About Leibniz cohomology and deformations of Lie algebras

ABOUT LEIBNIZ COHOMOLOGY AND DEF ORMA TIONS OF LIE ALGEBRAS A. FIALOWS KI, L. MAGNIN, AND A. MAND AL Abstract. W e co mpare the second adjoin t and trivial Leibniz cohomolog y spaces of a Lie algebra to the usual ones by a very el- ement ary appr oach. The co mparison gives some conditions, which are easy to verify for a given Lie alg e bra, for deciding whether it has more Leibniz deformations than just the Lie ones. W e also give the complete description of a Leibniz (and L ie ) versal deformation of the 4 - dimensional dia mond Lie algebra , a nd study the case o f its 5-dimensiona l a na logue. 1. Introd uction Leibniz algebras, along with their Leibniz cohomologies, were in- tro duced in [8] as a non antisymm etric v ersion o f Lie alg ebras. Lie algebras are sp ecial Leibniz alg ebras, and Pirash vili introduced [17] a sp ectral sequen ce, that, when applied to Lie algebras, measures the difference b et w een the Lie algebra cohomolog y and the Le ibniz coho- mology . Now , Lie algebras ha v e deformat ions as L eibniz alg ebras and those a re piloted by the adjoin t Leibniz 2-co cycles. In the presen t pap er, w e fo cus on the second Leibniz cohomology groups H L 2 ( g , g ) , H L 2 ( g , C ) for adjo in t and trivial represe n tations of a complex Lie al- gebra g . W e adopt a ve ry elemen tary approac h, not resorting to the Pirash vili s equence, to c ompare H L 2 ( g , g ) and H L 2 ( g , C ) to H 2 ( g , g ) and H 2 ( g , C ) resp ectiv ely . In bo th cases, H L 2 app ears to be the direct sum of 3 spaces: H 2 ⊕ Z L 2 0 ⊕ C where H 2 is the Lie algebra cohomol- ogy group, Z L 2 0 is the space of symmetric Leibniz-2-co cycles and C is a space o f c ouple d Leibniz-2-co cycles the nonzero elemen ts of whic h ha v e the prop ert y that their symme tric and an tisymmetric part s are not Leibniz co cycles. Our comparison giv es some useful practical infor- mation ab o ut the structure of Lie a nd Leibniz cocycles. W e a nalyse the case of Heisen berg algebras, the 4- dimensional diamond algebra and its 2000 Mathematics Subje ct Classific ation. P rimary: 17 A32, Seco ndary: 17 B56, 14D15. Key wor ds a nd phr ases. Leibniz algebra, Lie algebra , c o homolog y , versal deformation. The research of the first author was pa rtially supp orted by O TKA gr ants K 77757 and NK72523. The third author thanks the Luxembourgian NRF for suppo rt via AFR grant PDR-09- 062. 1 2 A. FIA LOWSKI, L. MAG NIN, AND A. MAN DAL 5-dimensional analo g ue. W e completely describ e a v ersal Le ibniz and Lie deformatio n of the diamond algebra. 2. Leibniz coh omology and de forma tions Recall that a (right) Leibniz algebra is an algebra g with a (non nec- essarily an tisymmetric) brack et, suc h that the righ t adjoin t op erations [ · , Z ] are required to b e deriv ations for an y Z ∈ g . In the presence of an tisymmetry , that is equiv alen t to the Jacobi identit y , henc e an y Lie algebra is a Leibniz alg ebra. The Leibniz cohomolo gy H L • ( g , g ) of a Leibniz algebra is defined from the complex C L • ( g , g ) = Hom ( g ⊗• , g ) = g ⊗ ( g ∗ ) ⊗• with the Leibniz-cob oundary δ defined for ψ ∈ C L n ( g , g ) by ( δ ψ )( X 1 , X 2 , · · · , X n +1 ) = [ X 1 , ψ ( X 2 , · · · , X n +1 )] + n +1 X i =2 ( − 1) i [ ψ ( X 1 , · · · , ˆ X i , · · · , X n +1 ) , X i ] + X 1 6 i 1) with bas is ( x i ) 1 6 i 6 2 N +1 and nonzero commutation relations (with an ticomm utativit y) [ x i , x N + i ] = x 2 N +1 (1 6 i 6 N ) is I - n ull since, for an y B ∈ ( S 2 H N ∗ ) H N , B ( x i , x 2 N +1 ) = B ( x i , [ x i , x N + i ]) = − B ([ x i , x i ] , x N + i ) = 0 (similarly with x N + i instead of x i ) (1 6 i 6 N ) , and B ( x 2 N +1 , x 2 N +1 ) = B ( x 2 N +1 , [ x 1 , x N +1 ]) = − B ([ x 1 , x 2 N +1 ] , x N +1 ) = 0 . If c denotes the cen ter of g , c ⊗ ( S 2 g ∗ ) g is the space of in v ariant c -v alued symmetric bilinear map and w e denote F = I d ⊗ I : c ⊗ ( S 2 g ∗ ) g → C 3 ( g , g ) = g ⊗ V 3 g ∗ . Then Im F = c ⊗ Im I . Theorem 2. L et g b e any finite dimen s ional c omplex Lie a l g e br a and Z L 2 0 ( g , g ) (r esp. Z L 2 0 ( g , C )) the sp ac e of symmetric adjo int (r esp. triv- ial) L eibniz 2-c o cycles. (i) Z L 2 ( g , g ) /( Z 2 ( g , g ) ⊕ Z L 2 0 ( g , g )) ∼ = ( c ⊗ I m I ) ∩ B 3 ( g , g ) . (ii) Z L 2 0 ( g , g ) = c ⊗ ker I . In p articular, dim Z L 2 0 ( g , g ) = c p ( p +1) 2 wher e c = dim c and p = dim g / C 2 g = dim H 1 ( g , C ) . (iii) H L 2 ( g , g ) ∼ = H 2 ( g , g ) ⊕ ( c ⊗ ker I ) ⊕ (( c ⊗ I m I ) ∩ B 3 ( g , g )) . (iv) Z L 2 ( g , C ) /( Z 2 ( g , C ) ⊕ Z L 2 0 ( g , C )) ∼ = Im I ∩ B 3 ( g , C ) . (v) Z L 2 0 ( g , C ) = k er I . (vi) H L 2 ( g , C ) ∼ = H 2 ( g , C ) ⊕ ke r I ⊕ ( Im I ∩ B 3 ( g , C )) . ABOUT LEIBNIZ COHOMOLOGY AND DEFORMA TIONS OF LIE A LGEBRAS 5 Pr o of. (i) The Leibniz 2-co c hain space C L 2 ( g , g ) = g ⊗ ( g ∗ ) ⊗ 2 decom- p oses as  g ⊗ V 2 g ∗  ⊕ ( g ⊗ S 2 g ∗ ) with g ⊗ S 2 g ∗ the space of symmetric elemen ts in C L 2 ( g , g ) . By definition of the Leibniz cob oundary δ , one has for ψ ∈ C L 2 ( g , g ) a nd X , Y , Z ∈ g (1) ( δ ψ )( X , Y , Z ) = u + v + w + r + s + t with u = [ X, ψ ( Y , Z )] , v = [ ψ ( X , Z ) , Y ] , w = − [ ψ ( X , Y ) , Z ] , r = − ψ ([ X , Y ] , Z ) , s = ψ ( X , [ Y , Z ]) , t = ψ ([ X , Z ] , Y ) . δ coincides with the usual cob o undary op erator on g ⊗ V 2 g ∗ . Now, let ψ = ψ 1 + ψ 0 ∈ C L 2 ( g , g ) , ψ 1 ∈ g ⊗ V 2 g ∗ , ψ 0 ∈ g ⊗ S 2 g ∗ . Supp ose ψ ∈ Z L 2 ( g , g ) : δ ψ = 0 = δ ψ 1 + δ ψ 0 = dψ 1 + δ ψ 0 . Then δ ψ 0 = − dψ 1 ∈ g ⊗ V 3 g ∗ is antisy mmetric. Then p erm uting X and Y in form ula (1) for ψ 0 yields ( δ ψ 0 )( Y , X , Z ) = − v − u + w − r + t + s. As δ ψ 0 is antisy mmetric, w e get (2) w + s + t = 0 . No w, the circular p erm utation ( X , Y , Z ) in (1) f or ψ 0 yields ( δ ψ 0 )( Y , Z , X ) = − v − w + u − s − t + r . Again, by antis ymmetry , (3) v + w + s + t = 0 , i.e. ( δ ψ 0 )( X , Y , Z ) = u + r . F rom (2 ) and (3), v = 0 . Applying t wice the circular p ermutation ( X , Y , Z ) to v , w e get first w = 0 a nd then u = 0 . Hence ( δ ψ 0 )( X , Y , Z ) = r = − ψ 0 ([ X , Y ] , Z ) . Note first that u = 0 reads [ X , ψ 0 ( Y , Z )] = 0 . As X , Y , Z a re arbitrary , ψ 0 is c -v a lued. No w the p ermutation of Y a nd Z changes r to − t = s (f rom (3)) . Again, b y antis ymmetry o f δ ψ 0 , r = t = − s. As X , Y , Z are arbitrary , one gets ψ 0 ∈ c ⊗ ( S 2 g ∗ ) g . No w F ( ψ 0 ) = − r = − δ ψ 0 = dψ 1 ∈ B 3 ( g , g ) . Hence ψ 0 ∈ Z L 2 0 ( g , g ) ⇔ F ( ψ 0 ) = 0 ⇔ ψ 1 ∈ Z 2 ( g , g ) ⇔ ψ 0 ∈ c ⊗ k er I . Consider now the linear map Φ : Z L 2 ( g , g ) → F − 1 ( B 3 ( g , g )) / k er F defined b y ψ 7→ [ ψ 0 ] (mo d k er F ) . Φ is on to: for any [ ϕ 0 ] ∈ F − 1 ( B 3 ( g , g )) / k er F , ϕ 0 ∈ c ⊗ ( S 2 g ∗ ) g , one has F ( ϕ 0 ) ∈ B 3 ( g , g ) , hence F ( ϕ 0 ) = dϕ 1 , ϕ 1 ∈ C 2 ( g , g ) , and then ϕ = ϕ 0 + ϕ 1 is a Leibniz co cycle suc h that Φ( ϕ ) = [ ϕ 0 ] . No w k er Φ = Z 2 ( g , g ) ⊕ Z L 2 0 ( g , g ) , since condition [ ψ 0 ] = [0] reads ψ 0 ∈ ker F whic h is equiv alen t to ψ ∈ Z 2 ( g , g ) ⊕ Z L 2 0 ( g , g ) . Hence Φ yields a n isomorphism Z L 2 ( g , g ) /( Z 2 ( g , g ) ⊕ Z L 2 0 ( g , g )) ∼ = F − 1 ( B 3 ( g , g )) / k er F . The latter is isomorphic to Im F ∩ B 3 ( g , g ) ∼ = ( c ⊗ Im I ) ∩ B 3 ( g , g ) . (ii) Results from the in v ariance of ψ 0 ∈ Z L 2 0 ( g , g ) . (iii) Results immediately f r o m (i), (ii) since B L 2 ( g , g ) = B 2 ( g , g ) as the Leibniz differen tial on C L 1 ( g , g ) = g ∗ ⊗ g = C 1 ( g , g ) coincides with the usual one. (iv)-(vi) are similar.  6 A. FIA LOWSKI, L. MAG NIN, AND A. MAN DAL Remark 1. Since k er I ⊕ (Im I ∩ B 3 ( g , C )) ∼ = k er h where h denotes I comp osed with the pro jection of Z 3 ( g , C ) on to H 3 ( g , C ) , the result (vi) is the same as in [14]. Remark 2. An y supplemen tary subspace t o Z 2 ( g , C ) ⊕ Z L 2 0 ( g , C ) in Z L 2 ( g , C ) consists of c ouple d Leibniz 2-co cycles, i.e. the nonzero el- emen ts ha v e t he prop erty that their symmetric and an tisymm etric parts are not co cycles. T o get suc h a supplemen ta ry subspace , pic k an y supplemen tary subspace W to ke r I in ( S 2 g ∗ ) g and take C = { B + ω ; B ∈ W ∩ I − 1 ( B 3 ( g , C )) , I B = dω } . Definition 4. g is said to b e adjoint (r esp. trivial) Z L 2 -unc oupling if ( c ⊗ Im I ) ∩ B 3 ( g , g ) = { 0 }  r esp. Im I ∩ B 3 ( g , C ) = { 0 }  . The class of adjoin t Z L 2 -uncoupling Lie algebras is rather extensiv e since it contains all zero-cen ter Lie algebras and all I - n ull L ie a lge- bras. F or non zero-cen ter, adjoin t Z L 2 -uncoupling implies trivial Z L 2 - uncoupling,Adjoin t Z L 2 -uncoupling implies trivial Z L 2 -uncoupling, since c ⊗ (Im I ∩ B 3 ( g , C )) ⊂ ( c ⊗ Im I ) ∩ B 3 ( g , g ) . The recipro cal holds ob- viously true f or I -exact Lie algebras. How eve r w e do not know if it holds true in general (e.g. w e do not kno w of a nilp oten t Lie algebra whic h is not I -exact). Corollary 1. (i) H L 2 ( g , g ) ∼ = H 2 ( g , g ) ⊕ ( c ⊗ k er I ) if and only if g is adjoint Z L 2 -unc oupling. (ii) H L 2 ( g , C ) ∼ = H 2 ( g , C ) ⊕ k er I if and only if g is trivial Z L 2 - unc oupling. Corollary 2. F or any Lie a l g e br a g with trivial c enter c = { 0 } , H L 2 ( g , g ) = H 2 ( g , g ) . Remark 3. This fact also fol lows fr om the c ohom olo gic al version of The or em A in [17] . Pr o of. Let g b e a Lie alg ebra and M b e a right g -mo dule. Consider the pro duct map m : g ⊗ Λ n g − → Λ n +1 in the exterior algebra. This map yields an epimorphism of c hain complexes C ∗ ( g , g ) − → C i ( g , K )[ − 1], where C ∗ ( g , K ) is t he reduced chain complex: C 0 ( g , K ) = 0, C i ( g , K ) = C i ( g , K ) for i > 0. Define the c hain complex C R ∗ ( g ) suc h t ha t C R ∗ ( g [1] is the kerne l of t he epimorphism C ∗ ( g , g ) − → C ∗ ( g , K )[ − 1]. Denote the cohomology of C R ∗ ( g ) by H R ∗ ( g ). Let us r ecall Theorem A in [17]. There exists a sp ectral sequence E 2 pq = H R p ( g ⊗ H L q ( g , M ) = ⇒ H r el p + q ( g , M ) . As the cente r of our Lie algebra is 0, it follows that E 2 00 = 0, and so w e g et H r el 0 ( g , g ) = 0. ABOUT LEIBNIZ COHOMOLOGY AND DEFORMA TIONS OF LIE A LGEBRAS 7 But then from the exact sequence in [1 7] 0 ← H 2 ( g , M ) ← H L 2 ( g , M ) ← H r el 0 ( g , M ) ← H 3 ( g , M ) ← ... w e get H L 2 ( g , M ) = H 2 ( g , M ) .  Corollary 3. F or any r e ductive algebr a Lie g with c enter c , H L 2 ( g , g ) = H 2 ( g , g ) ⊕ ( c ⊗ S 2 c ∗ ) , and dim H 2 ( g , g ) = c 2 ( c − 1) 2 with c = dim c . Pr o of. g = s ⊕ c with s = C 2 g semisimple. W e first pro v e that g is adjoin t Z L 2 -uncoupling. c ⊗ ( S 2 g ∗ ) g =  c ⊗ ( S 2 s ∗ ) s  ⊕ ( c ⊗ S 2 c ∗ ) = c ( S 2 s ∗ ) s ⊕ c ( S 2 c ∗ ) . Supp ose first s simple. Then a n y bilinear sym- metric in v ariant for m on s is some m ultiple of the Killing form K . Hence c ⊗ ( S 2 g ∗ ) g = c ( C K ) ⊕ c ( S 2 c ∗ ) . F or any ψ 0 ∈ c ⊗ ( S 2 g ∗ ) g , F ( ψ 0 ) is then some linear combination of copies of I K . As is well- kno wn, I K is no cob oundary . Hence if w e supp ose that F ( ψ 0 ) is a cob oundary , necessarily F ( ψ 0 ) = 0 . g is a djoin t Z L 2 -uncoupling when s is simple. Now , if s is not simple, s can b e decomposed as a direct sum s 1 ⊕ · · · ⊕ s m of simple ideals of s . Then ( S 2 s ∗ ) s = L m i =1 ( S 2 s i ∗ ) s i = L m i =1 C K i ( K i Killing form of s i . ) The same reason- ing then applies a nd sho ws that g is adjo in t Z L 2 -uncoupling. F ro m (ii) in theorem 2, Z L 2 0 ( g , g ) = c ⊗ S 2 c ∗ . Now, g = s ⊕ c with s = C 2 g semisimple. s can b e decomp osed as a dir ect sum s 1 ⊕ · · · ⊕ s m of ideals of s hence of g . Then H 2 ( g , g ) = L m i =1 H 2 ( g , s i ) ⊕ H 2 ( g , c ) . As s i is a non trivial g - mo dule, H 2 ( g , s i ) = { 0 } ([6], Prop. 11.4, page 154). Hence H 2 ( g , g ) = H 2 ( g , c ) = c H 2 ( g , C ) . By the K ¨ unneth for - m ula and Whitehead’s lemmas, H 2 ( g , C ) = ( H 2 ( s , C ) ⊗ H 0 ( c , C )) ⊕ ( H 1 ( s , C ) ⊗ H 1 ( c , C )) ⊕ ( H 0 ( s , C ) ⊗ H 2 ( c , C )) = H 0 ( s , C ) ⊗ H 2 ( c , C ) = C ⊗ H 2 ( c , C ) . Hence dim H 2 ( g , g ) = c 2 ( c − 1) 2 .  4. Examples F or ω , π ∈ g ∗ , ⊙ stands for the symmetric pro duct ω ⊙ π = ω ⊗ π + π ⊗ ω . Example 2. F or g = gl ( n ) , H L 2 ( g , g ) = Z L 2 0 ( g , g ) = C  x n 2 ⊕ ( ω n 2 ⊙ ω n 2 )  , where ( x i ) 1 6 i 6 n 2 is a basis o f g such that ( x i ) 1 6 i 6 n 2 − 1 is a basis o f sl ( n ) and x n 2 is the identit y matrix, and ( ω i ) 1 6 i 6 n 2 the dual basis to ( x i ) 1 6 i 6 n 2 . Hence there is a unique Leibniz deformation of gl ( n ) . Corollary 4. L et g = H N b e the (2 N + 1) -di m ensional c omp l e x Heisen- b er g Lie algebr a ( N > 1 ) as in example 1. (i) Z L 2 0 ( H N , H N ) has b asis ( x 2 N +1 ⊗ ( ω i ⊙ ω j )) 1 6 i 6 j 6 2 N with ( ω i ) 1 6 i 6 2 N +1 8 A. FIA LOWSKI, L. MAG NIN, AND A. MAN DAL the dual b asis to ( x i ) 1 6 i 6 2 N +1 ( ⊙ s tand s for the symmetric pr o duct ω i ⊙ ω j = ω i ⊗ ω j + ω j ⊗ ω i ) . (ii) dim Z L 2 0 ( H N , H N ) = dim B 2 ( H N , H N ) = N (2 N + 1); dim H L 2 ( H N , H N ) = dim Z 2 ( H N , H N ) = ( N 3 (8 N 2 + 6 N + 1) if N > 2 8 if N = 1 . Pr o of. (i) F ollows from ke r I = S 2 ( g / C 2 g ) ∗ . (ii) First H N is adjoint Z L 2 -uncoupling since it is I -n ull. The result then follo ws fr o m t he fa ct tha t ([9]) dim B 2 ( H N , H N ) = N (2 N + 1) and f o r N > 2 , dim H 2 ( H N , H N ) = 2 N 3 (4 N 2 − 1) .  Example 3. The case N = 1 has b een studied in [3]. In that case, dim Z L 2 0 ( H 1 , H 1 ) = 3 and the 3 Leibniz deformatio ns are nilp oten t, in con tradistinction with the 5 Lie defo r ma t io ns. The authors completely describe a Leibniz v ersal deformation o f the 3- dimensional Heisen b erg algebra. Example 4. The 4-dimensional solv able ”diamond” Lie algebra d has basis ( x 1 , x 2 , x 3 , x 4 ) and nonzero comm uta tion relations (with anticom- m utativit y) (4) [ x 1 , x 2 ] = x 3 , [ x 1 , x 3 ] = − x 2 , [ x 2 , x 3 ] = x 4 . The relations sho w that d is an extension of the one-dimensional ab elian Lie algebra C x 1 b y the Heisen b erg algebra n 3 with basis x 2 , x 3 , x 4 . It is a lso kno wn as the Nappi-Witten Lie algebra [1 5] or the cen- tral extension of the Poincar ´ e Lie algebra in t w o dimensions. It is a solv able quadratic Lie algebra, as admits a no ndegenerate bilinear symmetric inv a r ia n t form. Because of these prop erties, it pla ys an imp ortant role in conf o rmal field theory . W e can use d to construct a W ess-Zumino-Witten mo del, whic h describ es a homo g eneous four- dimensional Lorentz-signature space time [15]. It is easy to c hec k tha t d is I -exact. In fact, one v erifies that all o ther solv able 4-dimensional Lie alg ebras are I -n ull ( f or a list, see e.g. [16]). Consider d as Leibniz algebra with a differen t basis { e 1 , e 2 , e 3 , e 4 } o v er C . Define a bilinear map [ , ] : L × L − → L b y [ e 2 , e 3 ] = e 1 , [ e 3 , e 2 ] = − e 1 , [ e 2 , e 4 ] = e 2 , [ e 4 , e 2 ] = − e 2 , [ e 3 , e 4 ] = e 2 − e 3 and [ e 4 , e 3 ] = e 3 − e 2 , a ll other pro ducts of ba sis elemen ts b eing 0. W e get a basis satisfying the usual commu tation relat io ns (4) b y letting (5) x 1 = ie 4 , x 2 = e 3 , x 3 = i ( − e 2 + e 3 ) , x 4 = ie 1 . One should men tion that even though these tw o forms are equiv a len t o v er C , they represen t the t w o nonisomorphic real forms of the complex diamond a lgebra. W e fo und that b y considering L eibniz algebra deformation of d one gets more structures. Indeed it give s not only extra stucture but also ABOUT LEIBNIZ COHOMOLOGY AND DEFORMA TIONS OF LIE A LGEBRAS 9 k eeps track of Lie structures obtained by considering Lie algebra de- formations. T o get the precise deformations w e need to consider the cohomology groups. W e compute cohomologies necessary fo r o ur purp ose. First consider the Leibniz cohomology space H L 2 ( L ; L ). Our computation consists of the following steps: (i) T o determine a basis o f the space of co cycles Z L 2 ( L ; L ), (ii) to find out a ba sis of the cob oundary space B L 2 ( L ; L ), (iii) to determine the quotien t space H L 2 ( L ; L ). (i) Let ψ ∈ Z L 2 ( L ; L ). Then ψ : L ⊗ L − → L is a linear map a nd δ ψ = 0, where δ ψ ( e i , e j , e k ) = [ e i , ψ ( e j , e k )] + [ ψ ( e i , e k ) , e j ] − [ ψ ( e i , e j ) , e k ] − ψ ([ e i , e j ] , e k ) + ψ ( e i , [ e j , e k ]) + ψ ([ e i , e k ] , e j ) for 0 ≤ i, j, k ≤ 4 . Supp ose ψ ( e i , e j ) = P 4 k =1 a k i,j e k where a k i,j ∈ C ; for 1 ≤ i, j, k ≤ 4. Since δ ψ = 0 equating the co efficien ts of e 1 , e 2 , e 3 and e 4 in δ ψ ( e i , e j , e k ) w e g et the follo wing relations: ( i ) a 1 1 , 1 = a 2 1 , 1 = a 3 1 , 1 = a 4 1 , 1 = a 1 1 , 2 = a 3 1 , 2 = a 4 1 , 2 = 0; ( ii ) a 4 1 , 3 = a 3 1 , 4 = a 4 1 , 4 = a 1 2 , 1 = a 3 2 , 1 = a 4 2 , 1 = a 1 2 , 2 = a 2 2 , 2 = a 3 2 , 2 = a 4 2 , 2 = 0; ( iii ) a 4 3 , 1 = a 2 3 , 3 = a 3 3 , 3 = a 4 3 , 3 = a 3 4 , 1 = a 4 4 , 1 = a 2 4 , 4 = a 3 4 , 4 = a 4 4 , 4 = 0; ( iv ) a 2 1 , 2 = − a 2 2 , 1 = a 2 1 , 3 = − a 3 1 , 3 = − a 2 3 , 1 = a 3 3 , 1 ; ( v ) a 1 1 , 3 = − a 1 3 , 1 = a 2 1 , 4 = − a 2 4 , 1 ; ( v i ) a 3 2 , 3 = − a 3 3 , 2 = − a 4 2 , 4 = a 4 4 , 2 ; a 4 2 , 3 = − a 4 3 , 2 ; a 2 2 , 3 = − a 2 3 , 2 ; ( v i ) a 1 2 , 4 = − a 1 4 , 2 ; a 2 2 , 4 = − a 2 4 , 2 ; a 3 2 , 4 = − a 3 4 , 2 ; ( v ii ) a 1 3 , 4 = − a 1 4 , 3 ; a 2 3 , 4 = − a 2 4 , 3 ; a 3 3 , 4 = − a 3 4 , 3 ; a 4 3 , 4 = − a 4 4 , 3 ( ix ) a 3 3 , 4 = ( a 1 14 − a 2 24 ); a 4 3 , 4 = ( a 2 14 + a 2 23 ) ( x ) a 1 33 = 1 2 ( a 1 23 + a 1 32 ); a 1 41 = − ( a 1 14 + a 1 23 + a 1 32 ) . Therefore, in terms of the ordered basis { e i ⊗ e j } 1 ≤ i,j ≤ 4 of L ⊗ L and { e i } 1 ≤ i ≤ 4 of L ,transp ose of the matrix corresp onding to ψ is of the form 10 A. FIA LOWSKI, L. MAG NIN, AND A. MAN DAL M t =                            0 0 0 0 0 x 1 0 0 x 2 x 1 − x 1 0 x 3 x 2 0 0 0 − x 1 0 0 0 0 0 0 x 4 x 5 x 6 x 7 x 8 x 9 x 10 − x 6 − x 2 − x 1 x 1 0 x 11 − x 5 − x 6 − x 7 1 2 ( x 4 + x 11 ) 0 0 0 x 12 x 13 ( x 3 − x 9 ) ( x 2 + x 5 ) − ( x 4 + x 3 + x 11 ) − x 2 0 0 − x 8 − x 9 − x 10 x 6 − x 12 − x 13 − ( x 3 − x 9 ) − ( x 2 + x 5 ) x 14 0 0 0                            . where x 1 = a 2 1 , 2 ; x 2 = a 1 1 , 3 ; x 3 = a 1 1 , 4 ; x 4 = a 1 2 , 3 ; x 5 = a 2 2 , 3 ; x 6 = a 3 2 , 3 ; x 7 = a 4 2 , 3 ; x 8 = a 1 2 , 4 ; x 9 = a 2 2 , 4 ; x 10 = a 3 2 , 4 ; x 11 = a 1 3 , 2 ; x 12 = a 1 3 , 4 ; x 13 = a 2 3 , 4 and x 14 = a 1 4 , 4 are in C . Let φ i ∈ Z L 2 ( L ; L ) for 1 ≤ i ≤ 1 4 , b e the co cyle with x i = 1 and x j = 0 for i 6 = j in the ab ov e matrix of ψ . It is easy to c hec k that { φ 1 , · · · , φ 14 } f orms a basis of Z L 2 ( L ; L ). (ii) Let ψ 0 ∈ B L 2 ( L ; L ). W e ha v e ψ 0 = δ g for some 1- co c hain g ∈ C L 1 ( L ; L ) = Hom ( L ; L ). Supp o se the matrix a sso ciated to ψ 0 is same as the ab ov e matrix M . Let g ( e i ) = a 1 i e 1 + a 2 i e 2 + a 3 i e 3 + a 4 i e 4 for i = 1 , 2 , 3 , 4. The mat rix asso ciated to g is give n by     a 1 1 a 1 2 a 1 3 a 1 4 a 2 1 a 2 2 a 2 3 a 2 4 a 3 1 a 3 2 a 3 3 a 3 4 a 4 1 a 4 2 a 4 3 a 4 4     . F rom t he definition of cob oundary we get δ g ( e i , e j ) = [ e i , g ( e j )] + [ g ( e i ) , e j ] − ψ ([ e i , e j ]) for 0 ≤ i, j ≤ 4. The transp ose matrix of δ g can b e written as ABOUT LEIBNIZ COHOMOLOGY AND DEFORMA TIONS OF LIE A LGEBRAS 11                            0 0 0 0 − a 3 1 − a 4 1 0 0 a 2 1 − a 4 1 a 4 1 0 0 ( a 2 1 + a 3 1 ) − a 3 1 0 a 3 1 a 4 1 0 0 0 0 0 0 − ( a 1 1 − a 2 2 − a 3 3 ) − ( a 2 1 + a 4 2 − a 4 3 ) − ( a 3 1 − a 4 2 − a 4 1 − ( a 1 2 − a 3 4 ) ( a 3 2 + a 4 4 ) − 2 a 3 2 − a 4 2 − a 2 1 a 4 1 − a 4 1 0 ( a 1 1 − a 2 2 − a 3 3 ) ( a 2 1 + a 4 2 − a 4 3 ) ( a 3 1 − a 4 2 ) a 4 1 0 0 0 0 − ( a 1 2 − a 1 3 + a 2 4 ) − ( a 2 2 − 2 a 2 3 − a 3 3 − a 4 4 ) − ( a 3 2 + a 4 4 ) − ( a 4 2 − a 4 3 ) 0 − ( a 2 1 + a 3 1 ) a 3 1 0 ( a 1 2 − a 3 4 ) − ( a 3 2 + a 4 4 ) 2 a 3 2 a 4 2 ( a 1 2 − a 1 3 + a 2 4 ) ( a 2 2 − 2 a 2 3 − a 3 3 − a 4 4 ) ( a 3 2 + a 4 4 ) ( a 4 2 − a 4 3 ) 0 0 0 0                            . Since ψ 0 = δ g is also a co cycle in C L 2 ( L ; L ), comparing matrices δ g and M we conclude that t he transp o se matrix of ψ 0 is of the form M t =                            0 0 0 0 0 x 1 0 0 x 2 x 1 − x 1 0 0 x 2 0 0 0 − x 1 0 0 0 0 0 0 x 4 x 5 x 6 x 1 x 8 x 9 x 10 − x 6 − x 2 − x 1 x 1 0 − x 4 − x 5 − x 6 − x 1 0 0 0 0 x 12 x 13 − x 9 ( x 2 + x 5 ) 0 − x 2 0 0 − x 8 − x 9 − x 10 x 6 − x 12 − x 13 x 9 − ( x 2 + x 5 ) 0 0 0 0                            . Let φ i ′ ∈ B L 2 ( L ; L ) fo r i = 1 , 2 , 4 , 5 , 6 , 8 , 9 , 10 , 1 2 , 13 b e the cob ound- ary with x i = 1 a nd x j = 0 fo r i 6 = j in the ab o v e ma t r ix o f ψ 0 . It follo ws that { φ ′ 1 , φ ′ 2 , φ ′ 4 , φ ′ 5 , φ ′ 6 , φ ′ 8 , φ ′ 9 , φ ′ 10 , φ ′ 12 , φ ′ 13 } f o rms a basis of the cob oundary space B L 2 ( L ; L ). (iii) It is straigh tforw ard to chec k that { [ φ 3 ] , [ φ 7 ] , [ φ 11 ] , [ φ 14 ] } span H L 2 ( L ; L ) where [ φ i ] denotes the cohomology class represen ted b y the co cycle φ i . 12 A. FIA LOWSKI, L. MAG NIN, AND A. MAN DAL Th us dim( H L 2 ( L ; L )) = 4. The represen tativ e co cycles of the cohomology classes forming a basis of H L 2 ( L ; L ) are give n explicitely as the following. (1) φ 3 : φ 3 ( e 1 , e 4 ) = e 1 , φ 3 ( e 4 , e 1 ) = − e 1 ; φ 3 ( e 3 , e 4 ) = e 3 ; φ 3 ( e 4 , e 3 ) = − e 3 ; (2) φ 7 : φ 7 ( e 2 , e 3 ) = e 4 , φ 7 ( e 3 , e 2 ) = − e 4 ; (3) φ 11 : φ 11 ( e 3 , e 2 ) = e 1 , φ 11 ( e 3 , e 3 ) = 1 2 e 1 , φ 11 ( e 4 , e 1 ) = − e 1 ; (4) φ 14 : φ 14 ( e 4 , e 4 ) = e 1 . Here φ 3 and φ 7 are sk ew-symm etric, so φ i ∈ H om (Λ 2 L ; L ) ⊂ H om ( L ⊗ 2 ; L ) for i = 3 and 7 . Consider, µ i = µ 0 + tφ i for i = 3 , 7 , 11 , 14 , where µ 0 denotes the original brack et in L . This giv es 4 no n-equiv alen t infinitesim al deformations of the Leibniz brac k et µ 0 with µ 3 and µ 7 giving the Lie a lgebra structure on L [[ t ]] / < t 2 > . No w w e ha v e to compute the Massey br a c k ets [ φ i , φ j ] whic h are re- sp onsible for obstructions to extend infinitesimal deformatio ns. W e find [ φ 3 , φ 3 ] = 0 , [ φ 7 , φ 7 ] = 0 . That means that the t w o infinitesimal Lie deformations can b e ex- tended to real deformations, with the new nonzero bra ck ets (and their an ticomm utativ e v ersion) The first of the deformations represen ts a 2-parameter pro jectiv e family d ( λ, µ ), for whic h eac h pro jectiv e parameter ( λ, µ ) defines a nonisomorphic Lie algebra (in fact, t he dia mond algebra is a member of this family with ( λ, µ ) = (1 , − 1)): [ e 2 , e 3 ] λ,µ = e 1 [ e 2 , e 4 ] λ,µ = λe 2 [ e 3 , e 4 ] λ,µ = e 2 + µe 3 [ e 1 , e 4 ] λ,µ = ( λ + µ ) e 1 . The second deformation, [ e 2 , e 3 ] t = e 1 + te 4 [ e 2 , e 4 ] t = e 2 [ e 3 , e 4 ] t = e 2 − e 3 is isomorphic to sl (2 , C ) ⊕ C fo r ev ery nonzero v a lue of t , see [5]. F urthermore, w e a lso ha v e [ φ 14 , φ 14 ] = 0 whic h means that φ 14 defines ABOUT LEIBNIZ COHOMOLOGY AND DEFORMA TIONS OF LIE A LGEBRAS 13 a real Leibniz deformation: [ e 2 , e 3 ] t = e 1 [ e 2 , e 4 ] t = e 2 [ e 3 , e 4 ] t = e 2 − e 3 [ e 4 , e 4 ] t = te 1 . W e not e that this Leibniz algebra is not nilp otent. F or the brack et [ φ 11 , φ 11 ] w e get a nonzero 3-co cycle, so the infinites- imal Leibniz deformation with infinitesimal part b eing φ 11 can not b e extended ev en to the next order. The nontrivial mixed brack ets [ φ i , φ j ] determine relatio ns on the ba se of ve rsal deformation. Among the six p ossible cases [ φ 3 , φ 11 ], [ φ 3 , φ 14 ] and [ φ 11 , φ 14 ] are non- trivial 3 -co cycles, the ot hers are represen ted b y 3 -cob oundaries. Th us w e need to chec k the Massey 3-brack ets whic h are defined, namely < φ 3 , φ 3 , φ 7 > < φ 3 , φ 7 , φ 7 > < φ 7 , φ 7 , φ 11 > < φ 7 , φ 7 , φ 14 > < φ 7 , φ 14 , φ 14 > In these fiv e p ossible Massey 3- brac k ets, only < φ 3 , φ 3 , φ 7 > is rep- resen ted b y nontrivial co cycle. So we now pro ceed to compute the p ossible Massey 4-brack ets. W e get tha t four of them are non trivial: < φ 3 , φ 7 , φ 7 , φ 11 > < φ 3 , φ 7 , φ 7 , φ 14 > < φ 7 , φ 7 , φ 14 , φ 11 > < φ 7 , φ 7 , φ 14 , φ 14 > . A t the next step, w e get that a ll the 5-order Massey pro ducts are either not defined or are trivial. 14 A. FIA LOWSKI, L. MAG NIN, AND A. MAN DAL So w e can write the v ersal Leibniz deformation of our Lie algebra: [ e 1 , e 2 ] v = [ e 2 , e 1 ] v = [ e 1 , e 3 ] v = [ e 3 , e 1 ] v = 0 , [ e 1 , e 4 ] v = te 1 , [ e 4 , e 1 ] v = − ( t + u ) e 1 , [ e 2 , e 3 ] v = e 1 + se 4 , [ e 3 , e 2 ] v = ( u − 1) e 1 − se 4 , [ e 2 , e 4 ] v = e 2 , [ e 4 , e 2 ] = − e 2 , [ e 3 , e 4 ] v = e 2 + ( t − 1) e 3 , [ e 4 , e 3 ] v = − e 2 + (1 − t ) e 3 , [ e 1 , e 1 ] v = [ e 2 , e 2 ] v = 0 , [ e 3 , e 3 ] v = 1 / 2 ue 1 , [ e 4 , e 4 ] v = w e 1 . The base of the vers al deformat io n is C [[ t, s, u, w ]] / { tu , tw , uw ; t 2 s ; ts 2 u, ts 2 w , s 2 uw , s 2 w 2 } . Example 5. The quadratic 5- dimensional nilp oten t Lie algebra g 5 , 4 [11] has commutation r elat io ns [ x 1 , x 2 ] = x 3 , [ x 1 , x 3 ] = x 4 , [ x 2 , x 3 ] = x 5 . This is an extension of the trivial Lie algebra C x 1 b y the 4-dimensional Lie algebra C x 4 × n 3 ( n 3 the 3-dimensional Heisen b erg Lie algebra [ x 2 , x 3 ] = x 5 ). As it is moreov er the only 5- dimensional indecomp os- able nilp oten t Lie algebra whic h is not I -n ull, it can b e considered as a 5-dimensional a nalogue of the dia mo nd alg ebra d . Let us fir st compute its trivial Leibniz cohomo lo gy . W e here denote simply d for d C , a nd ω i,j for ω i ∧ ω j (see a lso [10],[12]). B 2 ( g , C ) = h d ω 3 = − ω 1 , 2 , dω 4 = − ω 1 , 3 , dω 5 = − ω 2 , 3 i , dim Z 2 ( g , C ) = 6 , dim H 2 ( g , C ) = 3 , Z 2 ( g , C ) = h ω 1 , 4 , ω 2 , 5 , ω 1 , 5 + ω 2 , 4 i ⊕ B 2 ( g , C ) , dim Z L 2 0 ( g , C ) = 3 , Z L 2 0 ( g , C )( ∼ = k er I ) = h ω 1 ⊗ ω 1 , ω 1 ⊙ ω 2 , ω 2 ⊗ ω 2 i , dim Z L 2 ( g , C ) = 10 , dim H L 2 ( g , C ) = 7 , a nd Z L 2 ( g , C ) = Z 2 ( g , C ) ⊕ Z L 2 0 ( g , C ) ⊕ C g 1 , H L 2 ( g , C ) = H 2 ( g , C ) ⊕ Z L 2 0 ( g , C ) ⊕ C g 1 with g 1 = B + ω 1 , 5 and B = ω 1 ⊙ ω 5 − ω 2 ⊙ ω 4 + ω 3 ⊗ ω 3 . (Here Im I = C I B = C dω 1 , 5 and Im I ∩ B 3 ( g , C ) = Im I is one-dimensional.) g 5 , 4 is not trivial Z L 2 -uncoupling (hence not adjoin t Z L 2 -uncoupling either), and g 1 is a coupled Leibniz 2-co cycle. No w let us turn to the adjoint Leibniz cohomology , which represen ts nonequiv alen t infinitesimal Leibniz deformatio ns. dim Z 2 ( g , g ) = 24; Z L 2 0 ( g , g ) = c ⊗ k er I has dimension 6, dim Z L 2 ( g , g ) = ABOUT LEIBNIZ COHOMOLOGY AND DEFORMA TIONS OF LIE A LGEBRAS 15 32 , Z L 2 ( g , g ) = Z 2 ( g , g ) ⊕ Z L 2 0 ( g , g ) ⊕ C G 1 ⊕ C G 2 , H L 2 ( g , g ) = H 2 ( g , g ) ⊕ Z L 2 0 ( g , g ) ⊕ C G 1 ⊕ C G 2 , where G 1 , G 2 are the following Leibniz 2-co cycles, each of whic h is coupled: G 1 = x 5 ⊗ ( B + ω 1 , 5 ) G 2 = x 4 ⊗ ( B + ω 1 , 5 ) Here H 2 ( g , g ) has dimension 9. Of course, these spaces are h uge to compute, but w e w ould lik e to p oint out some structural similarit y with the diamond alg ebra. One may observ e that the coupled co cycle φ 11 of d reads in the ba sis (5) φ 11 = − ix 4 ⊗ ( C − ω 2 , 3 + ω 1 , 4 ) with C = ω 1 ⊙ ω 4 + ω 2 ⊗ ω 2 + ω 3 ⊗ ω 3 the no n degenerate inv a r ia n t bilinear form, a similarit y with G 1 , G 2 . The similarity extends to the fact that G 1 , G 2 cannot b e extended to the second leve l. As of Lie deformations, g 5 , 4 has a n umber o f deformations. Without iden tifying all of them, w e list some: 1. A three-par a meter solv able pro jectiv e family d ( p : q : r ) where g 5 , 4 b elongs (it is its nilp oten t elemen t, with p = q = r = 0) with nonzero brack ets [ x 3 , x 4 ] p,q ,r = x 2 [ x 1 , x 5 ] p,q ,r = r x 1 [ x 2 , x 5 ] p,q ,r = ( p + q ) x 2 [ x 3 , x 5 ] p,q ,r = px 3 + x 1 [ x 4 , x 5 ] p,q ,r = x 3 + q x 4 . 2. A solv a ble Lie alg ebra with nonzero bra ck ets [ x 3 , x 4 ] = 2 x 4 [ x 3 , x 5 ] = − 2 x 5 [ x 4 , x 5 ] = x 3 [ x 1 , x 2 ] = x 1 . 16 A. FIA LOWSKI, L. MAG NIN, AND A. MAN DAL 3. Another solv able Lie a lgebra with nonzero brack ets [ x 3 , x 4 ] = 2 x 4 [ x 3 , x 5 ] = − 2 x 5 [ x 4 , x 5 ] = x 3 [ x 1 , x 3 ] = x 1 [ x 2 , x 5 ] = x 1 [ x 2 , x 3 ] = − x 2 [ x 1 , x 4 ] = x 2 . 4. A 2-pa rameter solv able pro jective family with nonzero brac k ets [ x 2 , x 5 ] p,q = x 1 + px 2 [ x 3 , x 5 ] p,q = x 2 + q x 3 [ x 4 , x 5 ] p,q = x 3 + ( p + q ) x 4 [ x 1 , x 5 ] p,q = ( p + q ) x 1 [ x 2 , x 3 ] p,q = pq x 1 [ x 2 , x 4 ] p,q = q x 1 [ x 3 , x 4 ] p,q = x 1 . 5. Another 2-par ameter solv able pro jectiv e family with nonzero brac k ets [ x 3 , x 4 ] p,q = x 2 [ x 2 , x 5 ] p,q = ( p + q ) x 2 [ x 3 , x 5 ] p,q = x 1 + px 3 [ x 4 , x 5 ] p,q = x 3 + q x 4 [ x 1 , x 5 ] p,q = ( q + 2 p ) x 1 [ x 2 , x 3 ] p,q = ( p − q ) x 1 [ x 2 , x 4 ] p,q = x 1 . Reference s [1] Fialowski, A., ”Defor mations of Lie algebr as,” Mat.Sb ornyik USSR, 127 (169), (1985), pp. 47 6–48 2; English translation: Math. USSR-Sb., 55 , (1986), no. 2 , 467–4 73 [2] Fialowski, A., ”An example o f forma l deformations of Lie alg ebras” NA TO Conference o n Deformation Theo ry o f Algebr as and Applications, Il Cio cc o, Italy , 1986, Pro ceedings. Kluw er, Dor drech t, 198 8, 375–401 [3] Fialowski, A., Ma ndal, A., Le ibniz algebra deformations of a Lie algebra , Jour- nal of Math. Physics , 49 , 2008, 093512 , 10 pp. [4] Fialowski, A., Mandal, A., Mukherjee, G., V ers al Deformations of Leibniz Algebras, Journal of K-The ory , 2008, doi:10.10 1 7/is00 8004027jkt049. [5] Fialowski, A., Penk av a, M., V ersal defo r mations of four dimensional Lie alge- bras, Commun. in Contemp or ary Math , 9 , 2007, 41–7 9 ABOUT LEIBNIZ COHOMOLOGY AND DEFORMA TIONS OF LIE A LGEBRAS 17 [6] Guic hardet, A., Cohomolo gie des gr oup es top olo giques et des alg ` ebr es de Lie , Cedic/F ernand Nathan, Paris, 198 0. [7] Koszul, J.L. Homologie et co ho mologie des a lg` ebr es de Lie, Bul l. So c. Math. F r anc e , 78 , 1950, 67-12 7. [8] Lo day , J.L., Une version non commutativ e des alg` ebres de Lie: les alg` ebr es de Leibniz, Ens. Math. , 39 , 1993, 269-293 . [9] Magnin, L., Cohomolo gie a djointe de alg` ebre s de Heisenberg, Comm. A lgebr a , 21 , 1993, 2101- 2129. [10] Magnin, L., Adjoint and trivial cohomolo gies of nilp otent c o mplex Lie algebra s of dimension 6 7 , Int. J. Math. math. Sci. , volume 2008, Ar ticle ID 80 5305 , 12 pages. [11] Magnin, L., Determination of 7-dimensio nal indecomp osable nilp otent com- plex Lie algebras by adjoining a deriv ation to 6- dimensional Lie algebras, Al- gebr as and R epr esent ation The ory , DOI: 10.10 07/s1 0468- 009-9172-3 (Online- First), 2009. [12] Magnin, L., A djoi nt and trivial c ohomolo gy tables for inde c omp osable nilp otent Lie algebr as of dimension ≤ 7 over C , online bo o k, 2d Co r- rected E dition 2007, ( Postcript, .ps file ) (810 pages + vi), acces sible a t http:/ /www. u-bourgogne.fr/monge/l.magnin or http:// math.u -bourgogne.fr/IMB/magnin/public html/i ndex. html [13] Magnin, L., O n I - null Lie alg e bras, arXiv, math.RA 1010.4 6 60, 201 0. [14] Hu, N., Pei, Y., Liu, D., A coho mological c hara cterization of Leibniz central extensions of Lie algebra s, Pr o c. Amer. Math. So c. , 136 , 2008, 437-47 7. [15] Nappi, C.R., Witten E ., W ess -Zumino-Witten mo del bas ed o n a nonsemisimple Lie group, Phys. R ev. L ett . , 71 , 1 993, 3751. [16] Ov ando, G., Complex, symplectic and K¨ ahler structure s on 4 dimensional Lie groups, R ev. Un. Mat. Ar gent ina , 45 , 2 004, 55-67 . [17] Pirashvili, T., On Leibniz homolog y , Ann. Instit. F ourier , 44 , 199 4, 401-411 . Institute o f Ma thema tics, E ¨ otv ¨ os Lor ´ and University, P ´ azm ´ any P ´ eter s ´ et ´ any 1/C, H-1117 Budapest, Hungar y E-mail addr ess : fialo wsk@cs .elte .hu Institut de Ma th ´ ema tiques de Bourgogne, UMR CN RS 5584 , Univer- sit ´ e de Bourgogne, BP 47870,, 21078 Dijon Cedex, France E-mail addr ess : magni n@u-bo urgog ne.fr University of Luxembourg, Campus Kirchbe rg, Ma th. Research Unit, 6, rue Richard Coudenhove-Keler gi, L-1359 Luxembourg City E-mail addr ess : ashis .manda l@uni .lu