Lie triple system central extensions of Lie algebras

LIE TRIPLE SYSTEM CENTRA L EXTENSIONS OF LIE ALGEBRAS R. KURDIANI Intr oduction. The presen t pap er deals with a ce rtain inte rrelationship b etw een Lie algebras, Lie triple syste ms and Leibniz algebras whic h sho ws up thro ugh the notion of univ ersal cen tral extension. Lie algebras ar e classical algebraic structures v ery widely used in mathematics and phy sics. Another, less widely used but equally classical structures are Lie triple systems (see [1, 2, 3, 4]). They are also used in differen t areas of mathematics. Leibniz algebra is y et another algebraic structure gaining increasing imp ortance. It is a s ort of “noncomm utativ e” generalization of the Lie a lg ebra structure recen tly in tro duced in [5]. Original reason f or in tro duction o f L eibniz algebras w as a new cohomolo gy theory (see [6, 7]). Later, the notion of Leibniz a lgebra found other applications in different a r eas of mathematics. It is well kno wn that on each Lie algebra one may define canonically the structure of a Lie triple system . W e use Leibn iz algebras to study Lie triple systems arising in this w a y , in a similar spirit to [8 ] where L eibniz algebras w ere applied to the study of Lie algebras. Namely , it is know n tha t eac h Lie a lgebra is also a particular case of a Leibniz algebra. Thu s sp eaking of univ ersal cen tral extensions , eac h p erf ect Lie algebra g , along w ith its univ ersal cen tral ex tension as a Lie algebra U Lie ( g ), also admits a Lie triple system universal c entr al extension U LT S ( g ) (if w e consider g as a Lie triple system) as w ell as the L eibniz universal c entr al extension U Leib ( g ) (if we consider g as a Leibniz alg ebra) . No w a nice unexp ected new fact is that the Lie triple sy stem U LT S ( g ) turns out to carry a natural Leibniz algebra structure (s ee Le mma 1). This fact allows us to us e properties of Leibniz and Lie algebras for Lie t riple system s. Namely , the main purp o se of the presen t article is to pro v e the following theorem: Theorem 1. F or a p erfe ct Lie algebr a g over a field F , w e have the i s o morphisms U LT S ( g ) ∼ = U Leib ( g ) , if cha r F = 2 , U LT S ( g ) ∼ = U Lie ( g ) , if cha r F 6 = 2 . In other wor ds, the universa l c entr al extension of g in the c ate gory of Lie triple systems is isomorphic either to the universal c entr al extensio n of g as a Lie algebr a (if cha r F 6 = 2 ) or to the unive rsal c entr al extension of g a s a L eibniz algebr a (if char F = 2 ). In the section 1 we recall some w ell kno wn facts ab out Lie a lgebras, Leibniz algebras and Lie triple systems. In particular, in 1.2 w e introduce non-ab elian tensor pro duct of Lie triple systems and in 1.3 w e give general definitions and standard fa cts ab out univ ersal central ex- tensions of Lie triple systems. The k ey lemma a b out natural Leibniz a lgebra structures on Lie triple system cen tral extensions of p erfect Lie a lgebras is prov ed in the section 2. Finally , using this lemma, w e obtain the pro of of the main theorem (Theorem 1) in section 3. Throughout this paper F is a field. All v ector spaces and homomorphisms are c onsidered o v er F if not sp ecified otherwise. 1 2 R. KURDIA NI A ckno wle dgement. The problem solved in the article w as prop osed b y T. Pirashv ili in a friendly con v ersation. 1. Definitions and Constr uctions. 1.1. Recollections. Let us b egin b y recalling some w ell kno wn definitions and facts. Namely , w e discuss tw o generalizations of the notion of a Lie algebra — the notions of a L ie triple system and a Leibniz alg ebra. W e also study a relationship b et w een these tw o generalizations. The notion of a Lie triple system is a classical one. The definition o f Lie triple systems can b e found in man y articles, but for the con v enience of the reader we recall it here. Definition 1. A Lie triple system is a v ector space L equipp ed with a ternary brac k et {− , − , −} : L ⊗ L ⊗ L → L, satisfying the follo wing { x, y , y } = 0 , { x, y , z } + { y , z , x } + { z , x, y } = 0 , {{ x, y , z } , a, b } = {{ x, a, b } , y , z } + { x, { y , a, b } , z } + { x, y , { z , a, b }} , for a ll x, y , z , a, b ∈ L . An y Lie algebra g can b e considered as a Lie triple system. Indeed, one can define a ternary brac k et {− , − , −} : g ⊗ g ⊗ g → g in the follo wing w ay: { x, y , z } = [ x, [ y , z ]] . Then g with the brack et {− , − , −} is a Lie triple system. An id e al in a Lie triple system L is a subspace L ′ suc h that the brac k et { x, y , z } ( x, y , z ∈ L ) is in L ′ whenev er at least one of the elemen ts x, y , z is in L ′ . A Lie tr iple system L is called p erfe ct if L = { L, L, L } . Leibniz algebras w ere in tro duced recen tly in [5], but they already sho w up in ma ny ar eas of mathematics. One of the asp ects of their imp ortance is the fact that t hey can b e used to study other algebraic ob jects. In this article we use Leibniz algebras to study Lie t r iple systems and in [8] they we re used to study Lie algebras. Let us recall the definition of a Leibniz algebra. Definition 2. A L eibn iz algebr a is a vector space h equipp ed with a binary brac k et [ − , − ] : h ⊗ h → h satisfying the Leibniz iden tity [ x, [ y , z ]] = [[ x, y ] , z ] − [[ x, z ] , y ] , for a ll x, y , z ∈ h . Ob viously , an y Lie algebra is a Leibniz algebra and con vers ely a Leibniz algebra g is a Lie algebra pro vided [ x, x ] = 0 for all x ∈ g . Note that, in this case, the Leibniz iden tit y is equiv alent to the Jacobi identit y [ x, [ y , z ]] + [ y , [ z , x ]] + [ z , [ x, y ]] = 0 . F or an y Leibniz algebra, one can define a ternar y brac k et a s it w as defined abov e for Lie algebras. In general, w e do not get a Lie triple system in this wa y . T o get a Lie triple system in this w a y , it is necessary and sufficien t tha t the Jacobi iden tit y b e satisfie d in the Leibniz algebra. Suc h Leibniz algebras will b e studied in what follow s. LIE TRIPLE SYSTEM CENTRAL EXTENSI ONS OF LIE ALGEBRAS 3 A b eauty o f Leibniz algebras is shown in the follo wing example. Namely , one can define a Leibniz algebra structure (but not a Lie algebra structure) on second tensor and exterior p ow ers of a Leibniz algebra or a Lie algebra. In fa ct, let g b e a L eibniz algebra (in particular a Lie algebra). The Leibniz algebra structure on g ⊗ g can b e defined by [ x ⊗ y , a ⊗ b ] = [ x, [ a, b ]] ⊗ y + x ⊗ [ y , [ a, b ]] , x, y , a, b ∈ g . The Leibniz a lgebra structure on g ∧ g is defined similarly , simply replace ⊗ by ∧ in the ab ov e form ula to get a brac k et on g ∧ g . These constructions can b e generalized to Lie triple systems. Let L b e a Lie triple system. The brac k ets on L ⊗ L and L ∧ L are giv en b y [ x ⊗ y , a ⊗ b ] = { x, a, b } ⊗ y + x ⊗ { y , a, b } , x, y , a, b ∈ L, [ x ∧ y , a ∧ b ] = { x, a, b } ∧ y + x ∧ { y , a, b } , x, y , a, b ∈ L. With these bra c kets L ⊗ L and L ∧ L are Leibniz algebras. T o go further w e need the definition of an actio n of a Leibniz algebra on a Lie triple system. W e say that a Leibniz algebra g ac ts on a Lie t riple system L if w e are given a map L ⊗ g → L, x ⊗ g 7→ x ∗ g , satisfying the follo wing relations ( x ∗ g ) ∗ h − ( x ∗ h ) ∗ g = x ∗ [ g , h ] , { x, y , z } ∗ g = { x ∗ g , y , z } + { x, y ∗ g , z } + { x, y , z ∗ g } , for a ll x, y , z ∈ L and g , h ∈ g . F or any Lie triple system L , one can define an action of the Leibniz algebra L ∧ L o n the Lie triple system L (in the ab o ve sense) b y x ∗ ( y ∧ z ) = { x, y , z } , where x, y , z ∈ L . Before moving to the non-a b elian tensor pr o duct of L ie triple systems, let us men tion an imp ortant construction whic h allows us to construct a Leibniz algebra structure o n a v ector space in some circumstances. Let g b e a Lie algebra and let M b e a g -mo dule. Assume t hat we are give n a map of g - mo dules f : M → g , where g is considered as a g -mo dule via the adjoint represen tatio n. W e call suc h a map a g -e quiva ria nt map . Then the brack et on M defined by [ m, n ] = m ∗ f ( n ) , m, n ∈ M giv es us a Leibniz algebra structure on M . 1.2. A non-abelian tensor pro duct for Lie triple systems. Let us in tro duce a non- ab elian tensor pro duct for Lie t r iple systems. This construction allo ws us to describ e univers al cen tra l extensions of Lie triple systems explicitly . Let L b e a Lie triple system. One can equip L ⊗ L ⊗ L with a ternary brac k et defined b y { x 1 ⊗ y 1 ⊗ z 1 , x 2 ⊗ y 2 ⊗ z 2 , x 3 ⊗ y 3 ⊗ z 3 } = { x 1 , y 1 , z 1 } ⊗ { x 2 , y 2 , z 2 } ⊗ { x 3 , y 3 , z 3 } , where x i , y i , z i ∈ L fo r all i = 1 , 2 , 3. Note that this brack et do es not determine a Lie triple system structure. But one can tak e a quotien t of L ⊗ L ⊗ L to g et a Lie triple system. Let I b e the subspace of L ⊗ L ⊗ L spanned b y t he elemen ts of the forms x ⊗ y ⊗ y , 4 R. KURDIA NI x ⊗ y ⊗ z + y ⊗ z ⊗ x + z ⊗ x ⊗ y , { x, a, b } ⊗ y ⊗ z + x ⊗ { y , a, b } ⊗ z + x ⊗ y ⊗ { z , a, b } − { x, y , z } ⊗ a ⊗ b, where x, y , z , a, b ∈ L . One can see immediately that I is an ideal of L ⊗ L ⊗ L and consequen tly the brac k et introduced ab o v e determines a brack et on the quotient L ⊗ L ⊗ L/I . One can straigh tforw ardly c hec k that the quotien t with this brac k et is a Lie triple system. W e call this Lie triple system the no n-ab elian tensor cub e of the Lie triple system L and denote it b y L ∗ L ∗ L . Note that, the origina l brack et of L determines a map of Lie triple systems b : L ∗ L ∗ L → L. W e define t he first homolog y group of L (with tr ivial co efficien ts) as the cok ernel of the map b and t he second homolo g y group of L (again with trivial co efficien ts) as the ke rnel of the map b . So, we hav e H 1 ( L ) = Cok er ( b ) , H 2 ( L ) = Ker ( b ) . Explicitly H 1 ( L ) = L/ { L, L, L } , H 2 ( L ) = Ker ( {− , − , −} ) /I , where {− , − , −} : L ⊗ L ⊗ L → L is the original brack et o f L a nd I is the subspace of L ⊗ L ⊗ L defined ab o v e. 1.3. Univ ersal cen t ral extensions of Lie triple systems. Next, let us study the univ ersal cen tral extension of a Lie triple system. Let us start with the definition of cen tral extensions of Lie triple systems . Definition 3. A c entr al extension of a Lie triple system g is a short exact sequence o f Lie triple systems 0 → K → L → g → 0 , where { L, L, K } = 0. Ob viously , in a cen tral extension w e ha v e { L, K , L } = { K , L, L } = 0 . In a usual w a y one can define the univ ersal cen tral extension of a Lie triple system a nd pro v e the f ollo wing Prop osition 1. L et L b e a Lie triple system. The universal c entr al e x tension of L exists if and only if L is p erfe ct. An explicit construction o f the univ ersal cen tral extension is obt a ined via the non-ab elian tensor pro duct whic h w as intro duced in the previous subsection. Let L b e a p erfect Lie triple system. Then b : L ∗ L ∗ L → L is the univ ersal cen tral extension of L . This construction sho ws tha t the k ernel of the univ ersal cen tral extension of L is the second homology group with trivial co efficien ts H 2 ( L ). LIE TRIPLE SYSTEM CENTRAL EXTENSI ONS OF LIE ALGEBRAS 5 One can thus consider a p erfect Lie algebra g as a Lie triple system o r as a Leibniz algebra and construct the appro priate univ ersal cen tral extensions. One obta ins three univ ersal central extensions for each p erfect Lie algebra g . Namely , let (1) 0 → K Leib → U Leib → g → 0 b e the univers al cen tral extension of Leibniz algebra g , (2) 0 → K LT S → U LT S → g → 0 b e the univers al cen tral extension of Lie triple system g and (3) 0 → K Lie → U Lie → g → 0 b e the univers al cen tral extension of Lie algebra g . It is obvious that the univ ersal cen tral extension of the Lie alg ebra g is a quotient of the univ ersal cen t ral extension o f the Leibniz algebra (or the Lie triple system) g . There is ho wev er further unexp ected relationship b et w een these univers al cen tra l extensions, which is the sub j ect of the next section. 2. Central Extensions of Lie Algebras Considere d as Lie Trip le Sys tems. In this section w e w or k with cen tral extensions of p erfect Lie algebras considered as Lie t r iple systems . Here w e prov e a lemma whic h is the k ey to our main result. Namely , surprisingly enough, one can construct a Leibniz algebra structure on the Lie t r iple system univers al cen tral extension of a p erfect Lie alg ebra g view ed as a Lie triple system and consider it as a cen tral extension of Leibniz algebras (see Lemma 1 b elo w). So, the univ ersal cen tral extension of the Lie triple system g is a quotien t of the univ ersal cen tra l extension of the Leibniz a lgebra g . This leads us to the description of the univ ersal cen tral extension o f the Lie triple system g via the univ ersal cen tral extension of the Leibniz algebra g . But the univ ersal cen t ral extensions of Leibniz algebras are studied ve ry w ell a nd thus w e can use this information. The precise form ulation of these fa cts are given in the follow ing lemma, whic h a s a corollary will yield our main theorem. Lemma 1. L et g b e a p erfe ct Lie alge br a and let 0 → K → L → g → 0 b e a c entr al extension of the Lie triple system g . Then ther e is a L eibniz a l g ebr a structur e on L [ − , − ] : L ⊗ L → L, such that the original Lie triple s ystem structur e on L is give n by the L eibniz br acket { x, y , z } = [ x, [ y , z ]] , ∀ x, y , z ∈ L. Pr o of. W e ha v e to construct a Leibniz algebra structure on the Lie t riple system L . First we construct a g -mo dule structure o n L , suc h that the map L → g is a g -equiv arian t map. Then the desired Leibniz a lg ebra structure can b e constructed as it w as describ ed in Section 1. It is we ll kno wn that L ∧ L is a Leibniz algebra and acts on L (see Section 1 ) . Since g is a p erfect Lie a lgebra, it can b e considered as a homomorphic imag e of the comp osition map L ∧ L → g ∧ g → g . Let us denote by Z the k ernel of this map. W e w a n t to sho w that Z acts trivially on L . One can easily v erify that the following statemen ts hold: (1) for an y elemen ts z ∈ Z and a ∈ L w e ha v e a ∗ z ∈ K , where ∗ denotes the actio n o f L ∧ L on L ; 6 R. KURDIA NI (2) a n y elemen t x ∈ L can b e represen ted as a sum of elemen ts x = k + y , where k ∈ K and y ∈ { L, L, L } ; (3) the a ction of L ∧ L (and in particular the action of Z ) on K is trivial. Th us, w e need to sho w that Z a cts trivially on { L, L, L } . This fo llo ws from the following: { a, b, c } ∗ z = { a ∗ z , b, c } + { a, b ∗ z , c } + { a, b, c ∗ z } = { k 1 , b, c } + { a, k 2 , c } + { a, b, k 3 } = 0 , where k 1 = a ∗ z , k 2 = b ∗ z , k 3 = c ∗ z ∈ K . So, the action of L ∧ L on L factors t hrough g . The desired Leibniz a lgebra structure o n L is giv en by the g -equiv arian t map L → g . The iden tity { x, y , z } = [ x, [ y , z ]] follo ws immediately from the definition of the Leibniz algebra structure on L .  Remark 1. Note t ha t the Leibniz brack et whic h app eared in Lemma 1 satisfies the Jacobi iden tity . 3. Proof of the Main Theorem Let J b e the linear subspace of U Leib generated b y the elemen ts of the form (4) [ x, [ y , z ]] + [ z , [ x, y ]] + [ y , [ z , x ]] , x, y , z ∈ U Leib and let I b e the linear subspace of U Leib generated by the elemen ts of the form [ x, y ] + [ y , x ] , x, y ∈ U Leib . It is o b vious that I and J are subspaces of K Leib and consequen t ly ideals of U Leib . Since the Jacobi iden tit y is equiv alen t to the Leibniz identit y mo dulo I and Leibniz identit y holds in U Leib , o ne can immediately conclude that J ⊂ I . More precise description is giv en by the follo wing lemma. Lemma 2. The e quation J = 2 I holds. Pr o of. The elemen ts of the form [ x, y ] + [ y , x ], x, y ∈ U Leib , are in the cen ter of U Leib since their image is 0 in g . Consequen tly we hav e [[ x, y ] , z ] = − [[ y , x ] , z ] f o r all x, y , z ∈ U Leib . Using this iden tity together with the Leibniz iden tity o ne can show the fo llo wing [ x, [ y , z ]] + [ y , [ z , x ]] + [ z , [ x, y ]] = [ x, [ y , z ]] + [[ y , z ] , x ] − [[ y , x ] , z ] + [ z , [ x, y ]] = [ x, [ y , z ]] + [[ y , z ] , x ] + [[ x, y ] , z ] + [ z , [ x, y ]] = [ x, [ y , z ]] + [[ y , z ] , x ] + [[ x, z ] , y ] + [ x, [ y , z ]] + [ z , [ x, y ]] = 2[ x, [ y , z ]] + [[ y , z ] , x ] + [[ x, z ] , y ] + [ z , [ x, y ]] = 2[ x, [ y , z ]] + [[ y , z ] , x ] − [[ z , x ] , y ] + [ z , [ x, y ]] = 2[ x, [ y , z ]] + [[ y , z ] , x ] − [[ z , y ] , x ] = 2[ x, [ y , z ]] + [[ y , z ] , x ] + [[ y , z ] , x ] = 2[ x, [ y , z ]] + 2[[ y , z ] , x ] = 2([ x, [ y , z ]] + [[ y , z ] , x ]) . This pro v es the lemma since U Leib is a p erfect Leibniz algebra.  LIE TRIPLE SYSTEM CENTRAL EXTENSI ONS OF LIE ALGEBRAS 7 Corollary 1. We have J = 0 , if cha r F = 2 , J = I , if cha r F 6 = 2 , wher e F is the gr ound fie ld. By the univers al prop erty of (1) and considering (2 ) as a cen tral extension o f Leibniz a lgebras w e get a map U Leib → U LT S . Similarly we hav e maps U Leib → U Lie and U LT S → U Lie . Aga in b y t he unive rsal prop erty we can a ssem ble these maps into a comm uta tiv e diagram U Leib − → U LT S ց ւ U Lie . Let I ′ b e the imag e of I in U LT S . Prop osition 2. With the ab ove notations we h ave the fol lowing is o morphisms U LT S ∼ = U Leib /J U Lie ∼ = U Leib /I , if char F 6 = 2 , U Lie ∼ = U LT S /I ′ , if char F 6 = 2 , wher e F is the gr ound fie ld. I n p articular 0 → K Leib /J → U Leib /J → g → 0 is the univers a l c entr al extension of the Lie triple system g . Pr o of. W e are go ing to pr ov e the first isomorphism, the other t w o can b e prov ed similarly and are left to the reader as an exercise . By Lemma 1 the extension (2 ) can b e considered as a cen tra l extension of the Leibniz algebra g . This implies existence of the unique map U Leib → U LT S o v er g . Moreo v er, the elemen ts of the fo r m (4) are in the kerne l of this map. Recall that J is an ideal of U Leib . So, the quotien t U Leib /J is a Leibniz algebra a nd the map U Leib → U LT S can b e factored as t he comp osition U Leib → U Leib /J → U LT S . On the o t her hand the ternary brack et on U Leib /J define d b y the form ula { a, b, c } = [ a, [ b, c ]] gives a Lie triple system structure and the extension 0 → K Leib /J → U Leib /J → g → 0 can b e considered a s a cen tral extension of the Lie triple system g . So, we hav e the unique map U LT S → U Leib /J . By the univ ersal prop erties w e can conclude that the maps U Leib /J → U LT S and U LT S → U Leib /J are inv erse to each other. So, w e ha v e U LT S ∼ = U Leib /J .  Com bining the prop osition ab ov e with Lemma 2 o ne immediately gets Theorem 1. 8 R. KURDIA NI Reference s [1] N. Ja c o bson, ”L ie and Jor dan triple sys tems”, Amer. J. o f Math., 7 1 (1949), 1 49-17 0 [2] W.G. Lister, ”A struc tur e theo r y of Lie tr iple systems ” , T rans. Amer. Math. So c., 72 (1952), 217-2 42 [3] O. Lo os, ”Symmetric spaces ”, Benjamin 19 69 [4] S. Helga son, ”Differential g eometry , Lie groups , and symmetric spaces” , Acad. Pr ess 19 78 [5] J.- L . Lo day , ”Une v ersion non comm utative des algebres de Lie: les algebr es de Leibniz”, Enseig n. Math. (2), 39 No . 3-4 (1 993), 269-2 93 [6] J.- L . Lo day , ”Cyclic ho mology”, Springer-V erlag 19 92 [7] J.- L . Lo day , T. Pira shvili, ”Universal env elo ping algebra s of Leibniz algebras and (co)homolo gy”, Math. Ann. 296, No .1 (1993), 139–1 58 [8] R. Kurdiani, T. Pir ashvili, ”A Leibniz algebr a str ucture on the second tensor p ow er”, J. Lie Theo ry 12, No. 2 (2002), 583–59 6