > (v) = (u)(w) > (v)(w) and (w)(u) > (w)(v) for any (w) X. Such an order s called deg-lex (degree-lexcographcal) order and we use ths order throughout ths paper. Gven a polynomal f M(X), t has the leadng word ( f) X accordng to the deg-lex order on X such that f = α(f) + α (u ), where (f) > (u ), α, α k, (u ) X. We call (f) the leadng term of f. f s called monc f α = 1. Defnton 2.1 Let S M(X) be a set of monc polynomals, s S and (u) X. We defne S-word (u) s by nducton: () (s) s = s s an S-word of S-length 1. () If (u) s s an S-word of S-length k and (v) s a non-assocatve word of length l, then are S-words of length k + l. (u) s (v) and (v)(u) s The S-length of an S-word (u) s wll be denoted by u s. Note that (asb) = (a sb) f (u) s = (asb), where a, b X. Let f, g be monc polynomals n M(X). Suppose that there exst a, b X such that ( f) = (a(ḡ)b). Then we set (w) = ( f) and defne the composton of ncluson (f, g) (w) = f (agb). It s clear that (f, g) (w) Id(f, g) and (f, g) (w) < (w). 3

4 Gven a nonempty subset S M(X), we shall say that the composton (f, g) (w) s trval modulo (S, (w)), f (f, g) (w) = α (a s b ), where each α k, a, b X, s S, (a s b ) an S-word and (a ( s )b ) < (w). If ths s the case, then we wrte (f, g) (w) 0 mod(s, (w)). In general, for p, q M(X), we wrte p q mod(s, (w)) whch means that p q = α (a s b ), where each α k, a, b X, s S, (a s b ) an S-word and (a ( s )b ) < (w). Defnton 2.2 Let S M(X) be a nonempty set of monc polynomals and the order < as before. Then S s called a Gröbner-Shrshov bass n M(X), f any composton (f, g) (w) wth f, g S s trval modulo (S, (w)),.e., (f, g) (w) 0 mod(s, (w)). Lemma 2.3 Let (a 1 s 1 b 1 ), (a 2 s 2 b 2 ) be S-words. If S s a Gröbner-Shrshov bass n M(X) and (w) = (a 1 (s 1 )b 1 ) = (a 2 (s 2 )b 2 ), then (a 1 s 1 b 1 ) (a 2 s 2 b 2 ) mod(s, (w)). Proof. We have a 1 s 1 b 1 = a 2 s 2 b 2 as assocatve words n the alphabet X { s 1, s 2 }. There are two cases to consder. Case 1. Suppose that subwords s 1 and s 2 of w are dsjont, say, a 2 a 1 + s 1. Then, we can assume that a 2 = a 1 s 1 c and b 1 = c s 2 b 2 for some c X, and so, w = (a 1 ( s 1 )c( s 2 )b 2 ). Now, (a 1 s 1 b 1 ) (a 2 s 2 b 2 ) = (a 1 s 1 c( s 2 )b 2 ) (a 1 ( s 1 )cs 2 b 2 ) = (a 1 s 1 c(( s 2 ) s 2 )b 2 ) + (a 1 (s 1 ( s 1 ))cs 2 b 2 ). Snce ((s 2 ) s 2 ) < ( s 2 ) and (s 1 (s 1 )) < ( s 1 ), we conclude that (a 1 s 1 b 1 ) (a 2 s 2 b 2 ) = α (u s 1 v ) + j β j (u j s 2 v j ) for some α, β j k, S-words (u s 1 v ) and (u j s 2 v j ) such that (u ( s 1 )v ), (u j ( s 2 )v j ) < (w). Thus, (a 1 s 1 b 1 ) (a 2 s 2 b 2 ) mod(s, (w)). Case 2. Suppose that the subword s 1 of w contans s 2 as a subword. We assume that ( s 1 ) = (a( s 2 )b), a 2 = a 1 a and b 2 = bb 1, that s, (w) = (a 1 a( s 2 )bb 1 ) for some S-word (as 2 b). We have (a 1 s 1 b 1 ) (a 2 s 2 b 2 ) = (a 1 s 1 b 1 ) (a 1 (as 2 b)b 1 ) = (a 1 (s 1 (as 2 b))b 1 ) = (a 1 (s 1, s 2 ) (w1 )b 1 ), 4

5 where (w 1 ) = (s 1 ) = (a( s 2 )b). Snce S s a Gröbner-Shrshov bass, (s 1, s 2 ) (w1 ) = α (c s d ) for some α k, S-words (c s d ) wth each (c ( s )d ) < (w 1 ) = ( s 1 ). Then, (a 1 s 1 b 1 ) (a 2 s 2 b 2 ) = (a 1 (s 1, s 2 ) (w1 )b 1 ) = α (a 1 (c s d )b 1 ) = β j (a j s j b j ) j for some β j k, S-words (a j s j b j ) wth each (a j ( s j )b j ) < (w) = (a 1 ( s 1 )b 1 ). Thus, (a 1 s 1 b 1 ) (a 2 s 2 b 2 ) mod(s, (w)). Lemma 2.4 Let S M(X) be a subset of monc polynomals and Red(S) = {(u) X (u) (a( s)b), a, b X, s S and (asb) s an S-word}. Then for any f M(X), f = α (u ) + β j (a j s j b j ), (u ) ( f) (a j (s j )b j ) ( f) where each α, β j k, (u ) Red(S) and (a j s j b j ) an S-word. Proof. Let f = α (u ) M(X), where 0 α k and (u 1 ) > (u 2 ) >. If (u 1 ) Red(S), then let f 1 = f α 1 (u 1 ). If (u 1 ) Red(S), then there exst some s S and a 1, b 1 X, such that ( f) = (u 1 ) = (a 1 ( s 1 )b 1 ). Let f 1 = f α 1 (a 1 s 1 b 1 ). In both cases, we have ( f 1 ) < ( f). Then the result follows from the nducton on ( f). The proof of the followng theorem s analogous to one n Shrshov [17]. For convenence, we gve the detals. Theorem 2.5 (Shrshov, Composton-Damond for non-assocatve algebras) Let S M(X) be a nonempty set of monc polynomals and the order < on X as before. Then the followng statements are equvalent. () S s a Gröbner-Shrshov bass. () f Id(S) ( f) = (a( s)b) for some s S and a, b X, where (asb) s an S-word. () f Id(S) f = α 1 (a 1 s 1 b 1 ) + α 2 (a 2 s 2 b 2 ) +, where α k, (a 1 (s 1 )b 1 ) > (a 2 (s 2 )b 2 ) > and (a s b ) S-words. () Red(S) = {(u) X (u) (a( s)b) a, b X, s S and (asb) s an S-word} s a bass of the algebra M(X S). Proof. () (). have Let S be a Gröbner-Shrshov bass and 0 f Id(S). Then, we f = n α (a s b ), where each α k, a, b X, s S and (a s b ) an S-word. Let =1 (w ) = (a (s )b ), (w 1 ) = (w 2 ) = = (w l ) > (w l+1 ) 5

6 We wll use the nducton on l and (w 1 ) to prove that (f) = (a(s)b) for some s S and a, b X. If l = 1, then (f) = (a 1 s 1 b 1 ) = (a 1 (s 1 )b 1 ) and hence the result holds. Assume that l 2. Then, by Lemma 2.3, we have (a 1 s 1 b 1 ) (a 2 s 2 b 2 ) mod(s, (w 1 )). Thus, f α 1 + α 2 0 or l > 2, then the result holds. For the case α 1 + α 2 = 0 and l = 2, we use the nducton on (w 1 ). Now, the result follows. () (). Assume () and 0 f Id(S). Let f = α 1 (f) +. Then, by (), (f) = (a 1 (s 1 )b 1 ). Therefore, f 1 = f α 1 (a 1 s 1 b 1 ), (f 1 ) < (f), f 1 Id(S). Now, by usng nducton on (f), we have (). () (). Ths part s clear. () (). Suppose that α (u ) = 0 n M(X S), where α k, (u ) Red(S). It means that α (u ) Id(S) n M(X). Then all α must be equal to zero. Otherwse, α (u ) = (u j ) Red(S) for some j whch contradcts (). Now, for any f M(X), by Lemma 2.4, we have f = α (u ) + So, () follows. (u ) Red(S), (u ) ( f) (a j (s j )b j ) ( f) () (). For any f, g S, by Lemma 2.4, we have (f, g) (w) = α (u ) + (u ) Red(S), (u )<(w) Snce (f, g) (w) Id(S) and by (), we have (f, g) (w) = (a j (s j )b j )<(w) (a j (s j )b j )<(w) β j (a j s j b j ). β j (a j s j b j ). β j (a j s j b j ). Therefore, S s a Gröbner-Shrshov bass. 3 Gröbner-Shrshov bass for unversal envelopng algebra of an Akvs algebra In ths secton, we obtan a Gröbner-Shrshov bass for unversal envelopng algebra of an Akvs algebra. 6

7 Theorem 3.1 Let (A, +, [, ], (,, )) be an Akvs algebra wth a lnearly ordered bass {e I}. Let [e, e j ] = m α m j e m, (e, e j, e k ) = n β n jk e n, where α m j, βn jk k. We denote m Let α m j e m and n β n jk e n by {e e j } and {e e j e k }, respectvely. U(A) = M({e } I e e j e j e = {e e j }, (e e j )e k e (e j e k ) = {e e j e k },, j, k I) be the unversal envelopng algebra of A. Let S = {f j = e e j e j e {e e j } ( > j), g jk = (e e j )e k e (e j e k ) {e e j e k } (, j, k I), h jk = e (e j e k ) e j (e e k ) {e e j }e k {e j e e k } + {e e j e k } ( > j, k j)}. Then () S s a Gröbner-Shrshov bass for U(A). () A can be embedded nto the unversal envelopng algebra U(A). Proof. (). It s easy to check that f j = e e j ( > j), g jk = (e e j )e k (, j, k I), h jk = e (e j e k ) ( > j, k j). So, we have only two knds of compostons to consder: (g jk, f j ) (e e j )e k ( > j, j k) and (g jk, f j ) (e e j )e k ( > j > k). For (g jk, f j ) (e e j )e k, ( > j, j k), we have (g jk, f j ) (e e j )e k =(e j e )e k e (e j e k ) + {e e j }e k {e e j e k } e (e j e k ) + e j (e e k ) + {e e j }e k + {e j e e k } {e e j e k } 0. For (g jk, f j ) (e e j )e k, ( > j > k), by notng that, n A, [[e, e j ], e k ] + [[e j, e k ], e ] + [[e k, e ], e j ] =(e, e j, e k ) + (e k, e, e j ) + (e j, e k, e ) (e, e k, e j ) (e j, e, e k ) (e k, e j, e ), 7

8 we have (g jk, f j ) (e e j )e k =(e j e )e k e (e j e k ) + {e e j }e k {e e j e k } e (e j e k ) + e j (e e k ) + {e e j }e k + {e j e e k } {e e j e k } e (e k e j ) e {e j e k } + e j (e e k ) + {e e j }e k + {e j e e k } {e e j e k } e j (e e k ) e k (e e j ) {e e k }e j + {e e j }e k e {e j e k } {e k e e j } + {e e k e j } + {e j e e k } {e e j e k } e j (e k e ) + e j {e e k } e k (e j e ) e k {e e j } {e e k }e j + {e e j }e k e {e j e k } {e k e e j } + {e e k e j } + {e j e e k } {e e j e k } e k (e j e ) + {e j e k }e + {e k e j e } {e j e k e } + e j {e e k } e k (e j e ) e k {e e j } {e e k }e j + {e e j }e k e {e j e k } {e k e e j } + {e e k e j } + {e j e e k } {e e j e k } {e j e k }e e {e j e k } + e j {e e k } {e e k }e j + {e e j }e k e k {e e j } + {e k e j e } + {e e k e j } + {e j e e k } {e j e k e } {e k e e j } {e e j e k } {e j e k }e e {e j e k } + {e k e }e j e j {e k e } + {e e j }e k e k {e e j } + {e k e j e } + {e e k e j } + {e j e e k } {e j e k e } {e k e e j } {e e j e k } {{e j e k }e } + {{e k e }e j } + {{e e j }e k } 0. + {e k e j e } + {e e k e j } + {e j e e k } {e j e k e } {e k e e j } {e e j e k } Thus, S s a Gröbner-Shrshov bass for U(A). () follows from Theorem 2.5. Ths completes our proof. Acknowledgement: The authors would lke to express ther deepest grattude to Professor L. A. Bokut for hs knd gudance, useful dscussons and enthusastc encouragement durng hs vst to the South Chna Normal Unversty. References [1] M. A. Akvs, The local algebras of a multdmensonal three-web (Russan), Sbrsk. Mat. Z., 17(1976), 1, Englsh translaton: Sberan Math. J., 17 (1976), 1, 3-8. [2] G. M. Bergman, The damond lemma for rng theory, Adv. n Math., 29 (1978), [3] L. A. Bokut, Imbeddngs nto smple assocatve algebras, Algebra Logka, 15(1976), [4] L. A. Bokut, Y. Fong, W.-F. Ke and P. S. Kolesnkov, Gröbner and Gröbner-Shrshov bases n Algebra and Conformal algebras, Fundamental and Appled Mathematcs, 6 (2000), 3, [5] L. A. Bokut and P. S. Konesnkov, Gröbner-Shrshov bases: from ther ncpency to the present, Journal of Mathematcal Scences, 116 (2003), 1,

9 [6] L. A. Bokut and P. S. Konesnkov, Gröbner-Shrshov bases, conformal algebras and pseudo-algebras, Journal of Mathematcal Scences, 131(2005), 5, [7] B. Buchberger, An algorthmcal crtera for the solvablty of algebrac systems of equatons, Aequatones Math., 4(1970), (n German) [8] D. Esenbud, Commutatve algebra. Wth a vew toward algebrac geometry. Graduate Texts n Mathematcs Berln: Sprnger-Verlag. [9] H. Hronaka, Rosoluton of sngulartes of an algebrac varety over a feld of characterstc zero. I, II, Ann. Math., 79(1964), , [10] D. E. Knuth and P. B. Bendx, Smple word problems n unversal algebras, Comput. Probl. abstract Algebra, Proc. Conf. Oxford 1967, (1970). [11] A. G. Kurosh, Nonassocatve free algebras and free products of algebras, Mat. Sb., N. Ser., 20(62)(1947), [12] C. Reutenauer, Free Le algebras. Oxford Scence Publcatons, [13] I. P. Shestakov, Every Akvs algebra s lnear, Geometrae Dedcata, 77(1999), [14] I. P. Shestakov and U. Umrbaev, Free Akvs algebras, prmtve elements and hyperalgebras, J. Algebra, 250(2002), [15] A. I. Shrshov, Subalgebras of free Le algebras, Mat. Sb., N. Ser., 33(75)(1953), [16] A. I. Shrshov, Subalgebras of free commutatve and free ant-commutatve algebras, Mat. Sbornk., 34(76)(1954), [17] A. I. Shrshov, Certan algorthmc problems for ǫ-algebras, Sb. Mat. Zh., 3(1962), [18] A. I. Shrshov, Certan algorthmc problems for Le algebras, Sb. Mat. Zh., 3(1962), [19] E. Wtt, Subrngs of free Le rngs, Math. Zet., 64(1956), [20] A. I. Zhukov, Reduced systems of dfnng relatons n non-assocatve algebras, Mat. Sb., N. Ser., 27(69)(1950),