Leibniz triple systems

LEIBNIZ TRIPLE SYSTEMS MURRA Y R. BREMNER AND JUANA S ´ ANCHEZ-OR TEGA Abstract. W e define Leibniz triple systems in a functorial manner using the algorithm of K olesniko v and Pozhidaev whi c h conv erts iden tities for algebras int o iden tities for dialgebras. W e v erify that Leibniz triple systems are the natural analogues of Lie triple syste ms in the con text of dialgebras b y showing that both the iterated brac k et in a Le ibniz algebra and the permute d associator in a Jor dan dialgebra satisfy the defining identities for Leibniz triple systems. W e construct the unive rsal Leibniz env elop es of Leibniz triple systems and prov e that ev ery iden tit y satisfied b y the iterated brac ket in a Leibniz algebra is a consequenc e of the defining identities for Leibniz tripl e systems. T o conclude, we present some examples of 2-dimensional Leibniz triple systems and their univ ersal Leibniz env elopes. 1. Introduction In this pap er we introduce Leibniz triple systems, which ar e related to Leibniz algebras in the same wa y that Lie tr iple systems are related to Lie algebras. Our motiv ation is to present a new t yp e of ter na ry algebr a with p otential a pplica tions in theo r etical physics. See our recent pap er [3] for a rela ted result on the partia lly alternating terna ry sum in an ass o ciative dia lgebra. W e sta rt by recalling in Section 2 the definitions of a sso ciative dialgebras and Leibniz algebr as. W e state the Kolesnikov-P ozhidaev (KP) alg orithm which takes as input the defining identities for a v ar iety of alge br as and pro duces as output the defining identities for the corres p o nding v ariety of dialg ebras. As exa mples, we reca ll how ass o ciative dialgebra s and Leibniz algebr as can be obtained from asso ciative a nd Lie algebr as by an applica tio n of this algor ithm. In Section 3 we apply the KP algor ithm to Lie triple systems, and o bta in a new v ariety of triple sys tems; we call these structures Le ibniz triple systems . W e show that this v a riety of structures may be characterized by tw o multilinear identities. In Section 4 we reformulate the defining identities for Le ibniz tr iple s ystems in terms of left and right m ultiplication op er ators. In Section 5 we use the s tructure theory for free Leibniz alg e bras to verify that any subspace of a Leibniz a lgebra closed under the itera ted Leibniz brack et is a Leibniz triple system. In Section 6 we prove that any subspac e of a Jor da n dialg ebra (quasi-Jor dan algebra) clos ed under the a s so ciator is a Leibniz tr iple system. In Section 7 we construct universal Leibniz env elo p e s for Leibniz triple systems . F rom this we obtain the co rollar y that every polynomia l identit y satisfied by the iterated br a ck et in a Leibniz algebr a is a consequence of the defining identit ies for Leibniz triple systems. In Section 8 we conclude the pap er with a co njectured classification of 2-dimen- sional Leibniz triple systems; we also co nstruct their universal Leibniz envelopes. 1 2 MURRA Y R. BREMNER AND JUANA S ´ ANCHEZ-OR TE GA In summa r y , our results demo ns trate that Leibniz triple systems are the natural analogue o f Lie triple sy stems in the context of dialg ebras. 2. Preliminaries 2.1. Di algebras and Leibniz algebras. Dialge bras were intro duced by L o day [8] (see also [9]) to provide a na tural setting for Leibniz alg ebras, a “ no ncommutativ e” version of Lie algebra s. Definition 1. (Cuvier [4], Lo day [7]) A (right) Leibniz algebra is a vector space L , together with a bilinear ma p L × L → L , denoted ( a, b ) 7→ h a, b i , sa tisfying the (righ t) Leibniz iden tit y , which says that r ight m ultiplications ar e deriv a tions: (1) hh a, b i , c i ≡ hh a, c i , b i + h a, h b, c ii . If h a, a i ≡ 0 then the L e ibniz identit y is the J acobi identit y and L is a Lie algebra . An asso cia tive a lgebra b ecomes a Lie alg ebra if the pro duct ab is replaced by the Lie brack et ab − ba . If we r eplace the pro duct ab and its opp osite ba by tw o distinct op erations a ⊣ b a nd b ⊢ a , the we obtain the notion of an a s so ciative dialg e bra, in which the Leibniz bracket a ⊣ b − b ⊢ a is not necessarily s kew-symmetric. Definition 2. An asso ciativ e dialgebra is a vector spa ce A with tw o bilinear maps A × A → A , denoted ⊣ a nd ⊢ and called the l eft and righ t pr o ducts, satisfying the l eft and righ t bar i den tities , and left, righ t and inner asso ci ativi t y : ( a ⊣ b ) ⊢ c ≡ ( a ⊢ b ) ⊢ c, a ⊣ ( b ⊣ c ) ≡ a ⊣ ( b ⊢ c ) , ( a ⊣ b ) ⊣ c ≡ a ⊣ ( b ⊣ c ) , ( a ⊢ b ) ⊢ c ≡ a ⊢ ( b ⊢ c ) , ( a ⊢ b ) ⊣ c ≡ a ⊢ ( b ⊣ c ) . The Leibniz bracket in an a s so ciative dialgebra satis fie s the Leibniz identit y . 2.2. K P Algorithm . Kolesnikov [6] intro duced a gener al ca tegorica l framework for transfor ming the defining identities of a v arie t y of binar y algebr as (asso ciative, Lie, J ordan, etc.) into the defining iden tities o f the cor resp onding v ariety of dialg e- bras. This pro cedure was extended by Pozhidaev (in an unpublished preprint) to v arieties of arbitrar y n -ary (m ultiop erator) a lg ebras. In this subs ection we present a simplified statement of the Ko lesniko v-Pozhidaev (KP) alg orithm. Algorithm 3 . The input is a m ultilinear p oly no mial identit y of degree d for an n -ary o p er ation; the output is a collection of d mult ilinear iden tities of degr ee d for n new n -ar y op erations . Part 1: W e consider a multilinear n -a r y op eratio n, deno ted by the symbol (2) {− , . . . , −} ( n a rguments) . Given a multilinear identit y of degre e d in this oper ation, we describ e the application of the algorithm to one monomial, and extend this by linea r ity to the entire identit y . Let a 1 a 2 . . . a d be a multilinear monomial of degre e d , wher e the ba r denotes some placement of n -ary o p er ation symbo ls (2 ). W e introduce n new n - ary o pe r ations, using the same op eration sy m b o l but distinguishe d by subscripts: (3) {− , . . . , −} 1 , . . . , {− , . . . , − } n . F or ea ch i = 1 , . . . , d we conv ert the mo no mial a 1 a 2 . . . a d in the n -ary op era tio n (2) into a new mono mial of the same degree in the n new n -a r y o per ations (3), according to the following rule, based on the p ositio n o f the indeterminate a i . F o r LEIBNIZ TRIPLE SYS TEMS 3 each occurr ence of ope r ation (2) in the monomia l, either a i o ccurs within o ne of the n arguments or no t, and we ha ve t w o cas es: • If a i o ccurs in the j -th arg ument then we c o nv ert this o ccurrence of { . . . } to the j - th new op eration symbol { . . . } j . • If a i do es not o ccur in a ny of the n ar guments, then either – a i o ccurs to the left o f this occur rence of { . . . } : w e con vert { . . . } to the fir st new op eratio n symbol { . . . } 1 , o r – a i o ccurs to the r ight of this o ccurrence of { . . . } : we con vert { . . . } to the la st new op eratio n symbol { . . . } n . In step i we c a ll a i the central indetermina te of the monomial. Part 2: W e also include the following identit ies, analogous to the bar identit ies for as so ciative dialg ebras, for all i, j = 1 , . . . , n with i 6 = j and all k , ℓ = 1 , . . . , n : { a 1 , . . . , a i − 1 , { b 1 , · · · , b n } k , a i +1 , . . . , a n } j ≡ { a 1 , . . . , a i − 1 , { b 1 , · · · , b n } ℓ , a i +1 , . . . , a n } j . This identit y sa ys that the n new op erations ar e in terchangeable in the i -th arg u- men t of the j -th new op era tion when i 6 = j . Example 4. The defining identities for asso cia tive dialg ebras can be obtained by applying the KP algor ithm to the asso cia tivit y identit y ( a ◦ b ) ◦ c ≡ a ◦ ( b ◦ c ). The op eration ◦ pro duces t wo new op erations ◦ 1 and ◦ 2 . Part 1 of the algor ithm g ives three identities of degree 3, making a , b , c in turn the central indetermina te: ( a ◦ 1 b ) ◦ 1 c ≡ a ◦ 1 ( b ◦ 1 c ) , ( a ◦ 2 b ) ◦ 1 c ≡ a ◦ 2 ( b ◦ 1 c ) , ( a ◦ 2 b ) ◦ 2 c ≡ a ◦ 2 ( b ◦ 2 c ) . Part 2 of the algorithm gives tw o ident ities: a ◦ 1 ( b ◦ 1 c ) ≡ a ◦ 1 ( b ◦ 2 c ) , ( a ◦ 1 b ) ◦ 2 c ≡ ( a ◦ 2 b ) ◦ 2 c. If we write a ⊣ b for a ◦ 1 b and a ⊢ b for a ◦ 2 b then we obtain Definition 2. Example 5. The defining iden tities for Leibniz a lgebras (Lie dialgebr as) can b e obtained by applying the K P alg orithm to the defining identities fo r Lie algebr a s: anticomm utativity (in its multilinear form) and the J acobi identit y , [ a, b ] + [ b, a ] ≡ 0 , [[ a, b ] , c ] + [[ b, c ] , a ] + [[ c, a ] , b ] ≡ 0 . Part 1 of the algorithm pro duce s the following fiv e identities: [ a, b ] 1 + [ b, a ] 2 ≡ 0 , [[ a, b ] 1 , c ] 1 + [[ b, c ] 2 , a ] 2 + [[ c, a ] 2 , b ] 1 ≡ 0 , [ a, b ] 2 + [ b, a ] 1 ≡ 0 , [[ a, b ] 2 , c ] 1 + [[ b, c ] 1 , a ] 1 + [[ c, a ] 2 , b ] 2 ≡ 0 , [[ a, b ] 2 , c ] 2 + [[ b, c ] 2 , a ] 1 + [[ c, a ] 1 , b ] 1 ≡ 0 . The t wo iden tities of degree 2 a re bo th equiv a lent to [ a, b ] 2 ≡ − [ b, a ] 1 , so the second op eration is supe rfluous. E liminating the s e cond o pe r ation from the three identit ies of degr e e 3 shows tha t each of them is equiv a le n t to the identit y [[ a, b ] 1 , c ] 1 + [ a, [ c , b ] 1 ] 1 − [[ a, c ] 1 , b ] 1 ≡ 0 . If w e w r ite h a, b i = [ a, b ] 1 then w e obtain an iden tity equiv a lent to (1). Part 2 of the K P algor ithm pro duces the following tw o identities: [ a, [ b, c ] 1 ] 1 ≡ [ a, [ b, c ] 2 ] 1 , [[ a, b ] 1 , c ] 2 ≡ [[ a, b ] 2 , c ] 2 . Eliminating the seco nd op era tion reduces thes e to right anticomm uta tivity: (4) h a, h b, c ii + h a, h c, b ii ≡ 0 . 4 MURRA Y R. BREMNER AND JUANA S ´ ANCHEZ-OR TE GA Setting b = c in (1 ) g ives h a, h b, b ii ≡ 0, and the linearizatio n of this is (4 ). W e conclude that the output of the KP algorithm is equiv a lent to identit y (1). 3. Leibniz triple systems Definition 6. A Lie triple system is a vector space T with a tr iline a r op e ration T × T × T → T , deno ted ( a, b, c ) 7→ [ a, b, c ], satisfying these identities: [ a, b, c ] + [ b, a, c ] ≡ 0 , (L1) [ a, b, c ] + [ b, c, a ] + [ c, a, b ] ≡ 0 , (L2) [ a, b, [ c, d, e ]] − [[ a, b, c ] , d, e ] − [ c, [ a, b, d ] , e ] − [ c, d, [ a, b, e ]] ≡ 0 . (L3) W e a pply Part 1 of the KP a lgorithm to (L1), (L2), (L3) and o btain 11 identities: [ a, b, c ] 1 + [ b, a, c ] 2 ≡ 0 , (5) [ a, b, c ] 2 + [ b, a, c ] 1 ≡ 0 , (6) [ a, b, c ] 3 + [ b, a, c ] 3 ≡ 0 , (7) [ a, b, c ] 1 + [ b, c, a ] 3 + [ c, a , b ] 2 ≡ 0 , (8) [ a, b, c ] 2 + [ b, c, a ] 1 + [ c, a , b ] 3 ≡ 0 , (9) [ a, b, c ] 3 + [ b, c, a ] 2 + [ c, a , b ] 1 ≡ 0 , (10) [ a, b, [ c, d, e ] 1 ] 1 − [[ a, b , c ] 1 , d, e ] 1 − [ c, [ a , b, d ] 1 , e ] 2 − [ c, d, [ a, b, e ] 1 ] 3 ≡ 0 , (11) [ a, b, [ c, d, e ] 1 ] 2 − [[ a, b , c ] 2 , d, e ] 1 − [ c, [ a , b, d ] 2 , e ] 2 − [ c, d, [ a, b, e ] 2 ] 3 ≡ 0 , (12) [ a, b, [ c, d, e ] 1 ] 3 − [[ a, b , c ] 3 , d, e ] 1 − [ c, [ a , b, d ] 1 , e ] 1 − [ c, d, [ a, b, e ] 1 ] 1 ≡ 0 , (13) [ a, b, [ c, d, e ] 2 ] 3 − [[ a, b , c ] 3 , d, e ] 2 − [ c, [ a , b, d ] 3 , e ] 2 − [ c, d, [ a, b, e ] 1 ] 2 ≡ 0 , (14) [ a, b, [ c, d, e ] 3 ] 3 − [[ a, b , c ] 3 , d, e ] 3 − [ c, [ a , b, d ] 3 , e ] 3 − [ c, d, [ a, b, e ] 3 ] 3 ≡ 0 . (15) Ident ities (5) and (6) a re b oth equiv ale nt to (16) [ a, b, c ] 2 ≡ − [ b, a, c ] 1 , which shows that the seco nd oper a tion is s uper fluous. Iden tities (8), (9), (1 0) are equiv alent, and applying (16) to eliminate the second op eration we obtain (17) [ a, b, c ] 3 ≡ − [ b, c, a ] 2 − [ c, a , b ] 1 ≡ [ c, b, a ] 1 − [ c, a , b ] 1 , which shows that the third op eration is sup erfluous. Applying (16) to identities (11)–(15) to eliminate the s econd op er ation we obtain [ a, b, [ c, d, e ] 1 ] 1 − [[ a, b , c ] 1 , d, e ] 1 + [[ a, b , d ] 1 , c, e ] 1 − [ c, d, [ a, b, e ] 1 ] 3 ≡ 0 , (18) − [ b, a, [ c, d, e ] 1 ] 1 + [[ b, a, c ] 1 , d, e ] 1 − [[ b, a, d ] 1 , c, e ] 1 + [ c, d, [ b, a, e ] 1 ] 3 ≡ 0 , (19) [ a, b, [ c, d, e ] 1 ] 3 − [[ a, b , c ] 3 , d, e ] 1 − [ c, [ a , b, d ] 1 , e ] 1 − [ c, d, [ a, b, e ] 1 ] 1 ≡ 0 , (20) − [ a, b, [ d, c, e ] 1 ] 3 + [ d, [ a, b, c ] 3 , e ] 1 + [[ a, b , d ] 3 , c, e ] 1 − [ d, c, [ a, b, e ] 1 ] 1 ≡ 0 , (21) [ a, b, [ c, d, e ] 3 ] 3 − [[ a, b , c ] 3 , d, e ] 3 − [ c, [ a , b, d ] 3 , e ] 3 − [ c, d, [ a, b, e ] 3 ] 3 ≡ 0 . (22) Ident ities (18) and (19) are eq uiv a lent, as ar e (20) and (2 1). Applying (17) to ident ities (18), (20), (22) to elimina te the third op eration we o btain [ a, b, [ c, d, e ] 1 ] 1 − [[ a, b , c ] 1 , d, e ] 1 + [[ a, b , d ] 1 , c, e ] 1 − [[ a, b , e ] 1 , d, c ] 1 + [[ a, b , e ] 1 , c, d ] 1 ≡ 0 , [[ c, d, e ] 1 , b, a ] 1 − [[ c, d, e ] 1 , a, b ] 1 − [[ c, b , a ] 1 , d, e ] 1 + [[ c, a , b ] 1 , d, e ] 1 LEIBNIZ TRIPLE SYS TEMS 5 − [ c, [ a , b, d ] 1 , e ] 1 − [ c, d, [ a, b, e ] 1 ] 1 ≡ 0 , [[ e, d, c ] 1 , b, a ] 1 − [[ e, c, d ] 1 , b, a ] 1 − [[ e, d, c ] 1 , a, b ] 1 + [[ e, c, d ] 1 , a, b ] 1 − [ e, d, [ c, b, a ] 1 ] 1 + [ e, d, [ c, a, b ] 1 ] 1 + [ e, [ c, b, a ] 1 , d ] 1 − [ e, [ c, a, b ] 1 , d ] 1 − [ e, [ d, b, a ] 1 , c ] 1 + [ e, [ d, a, b ] 1 , c ] 1 + [ e, c, [ d, b, a ] 1 ] 1 − [ e, c, [ d, a, b ] 1 ] 1 − [[ e, b, a ] 1 , d, c ] 1 + [[ e, a, b ] 1 , d, c ] 1 + [[ e, b, a ] 1 , c, d ] 1 − [[ e, a, b ] 1 , c, d ] 1 ≡ 0 . If we write h a, b, c i for [ a, b , c ] 1 then we obtain the identit ies h a, b, h c, d, e ii − hh a, b, c i , d, e i + hh a, b, d i , c, e i − hh a, b, e i , d, c i (23) + hh a, b, e i , c, d i ≡ 0 , hh c, d, e i , b, a i − hh c, d, e i , a, b i − hh c, b, a i , d, e i + hh c, a, b i , d, e i (24) − h c, h a, b, d i , e i − h c, d, h a, b , e ii ≡ 0 , hh e, d, c i , b, a i − hh e, c, d i , b, a i − hh e, d, c i , a, b i + hh e, c, d i , a, b i (25) − h e, d, h c, b, a ii + h e , d, h c, a, b ii + h e , h c, b, a i , d i − h e , h c, a , b i , d i − h e, h d, b, a i , c i + h e, h d, a, b i , c i + h e , c, h d, b, a ii − h e, c, h d, a, b ii − hh e, b , a i , d, c i + hh e, a, b i , d, c i + hh e, b, a i , c, d i − hh e, a, b i , c, d i ≡ 0 . W e apply Part 2 of the KP algo rithm and obtain 12 identities: [ a, [ b, c, d ] 1 , e ] 1 ≡ [ a, [ b, c, d ] 2 , e ] 1 , (26) [ a, [ b, c, d ] 1 , e ] 1 ≡ [ a, [ b, c, d ] 3 , e ] 1 , (27) [ a, b, [ c, d, e ] 1 ] 1 ≡ [ a, b, [ c, d, e ] 2 ] 1 , (28) [ a, b, [ c, d, e ] 1 ] 1 ≡ [ a, b, [ c, d, e ] 3 ] 1 , (29) [[ a, b, c ] 1 , d, e ] 2 ≡ [[ a, b, c ] 2 , d, e ] 2 , (30) [[ a, b, c ] 1 , d, e ] 2 ≡ [[ a, b, c ] 3 , d, e ] 2 , (31) [ a, b, [ c, d, e ] 1 ] 2 ≡ [ a, b, [ c, d, e ] 2 ] 2 , (32) [ a, b, [ c, d, e ] 1 ] 2 ≡ [ a, b, [ c, d, e ] 3 ] 2 , (33) [[ a, b, c ] 1 , d, e ] 3 ≡ [[ a, b, c ] 2 , d, e ] 3 , (34) [[ a, b, c ] 1 , d, e ] 3 ≡ [[ a, b, c ] 3 , d, e ] 3 , (35) [ a, [ b, c, d ] 1 , e ] 3 ≡ [ a, [ b, c, d ] 2 , e ] 3 , (36) [ a, [ b, c, d ] 1 , e ] 3 ≡ [ a, [ b, c, d ] 3 , e ] 3 . (37) Applying (16) and (17) to identities (26)–(3 7) to eliminate the se c ond and third op erations we obtain h a, h b, c, d i , e i ≡ −h a, h c, b, d i , e i , (38) h a, h b, c, d i , e i ≡ h a, h d, c, b i , e i − h a, h d, b, c i , e i , (39) h a, b, h c, d, e ii ≡ −h a, b , h d, c, e ii , (40) h a, b, h c, d, e ii ≡ h a, b, h e, d, c ii − h a, b, h e, c, d ii , (41) − h d, h a, b, c i , e i ≡ h d, h b, a, c i , e i , (42) − h d, h a, b, c i , e i ≡ −h d, h c, b, a i , e i + h d, h c, a, b i , e i , (43) − h b, a , h c, d, e ii ≡ h b, a, h d, c, e ii , (44) − h b, a , h c, d, e ii ≡ −h b, a, h e, d, c ii + h b, a, h e, c, d ii , (45) 6 MURRA Y R. BREMNER AND JUANA S ´ ANCHEZ-OR TE GA h e, d, h a, b, c ii − h e, h a, b , c i , d i ≡ −h e, d, h b, a, c ii + h e, h b, a, c i , d i , (46) h e, d, h a, b, c ii − h e, h a, b , c i , d i ≡ (47) h e, d, h c, b, a ii − h e, h c, b , a i , d i − h e, d, h c, a, b ii + h e, h c, a, b i , d i , h e, h b, c, d i , a i − h e, a, h b , c, d ii ≡ −h e, h c, b, d i , a i + h e, a, h c, b, d ii , (48) h e, h b, c, d i , a i − h e, a, h b , c, d ii ≡ (49) h e, h d, c, b i , a i − h e, a, h d, c, b ii − h e, h d, b , c i , a i + h e, a, h d, b, c ii . W e rewrite (38) and use it to rewrite (39) a s follows, h a, h b, c, d i , e i + h a, h c, b, d i , e i ≡ 0 , (50) h a, h b, c, d i , e i + h a, h c, d, b i , e i + h a, h d, b , c i , e i ≡ 0 . (51) Similarly (40) a nd (4 1) b ecome, h a, b, h c, d, e ii + h a, b, h d, c, e ii ≡ 0 , (52) h a, b, h c, d, e ii + h a, b, h d, e, c ii + h a, b, h e, c, d ii ≡ 0 . (53) W e r emark that (50)–(53) show that the inner triple in a mono mial of the second or third as s o ciation types, h− , h− , − , − i , −i and h− , − , h− , − , −ii , ha s pr op erties analogo us to identit ies (L1) and (L2). It is clear that (42)–(45) are e quiv alent to (50)–(53). W e note that (52)–(53) are immediate consequences of (23 ). W e hav e reduced the output of the K P a lg orithm to identities (23)–(25) and (50)–(51). W e show that (25) is redundant. Applying (50)– (5 3) to ident ity (25) we obtain  hh e, d, c i , b, a i − hh e, d, c i , a, b i − hh e, b, a i , d, c i + hh e, a, b i , d, c i − h e, h a, b, d i , c i − h e, d, h a, b, c ii  −  hh e, c, d i , b, a i − hh e, c, d i , a, b i − hh e, b, a i , c, d i + hh e, a, b i , c, d i − h e, h a, b, c i , d i − h e, c, h a, b, d ii  ≡ 0 , which follows fro m (24). This completes the pr o of of the following result. Theorem 7. Applying the KP algorithm to Lie triple s yst ems pr o duc es the variety of ternary alge br as with a triline ar op er ation h− , − , − i satisfying these identities: h a, h b, c, d i , e i + h a, h c, b, d i , e i ≡ 0 , (L TS1) h a, h b, c, d i , e i + h a, h c, d, b i , e i + h a, h d, b , c i , e i ≡ 0 , (L TS2) h a, b, h c, d, e ii − hh a, b, c i , d, e i + hh a, b, d i , c, e i − hh a, b, e i , d, c i (L TS-B) + hh a, b, e i , c, d i ≡ 0 , hh c, d, e i , b, a i − hh c, d, e i , a, b i − hh c, b, a i , d, e i + hh c, a, b i , d, e i (L TS3) − h c, h a, b, d i , e i − h c, d, h a, b , e ii ≡ 0 . W e remark that (L TS-B) shows that monomials in the third ass o ciation type h− , − , h− , − , −ii can b e expr essed as linear combinations of monomials in the first asso ciatio n type hh− , − , −i , − , − i . Moreov er , in the las t four terms of (L TS-B), the sig ns a nd p ermutations o f c, d, e corresp ond to the e x pansion o f the Lie tr iple pro duct − [[ c, d ] , e ] in an ass o ciative alg e bra. W e therefore int ro duce a n identit y analogo us to (L TS-B) but for the seco nd asso cia tion type h− , h− , − , −i , −i : h a, h b, c, d i , e i − hh a, b, c i , d, e i + hh a, c, b i , d, e i + hh a, d, b i , c, e i (L TS-A) − hh a, d, c i , b, e i ≡ 0 . LEIBNIZ TRIPLE SYS TEMS 7 This new identit y expres ses monomia ls in the seco nd a sso ciation t yp e as linea r combinations o f monomials in the firs t asso ciation type; in the last four terms of (L TS-A), the sig ns and p ermutations of b, c, d corr esp ond to − [[ b , c ] , d ]. Lemma 8. (L TS-A) , (L TS-B) ar e e quivalent to (L TS1) , (L TS2) , (L TS-B) , (L TS3) . Pr o of. W e wr ite S 1 , S 2 , S 4 , S A , S B for the the left sides o f ident ities (L TS1), (L TS2), (L TS3), (L TS-A), (L TS-B) r e sp ectively . The equations S 1 ( a, b, c, d, e ) = S A ( a, b, c, d, e ) + S A ( a, c, b , d, e ) , S 2 ( a, b, c, d, e ) = S A ( a, b, c, d, e ) + S A ( a, d, b , c, e ) + S A ( a, c, d, b, e ) , S 4 ( a, b, c, d, e ) = − S A ( c, a, b , d, e ) − S B ( c, d, a, b, e ) , S A ( a, b, c, d, e ) = S 1 ( a, b, c, d, e ) + S 4 ( c, b, a, d, e ) + S B ( a, d, c, b, e ) , can b e verified by direct calcula tion.  Definition 9. A Lei bn i z triple system is a v ector spac e T with a trilinear op eration T × T × T → T denoted h− , − , −i satisfying (L TS-A) and (L TS-B): h a, h b, c, d i , e i ≡ hh a, b, c i , d, e i − hh a, c, b i , d, e i − hh a, d, b i , c, e i + hh a, d, c i , b , e i , h a, b, h c, d, e ii ≡ hh a, b, c i , d, e i − hh a, b, d i , c, e i − hh a, b, e i , c, d i + hh a, b, e i , d, c i . W e rema rk that Lemma 8 shows that Leibniz triple sy s tems ca n also b e charac- terized by the four identities of Theor em 7 . W e will use without further commen t the mos t conv enient characterization for our purp o ses. Example 10. Let T b e a Lie triple system with pro duct [ − , − , − ]. It is e a sy to chec k that T is a Leibniz triple sys tem. If a, b, c, d, e ∈ T then [ a, b, [ c, d, e ]] − [[ a, b, c ] , d, e ] + [[ a, b, d ] , c, e ] − [[ a, b, e ] , d, c ] + [[ a, b, e ] , c, d ] (L3) ≡ [ c, [ a, b, d ] , e ] + [ c, d, [ a, b, e ]] + [[ a , b, d ] , c, e ] − [[ a, b, e ] , d, c ] + [[ a, b, e ] , c, d ] (L1) ≡ [ c, d, [ a , b, e ]] + [[ a, b, e ] , c, d ] + [ d, [ a, b , e ] , c ] (L2) ≡ 0 , which pr ov es (L TS-B ), and the pro o f of (L TS-A) is simila r way . Hence a n a sso cia- tive a lgebra gives a Leibniz triple system if we set h a, b , c i = abc − bac − cab + cba . Example 11. Let A b e a differential a sso ciative alg ebra in the sense of Lo day [9]: an asso ciative alg ebra A with a pro duct a · b and a linear map d : A → A such tha t d 2 = 0 a nd d ( a · b ) = d ( a ) · b + a · d ( b ) for all a, b ∈ A . O ne endows A with a dialgebra structure b y defining a ⊣ b = a · d ( b ) and a ⊢ b = d ( a ) · b . Then A b ecomes a Leibniz triple sys tem (see Corolla ry 18) if we set h a, b, c i = a · d ( b ) · d ( c ) − d ( b ) · a · d ( c ) − d ( c ) · a · d ( b ) + d ( c ) · d ( b ) · a. 4. Opera tor identities for Leibniz triple systems In this sectio n we present a more intuitiv e formulation of the defining identit ies for Leibniz triple systems. Definition 12. Let T be a triple sys tem with pro duct {− , − , −} . F or a, b ∈ T we define tw o endomor phis ms of T as follows: L a,b ( x ) = { a, b, x } , R a,b ( x ) = { x, a, b } − { x, b , a } . 8 MURRA Y R. BREMNER AND JUANA S ´ ANCHEZ-OR TE GA Let D be an endomorphism of T . W e call D a ternary deriv ation if it satisfies D  { a, b, c }  ≡ { D ( a ) , b, c } + { a , D ( b ) , c } + { a, b, D ( c ) } . W e call D a Lie triple deriv ation if it s atisfies D  { a, b, c }  ≡ { D ( a ) , b, c } − { D ( b ) , a, c } − { D ( c ) , a, b } + { D ( c ) , b, a } . The right side has the form of the expansion of the Lie triple pro duct [[ a, b ] , c ] using the Lie brack et [ a, b ] = ab − ba ; this motiv a tes our choice of terminolo gy . Corollary 13. Every L eibniz triple system satisfies t he fol lowing identities: R a,b ( h c, d, e i ) ≡ h R a,b ( c ) , d, e i + h c, R a,b ( d ) , e i + h c, d, R a,b ( e ) i , (OP1) L a,b ( h c, d, e i ) ≡ h L a,b ( c ) , d, e i − h L a,b ( d ) , c, e i − h L a,b ( e ) , c, d i (OP2) + h L a,b ( e ) , d, c i , [ R a,b , R c,d ] ≡ R R a,b ( c ) ,d − R R a,b ( d ) ,c , (OP3) [ R c,d , L a,b ] ≡ L L a,b ( c ) ,d − L L a,b ( d ) ,c . (OP4) In p articular, R ab is a derivation and L ab is a Lie triple derivation. Pr o of. W e rewr ite (L TS3) as hh c, d, e i , a, b i − hh c, d, e i , b, a i ≡ hh c, a, b i , d, e i − hh c, b, a i , d, e i − h c, h a, b , d i , e i − h c, d, h a, b, e ii , and note that h c, h a, b , d i , e i (L TS2) ≡ −h c, h d, a , b i , e i − h c, h b, d, a i , e i (L TS1) ≡ −h c, h d, a , b i , e i + h c, h d, b, a i , e i , h c, d, h a, b, e ii (53) ≡ −h c, d, h e, a, b ii − h c, d, h b, e, a ii (52) ≡ −h c, d, h e, a, b ii + h c, d, h e, b, a ii . Combining these equations gives (OP1), and (OP 2) is eq uiv alent to (L TS-A). Iden- tit y (OP3) follows dir e c tly from (25). W e rewr ite identit y (L TS-B) as hh a, b, c i , d, e i − hh a, b, d i , c, e i ≡ hh a, b , e i , c, d i − hh a, b, e i , d, c i + h a, b, h c, d, e ii . The left side equals L L a,b ( c ) ,d ( e ) − L L a,b ( d ) ,c ( e ) . Applying (52)–(53) to h a, b, h c, d, e ii the rig ht side b ecomes hh a, b, e i , c, d i − hh a, b, e i , d, c i − h a, b, h e, c, d ii + h a, b, h e, d, c ii , which eq uals ( R c,d ◦ L a,b )( e ) − ( L a,b ◦ R c,d )( e ) = [ R c,d , L a,b ]( e ) . This proves (OP 4).  LEIBNIZ TRIPLE SYS TEMS 9 5. Leibniz triple systems fr om Leibniz algebras In this section w e r ecall from Lo day and Pir ashvili [10] (see als o Lo day [9]) the structure of free Leibniz algebra s. F rom this we obtain a larg e class of examples of Leibniz triple systems. Let V be a vector space ov e r a field F . F o r m ≥ 1 we consider the m -th tensor power V ⊗ m = V ⊗ F · · · ⊗ F V ( m factors ) which is spanned by simple tensor s v 1 ⊗ · · · ⊗ v m . The (non-unital) tensor alge bra of V is A ( V ) = M m ≥ 1 V ⊗ m , with a s so ciative multiplication defined on simple tens ors b y conca tenation, ( v 1 ⊗ · · · ⊗ v m )( v m +1 ⊗ · · · ⊗ v m + n ) = v 1 ⊗ · · · ⊗ v m + n , and extended bilinearly . W e make A ( V ) in to a Leibniz algebra in whic h the pro duct, denoted x · y , is defined to b e the unique Leibniz pro duct for w hich (54) ( v 1 ⊗ · · · ⊗ v m ) · v m +1 = v 1 ⊗ · · · ⊗ v m ⊗ v m +1 . That is, w e define x · y inductively o n the de g ree n of y , using (54) for n = 1 and the following equation for n ≥ 2: ( v 1 ⊗ · · · ⊗ v m ) · ( v m +1 ⊗ · · · ⊗ v m + n ) = (55) v 1 ⊗ · · · ⊗ v m + n − ( v 1 ⊗ · · · ⊗ v m ⊗ v m + n ) · ( v m +1 ⊗ · · · ⊗ v m + n − 1 ) . If we write x = v 1 ⊗ · · · ⊗ v m , y = v m +1 ⊗ · · · ⊗ v m + n − 1 , z = v m + n . then (55) express es the Leibniz identit y in the form x · ( y · z ) = ( x · y ) · z − ( x · z ) · y . This inductive definition shows that simple tensor s corresp o nd to left-nor malized Leibniz pro ducts: v 1 ⊗ v 2 · · · ⊗ v m = (— (( v 1 · v 2 ) · v 3 ) — · v m − 1 ) · v m . W e may therefore o mit the tensor symbols, since we only need one a sso ciation type in each degree. Ro ughly sp eaking, rig ht (left) multip lication b y an element of V makes the left (right) factor an a s so ciative pro duct (left-normalized Lie pro duct). Example 14. F or a, b, c, d ∈ V , we hav e a · b = ab, ab · c = abc, a · bc = abc − acb , abc · d = ab cd, ab · cd = abcd − a b dc, a · bcd = abc d − acbd − adbc − adcb. Definition 15. W e write A ( V ) L for the v ector space A ( V ) with the Leibniz pr o duct defined by equation (54). Theorem 16. (Loday and Pirashvili [10]) The Le ibniz algebr a A ( V ) L is t he fr e e L eibniz algebr a on t he ve ctor sp ac e V . The iterated Lie brack et [[ − , − ] , − ] in a Lie a lgebra s atisfies the defining iden tities for Lie triple systems. An analogo us result ho lds in the L e ibniz s e tting. Prop ositi o n 17. Any subsp ac e of a L eibniz algebr a with pr o duct h− , − i which is close d under the triline ar op er ation hh− , −i , −i is a Le ibniz triple system. 10 MURRA Y R. BREMNER AND JUANA S ´ ANCHEZ-OR TE GA Pr o of. It suffices to v erify that identities (L TS-A) and (L TS-B) are sa tisfied b y the op eration h a, b, c i = hh a, b i , c i . Since every Leibniz algebra is a quotient o f a free Leibniz algebr a, we only need to pr ov e the claim for free L eibniz algebr as. The existence of a normal form for bas is monomials in the fre e L e ibniz algebr a implies that we ca n reduce the pro of to a str a ightforw ard computatio n. W e first no te that any ternary monomial in degree 5 in the firs t asso ciation type is a lready in normal form as a monomial in the free Leibniz a lgebra: hh a, b, c i , d, e i = ((( ab ) c ) d ) e. Applying the Leibniz identit y , for (L TS-A) and (L TS-B) we hav e h a, b, h c, d, e ii = ( ab )(( cd ) e ) = (( ab )( cd )) e − (( ab ) e )( cd ) = ((( ab ) c ) d ) e − ((( ab ) d ) c ) e − ((( ab ) e ) c ) d + ((( ab ) e ) d ) c = hh a, b, c i , d, e i − hh a, b, d i , c, e i − hh a, b, e i , c, d i + hh a, b, e i , d, c i , h a, h b, c, d i , e i = ( a (( bc ) d )) e = (( a ( bc )) d ) e − (( ad )( bc )) e = ((( ab ) c ) d ) e − ((( ac ) b ) d ) e − ((( ad ) b ) c ) e + ((( ad ) c ) b ) e = hh a, b, c i , d, e i − hh a, c, b i , d, e i − hh a, d, b i , c, e i + hh a, d, c i , b, e i . This completes the pro of.  Corollary 18. A subsp ac e of an asso ciative dialg ebr a is a L eibniz t riple system if it is close d under the triline ar op er ation h a, b, c i = a ⊣ b ⊣ c − b ⊢ a ⊣ c − c ⊢ a ⊣ b + c ⊢ b ⊢ a. Pr o of. Any such subspace is a Leibniz algebra with h a, b i = a ⊣ b − b ⊢ a .  6. Leibniz triple systems fr om Jordan dialgebras In this section we prov e that the p er mu ted ass o ciator in a Jordan dialgebra satifies the defining identities for Leibniz triple systems. Thus any subspace of a Jorda n dialgebra which is c lo sed under the a sso ciator b ecomes a Leibniz triple system. This generaliz es the cla ssical res ult that the ass o ciator in a Jorda n algebra satisfies the defining identities for Lie triple systems. Definition 19. (Koles niko v [6], V el´ asquez a nd F elip e [14], Br emner [1]) Over a field of characteristic not 2 or 3, a (righ t) Jordan dialgebra is a vector space with a bilinear op eration ab , s atisfying these p olynomial identities: righ t comm utativit y: a ( bc ) ≡ a ( cb ) , righ t Jordan iden tit y: ( ba 2 ) a ≡ ( ba ) a 2 , righ t Osb orn identit y: ( a, b, c 2 ) ≡ 2( ac, b, c ) . (Algebras satis fying the last identit y were sy stematically studied by Osb or n [11].) Lemma 20. The line arize d forms of the right Jor dan and Osb orn identities ar e RJ ( a, b, c, d ) = (56) ( d ( ab )) c + ( d ( ac )) b + ( d ( bc )) a − ( da )( bc ) − ( db )( ac ) − ( dc )( ab ) , RO ( a, b, c, d ) = (57) (( ac ) b ) d + (( ad ) b ) c − ( ab )( cd ) − ( ac )( bd ) − ( ad )( bc ) + a (( cd ) b ) . LEIBNIZ TRIPLE SYS TEMS 11 Pr o of. F o r a genera l discussio n o f linea rization of polyno mial identities in nonass o- ciative algebras, see Chapter 1 o f Zhevlakov et al. [15].  F ollowing Bremner and Peresi [2], we use the following order on the asso ciation t yp es in degre e 5 in a free right comm utative algebra: 1 : ((( ab ) c ) d ) e 2 : (( a ( bc )) d ) e 3 : (( ab )( cd )) e 4 : ( a (( bc ) d )) e 5 : (( ab ) c )( de ) 6 : ( a ( b c ))( de ) 7 : ( ab )(( cd ) e ) 8 : a ((( bc ) d ) e ) 9 : a (( bc )( de )) The consequence s of r ig ht commutativit y in deg ree 5 ca n b e e xpressed by the fol- lowing sy mmetries of the asso c iation types: 1 : ((( ab ) c ) d ) e has no symmetries 2 : (( a ( bc )) d ) e = (( a ( cb )) d ) e 3 : (( ab )( cd )) e = (( ab )( dc )) e 4 : ( a (( bc ) d )) e = ( a (( cb ) d )) e 5 : (( ab ) c )( de ) = (( ab ) c )( ed ) 6 : ( a ( bc ))( de ) = ( a ( cb ))( de ) = ( a ( bc ))( ed ) 7 : ( ab )(( cd ) e ) = ( ab )(( dc ) e ) 8 : a ((( bc ) d ) e ) = a ((( cb ) d ) e ) 9 : a (( bc )( de )) = a (( cb )( de )) = a (( bc )( ed )) = a (( de )( bc )) Theorem 21. L et L b e a subsp ac e of a Jor dan dialg ebr a J which is close d under the asso ciator ( a, b, c ) = ( ab ) c − a ( bc ) . Then L is a L eibniz triple system with the triline ar op er ation define d to b e the p ermu te d asso ciator h a, b, c i = ( a, c, b ) . Pr o of. It suffices to verify that iden tities (L TS1), (L TS2 ), (L TS-B), (L TS3 ) are satisfied by the per mut ed as so ciator . W e fir st consider (L TS1 ) and (L TS2); we show in fact that these iden tities fo llow from right comm utativit y , without using the Jordan and Osb or n identities. In (L TS1) and (L TS2) we replace each o ccurrence of the op eratio n h a, b, c i by the p er mu ted a sso ciator ( a, c, b ): ( a, e, ( b, d, c )) + ( a, e, ( c, d, b )) ≡ 0 , ( a, e, ( b, d, c )) + ( a, e, ( c, b, d )) + ( a, e, ( d, c, b )) ≡ 0 . Expanding the asso ciators gives ( ae )(( bd ) c ) − ( ae )( b ( dc )) − a ( e (( bd ) c )) + a ( e ( b ( dc ))) + ( ae )(( cd ) b ) − ( ae )( c ( db )) − a ( e (( cd ) b )) + a ( e ( c ( db ))) ≡ 0 , ( ae )(( bd ) c ) − ( ae )( b ( dc )) − a ( e (( bd ) c )) + a ( e ( b ( dc ))) + ( ae )(( cb ) d ) − ( ae )( c ( bd )) − a ( e (( cb ) d )) + a ( e ( c ( bd ))) + ( ae )(( dc ) b ) − ( ae )( d ( cb )) − a ( e (( dc ) b )) + a ( e ( d ( cb ))) ≡ 0 . Both equa tio ns ar e immediate c onsequences of rig ht co mmutativit y . W e next consider identit y (L TS-B). Replacing each o ccurre nce of h a, b, c i by the per muted a sso ciator ( a, c, b ) gives ( a, ( c, e , d ) , b ) − (( a, c, b ) , e, d ) + (( a, d, b ) , e, c ) − (( a, e, b ) , c, d ) + (( a, e, b ) , d, c ) . Expanding the asso ciators pro duces the following ex pression: ( a (( ce ) d )) b − ( a ( c ( ed ))) b − a ((( ce ) d ) b ) + a (( c ( ed )) b ) − ((( ac ) b ) e ) d + (( a ( cb )) e ) d + (( ac ) b )( ed ) − ( a ( cb ))( ed ) + ((( ad ) b ) e ) c − (( a ( db )) e ) c − (( ad ) b )( ec ) + ( a ( db ))( ec ) 12 MURRA Y R. BREMNER AND JUANA S ´ ANCHEZ-OR TE GA − ((( ae ) b ) c ) d + (( a ( eb )) c ) d + (( ae ) b )( cd ) − ( a ( eb ))( cd ) + ((( ae ) b ) d ) c − (( a ( eb )) d ) c − (( ae ) b )( dc ) + ( a ( eb ))( dc ) . W e stra ighten each ter m using right co mmu tativity , a nd so rt the ter ms b y asso ci- ation type and then by lex order of the p ermutation; some of the ter ms cancel: (58)          − ((( ac ) b ) e ) d + ((( ad ) b ) e ) c − ((( ae ) b ) c ) d + ((( ae ) b ) d ) c + (( a ( bc )) e ) d − (( a ( bd )) e ) c + (( a ( be )) c ) d − (( a ( be )) d ) c + ( a (( ce ) d )) b − ( a (( de ) c )) b + (( ac ) b )( de ) − (( ad ) b )( ce ) − ( a ( bc ))( de ) + ( a ( bd ))( ce ) − a ((( ce ) d ) b ) + a ((( de ) c ) b ) . Consider the following express ion, which clearly v a nishes in any Jorda n dialgebra: RJ ( ce, b, d, a ) − R J ( de, b, c, a ) + RJ ( b, c, e, a ) d − R J ( b, d, e, a ) c − RO ( a, b, ce, d ) + RO ( a, b, de, c ) − RO ( a, b, c, e ) d + R O ( a, b, d, e ) c. Expanding each term using equations (56) and (57) gives ( a (( ce ) b )) d + ( a (( ce ) d )) b + ( a ( bd ))( ce ) − ( a ( ce ))( bd ) − ( ab )(( ce ) d ) − ( ad )(( ce ) b ) − ( a (( de ) b )) c − ( a (( de ) c )) b − ( a ( bc ))( de ) + ( a ( de ))( bc ) + ( ab )(( de ) c ) + ( ac )(( de ) b ) + (( a ( bc )) e ) d + (( a ( be )) c ) d + (( a ( ce )) b ) d − (( ab )( ce )) d − (( ac )( be )) d − (( ae )( bc )) d − (( a ( bd )) e ) c − (( a ( be )) d ) c − (( a ( de )) b ) c + (( ab )( de )) c + (( ad )( be )) c + (( ae )( bd )) c − (( a ( ce )) b ) d − (( ad ) b )( ce ) + ( ab )(( ce ) d ) + ( a ( ce ))( bd ) + ( ad )( b ( ce )) − a ((( ce ) d ) b ) + (( a ( de )) b ) c + (( ac ) b )( de ) − ( ab )(( de ) c ) − ( a ( de ))( bc ) − ( ac )( b ( de )) + a ((( de ) c ) b ) − ((( ac ) b ) e ) d − ((( ae ) b ) c ) d + (( ab )( ce )) d + (( ac )( be )) d + (( ae )( bc )) d − ( a (( ce ) b )) d + ((( ad ) b ) e ) c + ((( ae ) b ) d ) c − (( ab )( de )) c − (( ad )( be )) c − (( ae )( bd )) c + ( a (( de ) b )) c. W e straighten ea ch term using right commutativit y , and sort the terms b y asso cia- tion t yp e and then by lex order of the p ermutation. Most of the terms cance l, and we o btain an expressio n identical to (58 ). W e finally cons ider (L TS3). Replacing ea ch o ccurrence of h a, b, c i by the p er- m uted a sso ciator ( a, c, b ) g ives (( c, e, d ) , a, b ) − (( c, e, d ) , b, a ) − (( c, a, b ) , e, d ) + (( c, b, a ) , e, d ) − ( c, e , ( a, d, b )) − ( c, ( a, e, b ) , d ) . Expanding the asso ciators pro duces the following ex pression: + ((( ce ) d ) a ) b − (( c ( ed )) a ) b − (( ce ) d )( ab ) + ( c ( ed ))( ab ) − ((( ce ) d ) b ) a + (( c ( ed )) b ) a + (( ce ) d )( ba ) − ( c ( ed ))( ba ) − ((( ca ) b ) e ) d + (( c ( ab )) e ) d + (( ca ) b )( ed ) − ( c ( ab ))( ed ) + ((( cb ) a ) e ) d − (( c ( ba )) e ) d − (( cb ) a )( ed ) + ( c ( ba ))( ed ) − ( ce )(( ad ) b ) + ( ce )( a ( db )) + c ( e (( ad ) b )) − c ( e ( a ( db ))) − ( c (( ae ) b )) d + ( c ( a ( eb ))) d + c ((( ae ) b ) d ) − c (( a ( eb )) d ) . LEIBNIZ TRIPLE SYS TEMS 13 W e stra ighten each ter m using right co mmu tativity , a nd so rt the ter ms b y asso ci- ation type and then by lex order of the p ermutation; some of the ter ms cancel: (59)          − ((( ca ) b ) e ) d + ((( cb ) a ) e ) d + ((( ce ) d ) a ) b − ((( ce ) d ) b ) a − (( c ( de )) a ) b + (( c ( de )) b ) a − ( c (( ae ) b )) d + ( c (( be ) a )) d + (( ca ) b )( de ) − (( cb ) a )( de ) − ( ce )(( ad ) b ) + ( ce )(( bd ) a ) + c ((( ad ) b ) e ) + c ((( ae ) b ) d ) − c ((( bd ) a ) e ) − c ((( be ) a ) d ) Consider the following express ion, which clearly v a nishes in any Jorda n dialgebra: cRJ ( a, d, e, b ) − cR J ( b, d, e, a ) + RO ( ce, a, b, d ) − RO ( ce, b, a, d ) − RO ( c, a, de, b ) + R O ( c, b, de, a ) + R O ( c, a, b, e ) d − R O ( c, b, a, e ) d. Expanding each term using equations (56) and (57) gives c (( b ( ad )) e ) + c (( b ( ae )) d ) + c (( b ( de )) a ) − c (( ba )( de )) − c (( bd )( ae )) − c (( be )( ad )) − c (( a ( bd )) e ) − c (( a ( be )) d ) − c (( a ( de )) b ) + c (( ab )( de )) + c (( ad )( be )) + c (( ae )( bd )) + ((( ce ) b ) a ) d + ((( ce ) d ) a ) b − (( ce ) a )( bd ) − (( ce ) b )( ad ) − (( ce ) d )( ab ) + ( ce )(( bd ) a ) − ((( ce ) a ) b ) d − ((( ce ) d ) b ) a + (( ce ) b )( ad ) + (( ce ) a )( bd ) + (( ce ) d )( ba ) − ( ce )(( ad ) b ) − (( c ( de )) a ) b − (( cb ) a )( de ) + ( ca )(( de ) b ) + ( c ( de ))( ab ) + ( cb )( a ( de )) − c ((( de ) b ) a ) + (( c ( de )) b ) a + (( ca ) b )( de ) − ( cb )(( de ) a ) − ( c ( de ))( ba ) − ( ca )( b ( de )) + c ((( de ) a ) b ) + ((( cb ) a ) e ) d + ((( ce ) a ) b ) d − (( ca )( be )) d − (( cb )( ae )) d − (( ce )( ab )) d + ( c (( be ) a )) d − ((( ca ) b ) e ) d − ((( ce ) b ) a ) d + (( cb )( ae )) d + (( ca )( be )) d + (( ce )( ba )) d − ( c (( ae ) b )) d. W e straighten ea ch term using right commutativit y , and sort the terms b y asso cia- tion t yp e and then by lex order of the p ermutation. Most of the terms cance l, and we o btain an expressio n identical to (59 ).  7. Universal Leibniz envelopes f o r Leibniz triple systems Suppo se that T is a Leibniz triple sys tem with pro duct h x, y , z i . W e co nstruct the fr ee Leibniz alg ebra A ( T ) L on the underlying vector space , a nd co ns ider the ideal I ( T ) ⊆ A ( T ) L generated b y the element s ( x · y ) · z − h x, y , z i , where x · y is the Leibniz pro duct in A ( T ) L . The quo tient algebra U ( T ) = A ( T ) L /I ( T ) is the universal Leibniz env elop e o f T . If a, b , c ∈ T then in U ( T ) we ha ve the iden tit y ( a · b ) · c ≡ h a, b, c i ; it follows that in U ( T ) every monomia l o f degr ee 3 or mo re is equal to a monomia l o f degree 1 or 2. F urthermor e, it is clear fr om the discussion in Section 5 that the intersection of I ( T ) with T ⊕ T ⊗ 2 is zero. Theorem 22. (a) The universal L eibniz envelop e of the L eibniz tr iple system T is the ve ct or sp ac e U ( T ) = T ⊕ ( T ⊗ T ) with the L eibniz pr o duct a · b = ab, a · b c = h a, b, c i − h a, c , b i , ab · c = h a, b, c i , ab · cd = h a, b, c i d − h a, b, d i c. (The simple tensor a ⊗ b is denote d ab .) (b) I f T has dimension n t hen U ( T ) has dimension n ( n +1) . The pr o of o f this theorem is immediate from the univ ersal proper t y of the fre e Leibniz alg ebra A ( T ) L . How ever, it is instr uctive to give a direct pro of tha t the equations o f the theorem satisfy the Leibniz iden tit y , esp e c ially in degree 5 where we nee d the defining ident ities for Leibniz triple sys tems. 14 MURRA Y R. BREMNER AND JUANA S ´ ANCHEZ-OR TE GA Degree 3. If a, b, c ∈ T then the definitions immediately give ( a · b ) · c − ( a · c ) · b − a · ( b · c ) = h a, b , c i − h a, c, b i −  h a, b, c i − h a, c, b i  = 0 . Degree 4. F or a, b, c, d ∈ T there a r e three cas e s for the first term of the Leibniz ident it y: ( ab · c ) · d, ( a · b c ) · d, ( a · b ) · cd. F or the fir st case, the definitions immediately give ( ab · c ) · d − ( ab · d ) · c − ab · cd = h a, b, c i d − h a, b, d i c − ( h a, b, c i d − h a, b, d i c ) = 0 . F or the s e cond cas e we ne e d to use the e quation h a, b, c i ≡ ( ab ) c in U ( T ): ( a · bc ) · d − ( a · d ) · bc − a · ( bc · d ) = ( h a, b, c i d − h a, c, b i d ) − ( h a, d, b i c − h a, d, c i b ) − a h b , c, d i = abcd − a c b d − adbc + adcb − a · bcd = 0 , using Ex ample 14. The third case is similar . Degree 5. If a, b, c, d, e ∈ T then a g ain there are thre e ca ses for the fir st ter m: ( ab · c d ) · e, ( ab · c ) · de, ( a · bc ) · de. F or the fir st case, we hav e ( ab · c d ) · e − ( ab · e ) · cd − ab · ( cd · e ) = ( h a, b, c i d − h a, b, d i c ) · e − h a, b, e i · cd − ab · h c, d, e i = hh a, b, c i , d, e i − hh a, b, d i , c, e i − hh a, b, e i , c, d i + hh a, b, e i , d, c i − h a, b, h c, d, e ii , which v a nishes by (L TS-B). F or the second case, we have ( ab · c ) · de − ( ab · de ) · c − ab · ( c · de ) = h a, b, c i · de − ( h a, b, d i e ) − h a, b, e i d ) · c − ab · ( h c, d, e i − h c, e, d i ) = hh a, b, c i , d, e i − hh a, b, c i , e, d i − hh a, b, d i , e, c i + hh a, b, e i , d, c i − h a, b , h c, d, e ii + h a, b, h c, e, d ii = 0 by (OP4). F or the third case, we hav e ( a · bc ) · de − ( a · de ) · b c − a · ( bc · de ) = ( h a, b, c i − h a, c, b i ) · de − ( h a, d, e i − h a, e, d i ) · bc − a · ( h b, c, d i e − h b, c, e i d ) = hh a, b, c i , d, e i − hh a, b, c i , e, d i − hh a, c, b i , d, e i + hh a, c, b i , e, d i − hh a, d, e i , b, c i + hh a, d, e i , c, b i + hh a, e, d i , b, c i − hh a, e, d i , c, b i − h a, h b, c, d i , e i + h a, e, h b, c, d ii + h a, h b, c, e i , d i − h a, d, h b, c, e ii . W e use (L TS- A) and (L TS-B ) to rewrite the la st four terms (with a sign change): h a, h b, c, d i , e i − h a, h b, c, e i , d i + h a, d, h b, c, e ii − h a, e, h b, c, d ii = hh a, b, c i , d, e i − hh a, c, b i , d, e i − hh a, d, b i , c, e i + hh a, d, c i , b, e i − hh a, b, c i , e, d i + hh a, c, b i , e, d i + hh a, e, b i , c, d i − hh a, e, c i , b, d i + hh a, d, b i , c, e i − hh a, d, c i , b, e i − hh a, d, e i , b, c i + hh a, d, e i , c, b i − hh a, e, b i , c, d i + hh a, e, c i , b, d i + hh a, e, d i , b, c i − hh a, e, d i , c, b i = hh a, b, c i , d, e i − hh a, c, b i , d, e i − hh a, b, c i , e, d i + hh a, c, b i , e, d i − hh a, d, e i , b, c i + hh a, d, e i , c, b i + hh a, e, d i , b, c i − hh a, e, d i , c, b i . LEIBNIZ TRIPLE SYS TEMS 15 This cancels with the fir st 8 terms, and co mpletes the pro of for degree 5. Degree 6. If a, b, c, d, e, f ∈ T then there is o nly one cas e: ( ab · c d ) · ef − ( ab · ef ) · cd − ab · ( cd · ef ) = hh a, b, c i , d, e i f − hh a, b, c i , d, f i e − hh a, b, d i , c, e i f + hh a, b, d i , c, f i e − hh a, b, e i , f , c i d + hh a, b, e i , f , d i c + hh a, b, f i , e , c i d − hh a, b, f i , e , d i c − h a, b , h c, d, e ii f + h a, b, f ih c, d, e i + h a, b, h c, d, f ii e − h a, b, e ih c, d, f i . Since this is an e x pression of even degree, we m ust r ewrite it entirely in ter ms o f the binary Leibniz pro duct. The first eight ter ms conv er t directly to this for m using the rule hh a, b, c i , d, e i f = (((( ab ) c ) d ) e ) f . W e rewr ite the last four terms (with a sign change) and rep eatedly apply the Leibniz identit y: h a, b, h c, d, e ii f − h a, b, h c, d, f ii e + h a, b, e ih c, d, f i − h a, b, f ih c, d, e i = (( ab )(( cd ) e )) f − (( ab )(( cd ) f )) e + (( ab ) e )(( cd ) f ) − (( ab ) f )(( cd ) e ) = (((( ab ) c ) d ) e ) f − (((( ab ) d ) c ) e ) f − (((( ab ) e ) c ) d ) f + (((( ab ) e ) d ) c ) f − (((( ab ) c ) d ) f ) e + (((( ab ) d ) c ) f ) e + (((( ab ) f ) c ) d ) e − (((( ab ) f ) d ) c ) e + (((( ab ) e ) c ) d ) f − (((( ab ) e ) d ) c ) f − (((( ab ) e ) f ) c ) d + (((( ab ) e ) f ) d ) c − (((( ab ) f ) c ) d ) e + (((( ab ) f ) d ) c ) e + (((( ab ) f ) e ) c ) d − (((( ab ) f ) e ) d ) c = (((( ab ) c ) d ) e ) f − (((( ab ) d ) c ) e ) f − (((( ab ) c ) d ) f ) e + (((( ab ) d ) c ) f ) e − (((( ab ) e ) f ) c ) d + (((( ab ) e ) f ) d ) c + (((( ab ) f ) e ) c ) d − (((( ab ) f ) e ) d ) c This cancels with the fir st 8 terms, and co mpletes the pro of. Theorem 23. Over a field of char acteristic not 2 or 3, every p olynomial iden- tity satisfie d by the iter ate d L eibniz br acket hh a, b i , c i in every L eibniz algebr a is a c onse quenc e of the defining identities for L eibniz triple systems. Pr o of. Supp ose to the co nt rary that there is a po ly nomial ident ity I = I ( a 1 , . . . , a n ) in n indeterminates, which is satisfied by the iterated Leibniz brack et, but is not a consequence o f the defining iden tities for Leibniz triple systems. Such an iden tit y I is a nonzer o element of the free L e ibniz tr iple system T = T n on n ge nera- tors a 1 , . . . , a n which is in the kernel of the natural ev alua tio n map η : T → L , h a, b, c i 7→ hh a, b i , c i , where L = L n is the free Leibniz algebra on the same n generator s. Let U ( T ) b e the universal Le ibniz env elop e o f T as cons tructed in Theorem 2 2; then there is an injective homomor phism of Leibniz triple systems ι : T → U ( T ) † where the da gger denotes the Leibniz tr iple sys tem o btained from the Leibniz algebr a U ( T ) by replacing the binary pro duct h a, b i by the iter ated Leibniz brack e t hh a, b i , c i . By the universal pr o p erty of the free Leibniz algebr a L , there is a (unique) s urjective homomo rphism φ : L → U ( T ), which is the identit y map on the gener a tors a 1 , . . . , a n , and which satisfies the condition φ † ◦ η = ι , where φ † : L † → U ( T ) † is the same as φ but reg arded as a homomor phism of Leibniz triple systems. If ker( η ) 6 = { 0 } then η is not injective, and hence ι is not injective, which is a contradiction. This prov es that such a p olyno mial identit y I cannot exist.  8. Tw o-dimensional Leibniz triple systems In this section we give s o me examples of 2-dimensio nal Leibniz triple systems, and constr uct their universal Leibniz env elop es. 16 MURRA Y R. BREMNER AND JUANA S ´ ANCHEZ-OR TE GA 8.1. Lei bniz tripl e s ys te ms. Let F b e an algebr a ically clo sed field of charac- teristic 0 , and let T b e a 2- dimensional Leibniz triple system with basis { x, y } and pro duct h− , − , −i . The system T has the following structure co nstants where α ij k , β ij k ∈ F : h x, x, x i = α 111 x + β 111 y , h x, x, y i = α 112 x + β 112 y , h x, y , x i = α 121 x + β 121 y , h x, y , y i = α 122 x + β 122 y , h y , x, x i = α 211 x + β 211 y , h y , x, y i = α 212 x + β 212 y , h y , y , x i = α 221 x + β 221 y , h y , y , y i = α 222 x + β 222 y . Impo sing ide ntities (L TS-A) a nd (L TS- B) w e obtain a s ystem of quadr atic equations in the indeterminates α ij k , β ij k . W e prese nt five s o lutions of these equations: four isolated cases , and a one-para meter family . By Example 10, any Lie tr iple system is a Leibniz triple system. J acobson [5, page 312] prese nts the three isomo rphism classes of 2- dimensional Lie tr iple systems ov er a n a lg ebraically closed field F of ch aracter istic not tw o. Ig noring the system with zer o multiplication, we hav e tw o cas es, where zero pro ducts are omitted: h x, y , x i = y , h y , x, x i = − y . (60) h x, y , x i = 2 x, h y, x, x i = − 2 x, h x, y , y i = − 2 y , h y , x, y i = 2 y . (61) The Leibniz triple systems which are not Lie triple systems ar e the following: h x, y , y i = x, h y , y , y i = x. (62) h x, y , y i = − x, h y , y , y i = x. (63) h x, y , y i = ζ x, h y , y , y i = (1 − ζ ) x. (64) The para meter ζ can be a ny element of the field F , including 0. Conjecture 24. Over an algebr aic al ly close d field of char acteristic 0, every 2- dimensional L eibniz triple system is isomorphic to one of the systems (60) – (64) . Let us verify , fo r exa mple, that system (6 4) satisfies (L TS-B): h a, b, h c, d, e ii − hh a, b, c i , d, e i + hh a, b , d i , c, e i + hh a, b, e i , c, d i − hh a, b, e i , d, c i ≡ 0 . W e make the following substitutions: a = a 1 x + a 2 y , b = b 1 x + b 2 y , c = c 1 x + c 2 y , d = d 1 x + d 2 y , e = e 1 x + e 2 y . F or the first term of the ident ity we note that (64) implies that h c, d, e i is a m ultiple of x , and hence that h a, b , h c, d, e ii = 0. F or the s econd term we obtain h h a 1 x + a 2 y , b 1 x + b 2 y , c 1 x + c 2 y i , d 1 x + d 2 y , e 1 x + e 2 y i = h a 1 b 2 c 2 ζ x + a 2 b 2 c 2 (1 − ζ ) x, d 1 x + d 2 y , e 1 x + e 2 y i =  a 1 b 2 c 2 ζ + a 2 b 2 c 2 (1 − ζ )  d 2 e 2 ζ x = ζ  a 1 ζ + a 2 (1 − ζ )  b 2 c 2 d 2 e 2 x. F or the third, four th and fifth terms we apply appropriate pe r mutations of c, d, e and obtain the same re s ult; hence the alternating sum of these four terms is zero . LEIBNIZ TRIPLE SYS TEMS 17 8.2. Uni versal Leibni z env elop es. By Theorem 22 we know that the univ er sal Leibniz en velope U ( T ) of the Leibniz triple sys tem T ha s dimension 6 and basis { x, y , x 2 , xy , y x, y 2 } where x 2 , xy , y x , y 2 denote resp ectively x ⊗ x , x ⊗ y , y ⊗ x , y ⊗ y . W e easily obtain the following structure constants for the universal Leibniz env elop es of the 2-dimensio nal Leibniz triple sys tems . W e note that for systems (60) and (61), even though the Leibniz triple system is a Lie triple system, the universal Leibniz env elop e is not a Lie algebra , since it is not a nt icommutativ e. System (60): . x y x 2 xy y x y 2 x x 2 xy . − y y . y y x y 2 . . . . x 2 . . . . . . xy y . . y 2 − y 2 . y x − y . . − y 2 y 2 . y 2 . . . . . . System (61): . x y x 2 xy y x y 2 x x 2 xy . − 2 x 2 x . y y x y 2 . 2 y − 2 y . x 2 . . . . . . xy 2 x − 2 y . 2( xy + y x ) − 2( xy + y x ) . y x − 2 x 2 y . − 2( xy + y x ) 2( xy + y x ) . y 2 . . . . . . System (62): . x y x 2 xy y x y 2 x x 2 xy . . . . y y x y 2 . . . . x 2 . . . . . . xy . x . − x 2 x 2 . y x . . . . . . y 2 . x . − x 2 x 2 . System (63): . x y x 2 xy y x y 2 x x 2 xy . . . . y y x y 2 . . . . x 2 . . . . . . xy . − x . x 2 − x 2 . y x . . . . . . y 2 . x . − x 2 x 2 . System (64): . x y x 2 xy y x y 2 x x 2 xy . . . . y y x y 2 . . . . x 2 . . . . . . xy . ζ x . − ζ x 2 ζ x 2 . y x . . . . . . y 2 . (1 − ζ ) x . ( ζ − 1) x 2 (1 − ζ ) x 2 . 18 MURRA Y R. BREMNER AND JUANA S ´ ANCHEZ-OR TE GA Ackno wl edgements The fir st a uthor w as s upp o r ted by a Discovery Gra nt fr om NSER C, the Natural Sciences and Engineering Resea rch Council of Canada. The seco nd author was sup- po rted by the Spanish MEC and F ondos FEDER join tly through pro ject MTM201 0- 15223 , and by the J unt a de Andaluc ´ ıa (pro jects FQM-336 a nd FQM2467). She thanks the Department of Mathematics and Statistics at the Univ ersity of Sask a t- chew an for its hospitality during her visit from March to June 20 1 1. References [1] M. R. Bremn er : O n the definition of quasi-Jordan algebra. Communic ations in Algebr a 38, 12 (2010) 4695–4704 . [2] M. R. Bremner, L. A. Peresi : Sp ecial iden tities for quasi- Jordan algebras. Communic ations in Algebr a (to app ear). Preprint : arXiv: 1008.2723 v1 [math.RA] [3] M. R. Bremn er, J . S ´ anchez-Or tega : The partially alternating ternary sum i n an ass o cia- tiv e dialgebra. Journal of Physics A: Mathematic al and The or etica l 43 (2010) 455215. [4] C. Cuvier : Alg` e bres de Leibnitz: d´ efinitions, propri´ et ´ es. A nnales Scientifiques de l’ ´ Ec ole Normale Sup´ erieur e (4) 27, 1 (1994) 1–45. [5] N. Jacobson : Structur e and R epr esentations of Jor dan Algebr as . American Mathematical Society , Providence, 1968. [6] P. S. Kolesnik ov : V arieties of dialgebras and conformal algebras. Sibirsk i ˘ ı Matematicheski ˘ ı Zhurnal 49, 2 (2008) 322–33 9; translation in Sib erian Mathematic al Journal 49, 2 (2008) 257–272. [7] J.-L. Loda y : Une v er sion non comm utativ e des alg ` ebres de Lie: les alg ` ebres de Leibniz. L’Enseignement M ath ´ ematique 39, 3-4 (1993 ) 269–293. [8] J.-L. Lod a y : Alg` ebres a y an t deux op ´ erations asso ciatives: les dig` ebres. Comptes R endus de l’A c ad ´ emie des Scie nces. S´ erie I. Math´ ematique 321, 2 (1995) 141–146. [9] J.-L. Loda y : Dialgebras. Dialgebr as and R e late d O p er ads , 7–66. Lectures Notes in Mathe- matics, 1763. Springer, Berlin, 2001. [10] J.-L. Loda y , T. Pirashvili : Universal en v eloping algebras of Leibniz algebras and (co)ho- mology . Mathematische Anna len 296, 1 (1993) 139–15 8. [11] J. M. Osborn : Lie triple algebras with one generator. Mathematische Zeitschrift 110 (1969) 52–74. [12] A. P. Pozhidaev : Dialgebras and rel ated triple systems. Sib irski ˘ ı Matematicheski ˘ ı Zhurnal 49, 4 (2008) 870–885; translation in Siberian Mathematic al Journal 49, 4 (2008) 696–708. [13] A. P. Pozhidaev : 0-dialgebras with bar-unity , Rota-Baxter and 3-Leibniz algebras. Gr oups, Rings, and Gr oup Rings , 245–256. Contemp or ary Mathematics 499. American Mathematical Society , Providence, 2009. [14] R. Vel ´ asquez, R. Felipe : Quasi-Jordan algebras. Communic ations i n Algebr a 36, 4 (2008) 1580–160 2. [15] K. A. Zhevlako v, A. M. Slinko, I. P. Shest akov, A. I. Shirshov : Rings That ar e Nea rly Asso ciative . T ranslated from the Russian by Harry F. Smith. Academic Press, New Y ork, 1982. Dep ar tm ent of Ma them atic s and St a tistics, University of Saska tchew an, Canada E-mail addr ess : bremner@math. usask.ca Dep ar tm ent of Algebra, G eometr y and Topology, University of M ´ alaga, Sp ain E-mail addr ess : jsanchez@agt. cie.uma.es