VietnamJ Math(2013)41:135-148 DOI 10.1007/S10013-013-0006-6
Two-Step Nilpotent Quadratic Lie Algebras and 8-Dimensional Non-commutative Symmetric Noviliov Algebras
Minh Thanh Duong
Received' 27 June 2012 / Accepted. 11 October 2012 / Pubhshed online- 30 January 2013
© Vietnam Academy of Science and Technology (VAST) and Springer Science+Busmess Media Singapore 2013
Abstract We study two-step nilpotent quadratic Lie algebras from the poini of view of dou- ble extensions and 7"*-extensions. As a consequence, an isomelrically isomorphic classifi- cation of two-step nilpolent quadratic Lie algebras can be obtained from the ciassificatiou of 3-vectors. Next, we apply the study of iwo-slep nilpolent quadratic Lie algebras to the clas- sification of non-commulative symmetnc Novikov algebras. We focus on the 8-dimensional case and give some examples which are indecomposable and not solvable
Keywords Quadratic Lie algebras • Two-step nilpotent r*-Extensions • Double extensions • Symmetric Novikov algebras
Mathematics Subject Classification (2010) 17A30 17B05 • 17B30 • 17D25
1 Introduction
Throughout the paper, the base field is the complex field C. All considered vector spaces are finite-dimensional complex vector spaces
Let 0 be a finite-dimensional algebra over C and (X, Y)\-^ XY be its product. A bilinear form S : g X 0 ^ - C is called invariant (or associative) if it satisfies:
B(XY.Z)^B(X.YZ)
for all X. Y, Z in 3 and non-degenerate if BiX. 0) ^ 0 implies X ^0. Such a bilinear form has arisen in several areas of Mathematics and Physics. It can be seen as a general- ization of the Killing form on a semisimple Lie algebra, the inner product of an Euclidean Jordan algebra, or simply as the Frobenius form of a Frobenius algebra. The associativ- ity of a bilinear form also can be found in ihe conditions of an admissible trace function
MT, Duong (Kl)
Depanmenl of Physics. Ho Chi Mmh City University of Pedagogy. 280 An Duong Vuong. Ho Chi Mmh City, Vietnam
e-mail ihanhdmi@hcmup,eiiu vn
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136 M,T Duong defined on a power-associative algebra. For details, the reader can refer to a paper by Bor- dematm [3].
If 0 is a Lie algebra with a non-degenerale invanant symmetnc bilinear form then it is called a quadratic Lie algebra. Quadratic Lie algebras are really interesting algebraic objects.
A strucmral theory of quadratic Lie algebras based on the notion of double extensions (a combination of a cenlral extension and a semidirect product) was introduced in [9] in the solvable case and [10] in the general case. In [6, 7], one-dimensional double extensions of an Abelian Lie algebra are completely classified up to isomorphism and up to isometncal isomorphism. They are regarded as ihe simplest case of double extensions.
Another construction, namely the T'-extension, was given in [3] for solvable quadratic Lie algebras. Il is generahzed from the notion of the semidirect product of a Lie algebra and ils dual space. We realize thai the simplest case of r*-extensions is two-step nilpotent quadratic Lie algebras. Such algebras wilh a characterization of isometrically isomorphic classes and isomorphic classes were introduced in a paper of Ovando [12], By studying die sel of linear transformations in o(t]) where 1) is a vector space with a fixed inner product, the author proved that if the dimension of the vector space ij is three or greater than four then there exists a reduced two-slep nilpotent quadratic Lie algebra. Moreover, there is only one six-dimensional reduced two-slep nilpotent quadratic Lie algebra (up to isometncal isomor- phism). In Sect, 3, once again, we want lo consider two-step nilpotent quadratic Lie algebras through the methods of double extensions and 7*-extensions. In terms of double extensions, we have a rather obvious result: every two-step nilpolent quadratic Lie algebra can be ob- tained from an Abelian algebra by a sequence of double extensions by the one-dimensional algebra In terms of T*-extensions, we give a definition but with some more restricted con- ditions ihai can represent all Iwo-slep nilpolent quadratic Lie algebras. As a consequence, we obtain isomorphic and isometncally isomorphic characterizations of reduced two-step nilpolent quadratic Lie algebras. By defining a 3-form associated to a iwo-step nilpotent quadrafic Lie algebra, we also gel the same result as in [12] and further that there exist only one reduced Iwo-slep nilpolent quadratic Lie algebra of dimension 10, one indecompos- able reduced two-step nilpolent quadrafic Lie algebra of dimension 12 and several two-step nilpotent quadratic Lie algebras of dimension 14.
Another reason leading us lo study Iwo-slep nilpolent quadratic Lie algebras is thai the sub-adjacenl Lie algebra of a symmetric Novikov algebra is a two-slep nilpotent quadratic Lie algebra [1]. Novikov algebras appeared in the study ofthe Poisson brackets of hydrody- namic type and Hamiltonian operators in the formal variational calculus [2, 8|, A Novikov algebra is called sytrunetric if il is endowed with a non-degenerate associative bilinear form.
We hope lhat the classification of two-slep nilpotent quadratic Lie algebras would supply some useful information on the structure of iheir associated symmetric Novikov algebras. In particular, symmetnc Novikov algebras up lo dimension 6 are either commutative or two- step nilpotent since Iwo-step nilpolent quadratic Lie algebras up to dimension 6 are either Abelian or reduced. Furthermore, it has been proven thai if a non-commutative symmetric Novikov algebra of dimension 7 is indecomposable then it must be three-step nilpolent [5].
In Sect. 4, we study 8-dimensionai non-commutalive symmetric Novikov algebras and give several examples which are indecomposable and not solvable. Appendix is a short survey in classifying of alternating 3-vectors on a vector space up to dimension 7 lhat is useful to Seel. 3.
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Two-Step Nilpotent Quadratic Lie Algebras and S-Dimensional 2 Quadratic Lie Algebras
Definition 1 A quadratic Lie algebra (g, B) is a vector space g equipped with a non- degenerate symmetric bilinear form B and a Lie algebra strucmre on g such that B is invan- anl (thai means, B([X, Y], Z) = B(X, [Y, Z]) for all X, Y, Zs g).
Let (g, B) be a quadratic Lie algebra. Smce B is non-degenerate and invariant, we have some simple properties of g as follows:
Proposition 1 I. If I is an ideal ofg then /-'- is also an ideal ofs Moreover, if I is non- degenerale then so is I^ and 0 — / © /"'". Conveniently, in this case we use the notation 0 = / © / ^ .
2 Z(g) — [g, g]^ where Z(g) is the center ofg. And then dim(Z(0)) -b dim([0, g]) - dim(g).
We say that two quadratic Lie algebras (g, B) and (g', B') are isomelrically isomorphic (or i-isomorphic, for short) if there exisis a Lie algebra isomorphism A from g onto g' safisfying
B'{A(X), A(Y)) - BiX. Y) VX, K e g
In this case, A is called an i-isomorpbism. Nole lhat two isomorphic quadratic Lie algebras are not necessarily i-isomorpbic (see an example in [6]),
Proposition 2 [13] Let (g, B) be a non-Abelian quadratic Lie algebra. Then there exist a central ideal 3 and an ideal t / (0) such that:
1 0 = J © I where (j, Sl^x^) and (l, B\iy_{) are quadratic Lie algebras. Moreover. I is non- Abelian
2 The center Z(l) of I is totally isotropic, equivaiently Z(l) C [I, I], and dim(2{l)) < ^ dim(t) < dim([l, 1]).
3. Let g' be a quadratic Lie algebra and A:Q-^ g' bea Lie algebra isomorphism Then
where 3' ^ A(i) is central, l' = A(j)-^, Z(t') is totally isotropic and i and I' are isomor- phic. Moreover, if A is an i-isomorphism, Ihen I and I' are i-isomorpbic.
It results that the study of quadratic Lie algebras can be reduced to the study of quadratic Lie algebras satisfying the condiiion: the center Z(g) is totally isotropic Therefore, we have the following definition.
Definition 2 A quadratic Lie algebra g is reduced if 0 / [0} and Z(g) is totally isotropic Let (0, B) be a quadratic Lie algebra and C be an endomorphism of g. We say lhat C is skow-symmelric (or B-antisymmetnc) if B(C(X). Y) = -BiX. C(Y)) for all X, Ye g, Denoic by Dero(0) the space of skew-symmetnc denvalions of 0 We recall the notion of double extensions given by V. Kac, A Medina and Ph. Revoy as follows
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13g M T. Duong Defimtion 3 [9, 10] Let (g, B) be a quadratic Lie algebra, t) be another Lie algebra and TT : f) -»• Dero(0) be a representation of f) by means of skew-symmetnc denvations of g Define the map ^ : g x g ^ Ij* by (p(X, Y)Z = B(ii(Z)X, Y) for all X, K e g , Z € I).
Denote by a d ' the coadjoint representation of f). Then the vector space 0 = f) ® g ® 1)' with the product
[X -\-Y -i- f,X' + Y' -\- f'] = [X.X\-\-[Y,Y\-\-7T(X)Y' -7t{X')Y + a d ' ( X ) / ' - ad*{X')/ + 9{Y. Y') for all X, X' e i), Y, Y' s g, / , / ' 6 fj* is a Lie algebra and il is called the double extension of g by I) by means of jr. It is easy to show that g is also a quadratic Lie algebra with the bilinear form B defined by
B(X -\-Y-\-f X'+ Y'-\-f) ^ B{Y. Y') + f{X') + f'(X) for all X, X' e f], Y, Y' 6 g, / , / ' € !)*.
A quadratic Lie algebra (g, B) is called indecomposable if 0 = 0i 0 Qt, with gi and 02 ideals of 0, implies 01 or02 —jO],
Propositions [9,10] Lei (g, B) be an indecomposable quadratic Lie algebra such lhat it is neither simple nor one-dimensional. Then g is a double extension of a quadratic Lie algebra by a simple or one-dimensional algebra.
Let us present now the second construction which is given by Bordemann in [3].
Definition 4 Let g be a Lie algebra over C and g* its dual space. Consider a 2-cocycle
^ : 0 X g ^ - g ' and define on the vector space Tg(g) = g ffi g* the product as follows:
[X-\-f,Y-¥g] = [X, y]g + / o adg(y) - g o ade(X) + e(X, Y) for all X, F € g, / , g e g ' . Then r / ( g ) becomes a Lie algebra and it is called the T'-exlensionof g by means of &.
Moreover,if6'satisfies6'(X, y ) Z ^ 6 ' ( l ' , Z ) X for all X. Y, Z e g (the cyclic condition), then Tg(g) becomes a quadratic Lie algebra with the bilinear form B defined by
B(X + fY + g) = f(Y) + g(X) VX, Yeg, / , geg*.
Proposition 4 [3] Let (g, B) be an even-dimensional quadratic Lie algebra over C. //g is solvable then Q is i-isomorphic to a T'-extension T^(i})ofi) where i) is the quotient algebra
"/S ^>'" totally isotropic ideal.
3 Two-Step Nilpotent Quadratic Lie Algebras
Conveniently, we redefine a two-slep nilpotent Lie algebra in another way as follows:
Definition 5 An algebra 0 over C with a bibnear product 8 X 0 -* g, (x v)\-^ \x y] is called a two-step nilpolent Lie algebra if it satisfies [x, y] = —[y, x] and Ux vl ^1 = 0 for all JT. y, t e 0. Somedmes, we use the notion 2SN-Lie algebra as an abbreviation.
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Two-Siep Nilpotent QuadraLc Lie Algebras and 8-Dimensional
Let g be a 2SN-Lie algebra. V be a vector space a n d ^ : g x g - > V b e a biUnear map.
On the space g = 0 ffi V we define the following product:
lx-\-u,y-\-v] = [x,y]-i-tpix.y) Vx, y e 0, «, veV.
Then it is easy lo see that g is a 2SN-Lie algebra if and only if tp is skew-symmetric and (p([x.y],z) = OfoTa]lx, y, z € g. In this case, V is contained in the center Z ( g ) of g so Ihe Lie algebra g is called the 2SN-central extension of g by V by means of tp.
Proposition S Let Q be a 2SN-Lie algebra then g is a 2SN-central extension of an Abelian algebra \) by some vector space V.
Proof Denote V = [g, g] and let V\ = g/[g, g]. Then i) is Abelian. Sel ip : f] x b ^ V by
<p{pM^piy)) = i^-y'\ ^^' yeg,
where p : g -*• f) is the canonical projection This map is well-defined since g is two-slep
nilpotent. So g is the 2SN-cenlral extension of t) by K by means of ip. D Let g be a 2SN-Lie algebra, V be a vector space and TT : g -* End(V) be a linear map
On the space g = g ffi V we define the following product:
\x + u. y + v] = [x. y] + ir(x)v - niy)u VJT, y€g. U. V^V The proposition below follows Definition 5,
Proposition 6 The vector space g is a 2SN-Lie algebra if and onlv ifrt satisfies the condi- tion:
7t{[x.y]) = n{x)niy) = 0 Vx. y e g In this case, n is called a 2SN-representation ofg in V.
Remark I The adjoint represeniation and the coadjoint representation are 2SN-repre- .sentafions of a 2SN-Lie algebra g. Therefore, the extensions of g by itself or its dual space with respect to these representations are iwo-step nilpoienl.
Let g be a 2SN-Lie algebra, V and IV be vector spaces If JT • g -•• End(V) and /3: g - • End(H') are two 2SN-representations ofg then the mapjr ffip:0-> End(Vffi W) defined by
(7T®p)(x)(v-\-w) = jr(x)v-\-p(x)w V x e g , vsV. m^W IS also a 2SN-representaiion ofg and it is called the direct sum of the two representations JT and p.
Proposition 7 Le; 0,, 02 fee 2SN-Lie algebras and n • B\ -^ End(g:) bea linear map We define on the vector space g = gi ©02 the following product
[. + v.x -\-y'] = [..x']^^+K(x)y- -Jt(x')y + [y.y\
forallx. \ ' € g i , v. v'eg2. Then g with this product is a 2SN-Lie algebra if and only if rr .satisfies the following conditions
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1. 7T([x,x']g^)^7r(x)7i:(x-) = 0, 2. Ji(x)([y,y\^) = ljzix)y,y%^^0 forallx, x' £ 0 1 , y, y ' e g 2 .
Proof The proof is a straightforward computation. • Clearly, the map TT in Proposition 7 is a 2SN-representation of 01 in 02. Hence, we obtain
the following proposition:
Proposition 8 Let gi, g? be 2SN-Lie algebras and ;r . gi ^ End(g2) be a 2SN-represent- ation of Q[ i l 02- Then Ihe vector space 0 = 0i ©02 with the product:
[x + y.x' + y']^[x,x\+7Tix)y' -n{x')y+[y,y\^
forallx, x ' e 0 i , y, y'e g2 becomes a 2SN-Lie algebra if and only if J t ( x ) ( [ y , / ] ^ ^ ) - [ 7 r ( A : ) y , / ] g ^ ^ 0 Vj;e0i, y, / e 02
In this case, TZ is called a 2SN-admissible representation ofg\ in 02 and we say that g is the semi-direcl product of g2 by gj by means of TT.
Remark 2 1. The condition in the above proposition ensures 7T(X) s Der(g2) forallx 6 0|.
2. The adjoint representation of a 2SN-Lie algebra is a 2SN-admissible representation.
Let (g, B) be a quadratic Lie algebra and g — I) © g ffi I)' be the double extension of g by h by means of TT as in Definition 3 If g is a 2SN-Lie algebra then g and f) should be also two-step mlpotent
Proposition 9 Let (g, B) be a two-step mlpotent quadratic Lie algebra (or 2SNQ-Lie alge- bra, for short), I] be another 2SN-Lie algebra and JT : I) —»• Dero (g) be a representation of i) by means of skew-symmetric derivations ofg. Then the double extension ofg by i) by means of 71 IS two-step nilpotent if and only if TT isa 2SN-admissible representation of \:\ in g.
Proof We can prove directly by checking the conditions of Defimtion 5 for the Lie algebra g However, it is easy to see that 0 is the semi-direct product of 1) by 0 ffi h* by means of TT © ad' where g ffi !i* is the central extension of g by \f by means of ip (see Definifion 3), Therefore,
the result follows. D Combining this with Propositions 7 and 8, we obtain the following result:
Corollary 1 Let (g, B) be a 2SNQ-Lie algebra and D e Deru(0) be a skew-symmetric derivation of g. Then the double extension of Q by means of D is a 2SNQ-Lie algebra if and only ifD'^^Q and [D(x), y] ^ Ofor allx. y € g.
Theorem 1 Let (g, B) be a 2SNQ-Lie algebra of dimension « -|- 2, H > 0. Then g is a double extension of a 2SNQ-Lie algebra of dimension n. Consequently, every 2SNQ-Lte algebra can be obtained from an Abelian algebra by a sequence of double extensions by the one-dimensional algebra.
Two-Step Nilpoieni Quadratic Lie Algebras and 8-Dimensional
Proof If g is Abelian then g is the double extension of an Abelian algebra by means of the zero map. If g is non-Abelian then there exists a central element x such that x is isotropic.
Then there is an isotropic element y such that Bix. y) = 1 and g is a double extension of (h = ( C T ffi C y ) ^ . fi') where fi' = fl|i,xb- Certainly, I) is still two-step mlpotent D
Next, we will consider a 2SNQ-Lie algebra from the point of view of T'-extensions. We have the first result as follows
Proposition 10 Let Q be a Lie algebra, d be a cyclic 2-cocycle of g wilh value in g* and T,'(g) be Ihe T'-extension of Q by means of 6. Then T^ig) is a 2SNQ-Lie algebra if and only ifg is two-step nilpotent and 6 satisfies
e(x,y)oadg(z)-\-9([x.y]g.z) = 0 Vx. y, zeg.
Proof The proof is obtained by checking directly Definition 5 for Tyig). G By the above proposition, if Tgig) is a 2SNQ-Lie algebra then 0 should be two-step
mlpotent. However, we can only consider TJ"-extensions of an Abelian algebra by the fol- lowing proposition.
Proposition 11 Let (g, B) be a reduced quadratic Lie algebra. Then g is rwo-step nilpotent if and only if il is i-isomorphic to a Tg-extension of an Abelian algebra by means of a non-degenerate cyclic 2-cocycle 0.
Proof Assume lhat g is two-step nilpolent then [g, g] C Zig) Since g is reduced one has lg,g] = Z(g) and dim(g) even By Proposition 4 (given in [3]) and for Z(g} a totally isotropic ideal, we write g = V © Z(g) wilh V lolally isotropic. We can identify V wilh the quoUent algebra f) ~ 0/2(g) and Z(g) with \f. Then g is i-isomorphic to the T^-extension of h by e defined by
9{poix).Poiy))=<t>{pi{U.y])) Vx, y € 0 .
where po and p\ are respectively the projections from g into V and 2(0) Cenainly, h is Abelian since [0, g] = 2 ( 0 ) . We wnte g = h ffi h* and the bracket on g becomes
[X + / , y + g]= eix. y) Wx. y e il. / . g e l ) ' Since 2(0) = h" then 0 is non-degenerale on h x h-
Conversely. if 0 is i-isomorphic to the T^"-extension T^il]) of an Abelian algebra h by means of a non-degenerate cyclic 2-cocycle 0, il is obvious that 0 is iwo-step mlpotent and 2(0} 2: Z(7;;(li)) = h"- Since b* is lolally isotropic then 2(0) is also lolally isotropic.
Therefore, g is reduced. ^ As a consequence, we have a restricted definilion for the reduced two-slep nilpolent case
as follows.
Definition 6 Lei h be a complex vector space and ^ h x I) -^ h' be a non-degenerate cyclic skew-symmetric bilinear map. Let g = h ffi fi' be the vector space equipped with the bracket
\x-\-fs+g]=0(x.y)
and thebilinearform B(.x+ / . y - f - g ) = / ( v ) - l - g ( x ) forall-\. y e t ] - / . g e I)'-Then (g, B) IS a 2SNQ-Lie algebra. We say that g is the 7"*-exlension of h by 9.
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T h e o r e m ! Letg andg' be T'-extensionsof i] by 6] and02 respectively. Then 1. There exists a Lie algebra isomorphism between g and g' if and only if there exist an
isomorphism A\ off) andan isomorphism Ai ofi)' such that A2{edx,y))^62{Ai{x),A,(y)) Vx, y £ f)-
2. There exists an i-isomorphism between g and g' if and only if there exists an isomorphism A\ of\) such that
e , ( x , y ) - S 2 ( A , ( x ) , A , ( y ) ) o A i Vx, y e f i .
1 Let A 0 ^ - g' be a Lie algebra isomorphism. Since f)* — 2(g) — Z(g') is stable by A then there exist linear maps Ai : i] ^> i), A'y : i) —^ f)* and A2 :1)* ^ [)* such that
A(x-\-f)^Atix)-\-A\(x) + A2(f) Vxef), f€i)\
It is obvious that A2 is an isomorphism of t)*. We show that Ai is also an isomor- phism of h Indeed, if there exists XQ €i} such that Ai(xo) = 0 then 0 — [A(xo),g']' = A([xo, A"'(g')]) —A([xo,g]) It means that [XQ, g] —0 Since [)* — 2(g) thenxo = 0.
Therefore, A| is an isomorphism of f).
F o r a l l x , y e i ) , / , g e t ] ' , one has
A{[x + f,y + g])^A{Odx,y))=A2{0dx,y)) and [A(x + / ) , A(y + g)]' - 02(Ai(x), A,(y))
Therefore, A2(6'i (x, y)) ^ 92(Ai (x). A, (y)) for all x, y e 1).
Conversely, if there exist an isomorphism Ai of f) and an isomorphism A2 of ^*
such that A2(^i(jc, y)) = 6'2(Ai(x), Ai(y)) for all x, y e f], we define A : g ^ g' by A ( x - f / ) ^ Ai(x) -b A 2 ( / ) for all X 4 - / € g. Thenit IS easy to check lhat A is a Lie algebra isomorphism.
2 Assume A . g ->- g' is an i-isomorphism then there exist A] and A2 defined as in Part 1.
Let A: € h, / e r then B'(A(x), A ( / ) ) - B(x, f) implies A2(/)(A|(x)) = f(x).
Therefore, A2(/) ^ / o A 7 ' for all / e ij'.
On the other hand, since A2i8i(x,y)) — &2(Ai(x), Ai(y)) we obtain Si (X, y) - ^2(Ai(x), Ai(y)) o A, Vx, y e f]
Conversely, define A(x -1- / ) = A[(x) -I- / o A^' for all x e f), / e tj* then A is an i-isomorphism
P Example 1 We keep the notations as above. Let 0' be the r''-extension of i] by 9' = Xfl where A 5^ 0. Then g and 0' are i-isomorphic by A : 0 ^ g' defined by A(x + f)= k'^'^x -1- X"'V f o r a l l x - b / e g
For a non-degenerale cyclic skew-symmetnc bilinear map 9 of Ij, define the 3-fonn I{x,y,z)=9(x,y)z Vx, j - , z e l ) .
Then / e A^(\]} where ^^(h) is the space of alternating 3-forms on ^. The non-degenerate condition of 6 is equivalent lo (j(/) ^ 0 for all x e 1} \ {0) where
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Two-Step Nilpotent Quadratic Lie Algebras and 8-Dimensional
ixil)iy,z) = I(x,y,z) Ny,z€l).
Conversely, let 1] be a vector space and / e A^(i)) such that i j ( / ) / 0 for every non- zero vector X e [)• It means that the 3-form I has rank(/) = dim(h). Define S : h x E] -*•
I)' by 0(x, y) = / ( x , y, •) for all x, y e h then 9 is skew-symmetric and non-degenerale.
Moreover, smce / is alternating, 0 is cyclic and therefore we obtain a reduced 2SNQ-Lie algebra T^(\}) defined by 6. It implies that there is aone-lo-one map from the sel of all T*- extensions of 1} onto the subset [/ e A^(i}) \ i j ( / ) / 0 V x e f] \ {0|| We have a corollary of Theorem 2 as follows
Corollary 2 Let g and g' be T*-extensions ofi\ with respect to I\ and f;. Then g and g' are i-isomorphic if and only if there exisis an isomorphism A off] such that I\(x,y,z) — h{Aix),A(y),A(z)) for allx, y, z e I).
Notice that the 3-form / above is the 3-form associated to Tg (f)) which has been defined in [13]. In this case, the Lie bracket on Tg(i)) is given by
[x,y]^h^yil} Vx, y e t ] .
Lemma 1 Let i) be a vector space and I e A^it)) satisfying !j(/) 7^ 0 V x e 1) \ (0). / / there are nontrivial subspaces i]|, hi "/1] such that t] — I)] ffi 1)2 ' " " ' 11^ decomposed by I = Iy-\-12 where I] € A^(i}]), h e J^i^i)^ then the T'-extension ofi) with respect lo I is decomposable.
Proof Let a — t]i ffit]^ be the T^-extensionof f)i with respect to /] then a is non-degenerale.
Denote by g the T'-extension of I] with respect to / . We wil! show lhat a is an ideal of g Indeed, one has
[1)1,1)1 ffiil2]-/(f)i. 1)1*1)2, ) = / ( b i . l ) i , 0 + /(hi,[]2,-).
Since /(t)i, 1)2, 0 = 0 then [f)i. [)i ffi I);] = / ( I ) i , t ] , , - ) - / | (hi, hi-0 C ht-Therefore, a is
an ideal of g and then g is decomposable Q Denote by N(2n) the sel of i-isomorpbic classes of 2/1-dimensional reduced 2SNQ-Lie
algebras. By Appendix and Example 1, il is obvious that ^ ( 2 ) = N(4) = 0 and ^ ( 6 ) bas only an element (also given in [12, 13]) as follows. Let ge be a 6-dimensional Lie algebra spanned by (X,, X ' ) , 1 < i < 3, The 3-form / associated 10 06 is defined by / - X ; A X J A Xy Then the Lie bracket is given by
[Xj,X2]-X3*, [X2,Xi] = X1 and [ X 3 , X i ] - x ; By Appendix combined wilh Lemma 1, A'(8) ^ 0 and A'(IO) contains only an element gio defined as follows: let {X,, X;], 1 < i < 5, be a basis of g.o such lhat its associated 3-form is / = X* A (XJ AX^ + X^A X | ) . Then tiie Lie bracket defined on g.o is given by
[X|, X2] = X;, [X,. Xj] = XI [X.,, X|] - X,-, [X,,X4] = X5", [ X 4 , X 5 ] - X ;
and\Xs.Xi] = X;.
Remark thai in the 12-dimensional case, we have two reduced 2SNQ- Lie algebras not i-isomorphic but there is only an indecomposable one corresponding to the 3-form / - P] A £2 A 64 -I- 62 A ej A es -b ei A £3 A Cfy in Appendix. In higher dimensions, N(2n) # 0 if
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4 Non-commutative Symmetric Novikov Algebras of Dimension 8
Definition 7 An algebra 91 over C widi a bilinear product 01 x 01 - * 5T, (x, y) i-> x_v is called a Novikov algebra if it satisfies
(xy)z -x(yz) = (yx)z - yixz)
(xy)z = {xz)y
foralix, y, 2 € 01. In this case, the commutator [x, y] — xy-yx of 51 defines a Lie algebra, denoted by 0(01), which is called the sub-adjacent Lie algebra of 01.
If 01 IS also endowed with a non-degenerate associative symmetric bilinear form then it is called a symmetric Novikov algebra
Given an algebra 01, denote by Z(01) — (x e 911 xy — yx V y e 01) the center of 01 and Ann(Ol) — [x e 01 | xOl — Olx ~ 0] the annihilator of 01. Then we have the following result.
Lemma 2 /fOl is a symmetric Novikov algebra thenZ(^) — [g(01),g(01)]-'- and AnTi{% = (01^)^.
We recall here a remarkable properly of symmetnc Novikov algebras given in [1] as follows.
Proposition 12 Let (01, B) he a symmetric Novikov algebra, then the product on 01 is as- sociative and the sub-adjacent Lie algebra g(Ol) of 01 with the bilinear form B becomes a two-step nilpotent quadratic Lie algebra.
As quadratic Lie algebras, we have the notion of reduced symmetric Novikov algebras as well as the work on symmetnc Novikov algebras can be only done on reduced ones [5], DeHnition 8 A symmetric Novikov algebra 01 / | 0 | is reduced if Ann(OT) is totally isotropic, or equivaiently, Ann{Ol) C 01^.
In classifying non-commutative symmetric Novikov algebras, we need the following lemma.
Lemma 3 [5] Lei 'Ti he a non-commutative symmetric Novikov algebra, /f 01 is reduced then
3 < dim{Ann(Ol)) < dim(Ol^) < dim(Ol) - 3
Since symmetric Novikov algebras up to dimension 7 have been studied in [1, 5, 14]
then we begin with (01, B) an 8-dimensional non-corrunutative symmetric Novikov algebra and assume 01 reduced By the lemma above, we consider the first case: dim(Ann(01)) = 3, Then dim (01^) = 5
Lemma 4 Lef a be a complex 2-dimensional associative commutative algebra equipped with a non-degenerale associative symmetric bilinear form B Then a is i-isomorphic to one of the following algebras:
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Two-Step Nilpotent Quadratic Lie Algebras and 8-Dimensional (i) 0 ^ span(ei. ^2) with a^ = (0).
(ii) a = span{eue2] where el = et, B(e,,e2) = 1 arw/fi(e,,e,) = fi(e2, ^3) = 0 . (iu) a = span{ei,^2} where e,e2 = cj, e^ = Xe, -|- ej with A e C, 8(61.^2) = 1 and
B(et.ej) = B(e2.e2) = 0.
(iv) a = spaniel.^2) where e,e2^eu el = £ 3 , e\ = Xe2 with A. e C, B(ey,e2) = 1 and B(et,e\) = Bie2.e2) = Q.
(v) a is decomposed by a = Cej ffi Ce2. where e] = Xe^ el ^ i^e-, with k, p € C, fi(e,,£i) = S ( e 2 . e 2 ) = l .
Proof The first assertion corresponds to dim(o^) — 0. We assume dim(a^) = 1 and a- — Ce,.
If fi(ei,ei) = 0 tiien there exists 62 G a such that e(e2,f2) = 0 and fi(ei,e2) = 1. In this case, by fi associative and non-degenerate, it is easy to show e\e2—e} — 0 and ej — Ae,, where >. e C and A ?t 0, SetX'/^e, by^, and X"''^€2 by ^2 then we obtain (ii).
If fi(e|,ei) 5^0 then we can choose e, such that Bie\,e\) — 1 and ^2 e e-j- such that fi(^2, ^2) = L In this case, it is easy to check that el = e2ej = 0 and ^f = Xe, with A e C and A 7^ 0, That is a particular case oflhe last statement.
It remains to consider dim(a^) = 2 . Choose a basts {fi.f:} of a satisfying fi(ei,e,) = fi(^i, ^2) = 0 and B(e,, e2) = 1 We assume that
e,^, =xe, -\- yci. ^1^2 = ze\ -b/ci and ^2^2 — MSI + ve2.
where x, y. z, f, «, r s C. Since fi is associative, one has x = r and s — 1; We rewrite e\e\=xe\-^ ye2, e\e2 = ze\ •\-xe2 and ^3^2 = " ^ i + ' ' ' 2 By the associativity of the product on a, one has xz—\u
• If X = 0 then y = 0 or w = 0. If y — 0 then we obtain e\e2 = ;ei and c^c; = we, -H ;e2 where z must be nonzero. Set ze[ by e\ and E~'CI by e'2, then one has the assertion (iii).
If « = 0 then we have eie, = ye2, ^1^3 = zei and 63^3 = 263 where z must be nonzero.
Set ze\ by e\ and z~'e2 by e'2. then one has the assertion (iv). Note that the case z — 0 is equivalent to the case x = 0.
• Ifx # O a n d z ? ^ O t h e n y / O a n d u ^ O Replacee, byw'^^f, -x'-'^ei ande2 by w'/^e, -\- x"^e2. Then one has Bie\. ei) = 0, cic? = 0, ef = Ae, and c; = /ie;. where A and p. are nonzero in C.
n
We tum to (01, 6 ) an 8-dimensional reduced non-commutative symmetrii. No\ ikov alge- bra having dim(Ann(01)) = 3. We have Ann(Ol) c 01". Let {z,. Z2. z j | be a basis of Ann(Ol), By [4], there exist isotropic elements xi, .ii. X3 and a 2-dimensional subspace W of 01 such lhat B(x,.z,) =&,j, I <i < 3, and 01 ^ (Ann(Ol) ffi V) ffi IV, where V = span{xi,.1:2,^:3}
and W = (Ann(Ol)ffi V)-^, Assume W = span|yi,y2} dien 0^- ^ Ann(01)-^ = span|j',,Ej}, 1 < I < 2 and 1 < ;• < 3. Since 01 is non-commutative, g(01) is non-Abelian. By the classi- fication of two-slep nilpoieni quadratic Lie algebras given in Sect. 3. 0(01) = g^ ffi [12 where 06 is Ihe reduced two-step nilpotent quadratic Lie algebra of dimension 6 and 1)2 's the Iwo-dimensional Abelian Lie algebra Therefore, dim(Z(01)) = 5, By Lemma 4,24 in [5], 0!^ C Z(01) and then Z(01) = 01".
We consider the quotient algebra n = Ol-/Ann(01). then 0 is two-dimensional and com- mutative. Moreover, there is a non-degenerale associative symmetric bilinear form B on a
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defined by B(x. y) = B(x, y), where the notion x denotes the class of an element x e Or According to Lemma 4(i)-(v), we have
1 a- = (0), tiiat is, (Ol-)(01-) c Ann(Ol). Therefore. ((01^)01)01 - (01-)(01^) C Ann(Ol).
That means Ot is t-step mlpotent with it < 4. We give an example in the 3-step nilpolent case as follows. Define the product on 01 by
x\^y\. -Tiyi = y i X | = 2 , . x , x 2 ^ z 3 . -r| ^ >:• Jr2 V2 ^ .^2-V2 = Z2, ^2-^3 ^ Z l . XzXi = Z2
with fi(yi. yi) = B(y2. yj) = 1, B(_yi. v;) - 0. _
2. a ^ s p a n { y , , y 2 | where jS" = yi, fi(yi.yi) = fi(y3.y2) = O a n d fi(yi, i ' ^ ) ^ 1. We will prove that y2.'^ € Cyj ® Ann(Ol) for all x e 01. Indeed, assume y3X = ay, -h feyi H- z with z G Ann(Ol) then
(i2x)x = (ayi -\-by2-\-z)x=aytx-\-b\2X=ayiX -i-abyi -i-b^y2-)-bz.
Note that (y2x)x = y2X^ e Cy, ffi Ann(Ol) If a = 0 then 6 = 0 If a ?^ 0 Uien yiX = -b^y2-\-y.
where y e Cy, ffi Ann(Ol) So 0 = (yiy2)^ = iyix)y2 = -b-yl -b y.V2. and there- fore h — 0. We can conclude y2^ e Cy, ffi Ann(Ol) for all j : e 01, As a consequence, B(y2X.y'2y) — 0 for all x, y e 01. By tiie invariance of B, one has Biyj.xy) — B(yt. xy) = 0. This is a contradiction since y; e 01^. Therefore, this case does not hap- pen.
3 Let us give a not solvable example corresponding to Lemma 4(iii) as follows:
-ri.V3=y2-ri = y i . y3- = Ay, -f-y2 + zj +Z2. J:2-<:3=C,.
witii S(y,, y2) = 1. S(yi, y,) ^ B(.i2, y:) = 0.
4 Let us give a not solvable example corresponding to Lemma 4(iv) as follows:
-»^i.v2 = V2J:I = j 2 . yty2=y2y\ =y}+zi. jr2X3=zi, yi = >':• v'f = Ayi. XiX\ = Z2 witii B(y,.y2) = 1. fi(y,,y,)- fi(y2.y2) - 0 .
5, a = span(y,,y2} where y , ' = Ay,, y3-=/fy2, >-, fisC, fi(yi, y,) ^ fi(f2. ^2) = 1 and Biyi. V2) = 0 . We can assume A ^ 0. We will prove y]y3 = 0. Indeed, if v,y2 = v ?^0 with y e Ann(Ol) tiien tiiere is x such that fi(yiy2.-<:) ^ 0 . It results in 6(y].y2X) ?t 0, and therefore yzx = ay, -f by2 H- z. where a ?^ 0, fo e C and z € Ann(Ol). Right- multiplying by V], we have (y2J:)yi = ay^ -|- foy^y, =a(Xy^ +t)-^by with t e Ann(Ol).
But (y2-v)yi — (y2yi)x = yx — 0, which is a contradiction So y,y2 = 0. As a conse- quence, y, (01-) c C v f .
Set u = yf, then for all r e 01 we have tu = t(yj) = (ry,)yi € Cyf = CM. That means Cii is an ideal of 01. Moreover, CM is non-degenerate, so 01 is decomposable (see Lemma 4.5 in [5]).
It remains to consider dim(Ann(01)) = 4. Then dim(Ol-) = 4 and therefore Ann(Ol) = 01' Ii means that 01 is iwo-step mlpotent. This case is rather tnvial since a two-slep nilpotent algebra is obviously a Novikov algebra.
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Two-Step Nilpotent Quadratic Lie Algebras and S-Dimensioaal
Acknonledgemeals I would like to thank D. Amal and R. Ushirobira for many valuable remaAs and dis- cussions conceming results in Sect 3. Moreover, I am very grateful to them for Iheu fnendly encouragements and help.
This work is supported by the Foundation for Science and Technology Project of Vietnam Minisuy of Education and Traimng
Appendix
Let V be a vector space over C. We recall here a classification of 3-vectors in /\^ l^ (up to an isomorphism in the linear group GL(V)) with 1 <dim(V') < 7 that can be applied in the classification of two-step nilpolent quadratic Lie algebras in low dimensions These results can be found in [ I I ] .
It is obvious lhat / = 0 if dim(V) = 1 or 2. If dim(V) = 3 and / e / \ V is a non-zero 3-veciorthen there exists a basts [ a i , « 3 , a 3 | of V such that / — a a , Aa2 Aas, Replacing a, by a " ' a , , we get the result
Assume dim(V) = 4, It is easy to check lhat every 3-veclor in / \ V is decomposable Hence /\ V has only a non-zero 3-vector up to isomorphism. In the case of dim(V) = 5, one has the following proposition.
Proposition 13 If I is an indecomposable 3-vector on V with din\(V) — 5 then there exists a basis [a\,a2,a3,04,05] of V such that
I —a\ A (0:3 AQ:3 -|-a4 Aas), In the case of higher dimensions, we have
Proposition 14 Let V bea vector space such that dim(V) > 6, Then there exists an element / 6 A ' ^ having rank.(l) =dim(V)
Proof We denote n = dim(V) and fix a basis (a,, .,.ar„| of V. Then the element / is defined as follows:
• \fn=Zk where i > 2 then we sel
I =a\ AQ-T Aofj -I i - O n - 2 ^ " F I - I A Q : „ .
• lfn = 3k + l = 3(k~2) + l where k>2 tiien we sel / = 0 ! , A O ; AQIj -b • •-'r a„-<) Aa„-g A a„-7
-(-a„-6 A (ar„_5 Aa:„_4-|-a„_3 A0'„_2 -t-ffn-i A a.,).
• Ifn =3Jt + 2 = 3 ( i - 1 ) + 5. where* > 2 then we sel / = « , A0;2 A t t 3 -1 \-a„-2 A a n _ 6 Affn_5
-i-a„^i A (a„-3 AHn-i -l-a„_i Aof^).
D
Finally, a classification of 3-veciors in the 6- and 7-dimensional cases is given as follows:fi Spnnger
148 M T. Duong
P r o p o s i t i o n I S Let V bea 6-dimensional vector space over C . Then there exists a basis e,-, 1 < I < 6 , o f V such lhat every 3-veclor of rank 6in / \ V has tivo non-equivalent types as follows:
1 / i — e , A e2 A e3 -(- e4 A e , A e^, 2. / ; — e, A e2 A e4 -f- e2 A e3 A e^ -b e , A £3 A eg.
P r o p o s i t i o n 1 6 ( S c h o u i e n ) Let V be a 7-dimensional vector space over C Then there exist 5 non-equivalent types of 3-vectors of rank 1 in f\ V as follows:
1. / , — e , A (e2 A e 3 - | - e 4 A e , - l - f 6 A e ? ) , 2. /2 — / , H-€2 A e 4 A C e ,
3. h — e\ A e3 A e3 -b e j A e4 A fij -b es A Cfi A e-;.
4. U = e\ A (e2 A 63 -I- ^4 A eg) H- e i A e4 A eg -)- e3 A e j A ey, 5 . /g — /2 H- e3 A eg A e ? ,
where e,, 1 <i <l,is a basis of V.
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