CC BY
12
УДК 512.572
On Varieties of Leibniz-Poisson Algebras with the Identity {x,y} • {z,t} = 0
Sergey M. Ratseev*
Department of Mathematics and Information Technologies,
Ulyanovsk State University, Lev Tolstoy, 42, Ulyanovsk, 432063
Russia
Received 12.11.2012, received in revised form 12.11.2012, accepted 15.11.2012 Let K be an arbitrary field and let A be a K-algebra. The polynomial identities satisfied by A can be measured through the asymptotic behavior of the sequence of codimensions of A. We study varieties of Leibniz-Poisson algebras, whose ideals of identities contain the identity {x, y} • {z, t} = 0, we study an interrelation between such varieties and varieties of Leibniz algebras. We show that from any Leibniz algebra L one can construct the Leibniz-Poisson algebra A and the properties of L are close to the properties of A. We show that if the ideal of identities of a Leibniz-Poisson variety V does not contain any Leibniz polynomial identity then V has overexponential growth of the codimensions. We construct a variety of Leibniz-Poisson algebras with almost exponential growth.
Keywords: Poisson algebra, Leibniz-Poisson algebra, variety of algebras, growth of a variety.
Introduction
Let A be an algebra over an arbitrary field. A natural and well established way of measuring the polynomial identities satisfied by A is through the study of the asymptotic behavior of it's sequence of codimensions cn(A), n = 1,2,... . The first result on the asymptotic behavior of cn (A) was proved by A.Regev in [1]. He showed that if A is an associative algebra cn(A) is exponentially bounded. Such result was the starting point for an investigation that has given many useful and interesting results.
For associative algebras A.R.Kemer in [2] proved that the sequence cn(A) is either polynomi-ally bounded or grows exponentially. Then A.Giambruno and M.V.Zaicev in [3] and [4] showed that the exponential growth of cn(A) is always an integer called the exponent of the algebra A.
When A is a Lie algebra, the sequence of codimensions has a much more involved behavior. I.B.Volichenko in [5] showed that a Lie algebra can have overexponential growth of the codimensions. Starting from this, V.M.Petrogradsky in [6] exhibited a whole scale of overexponential functions providing the exponential behavior of the identities of polynilpotent Lie algebras.
In this paper we study Leibniz-Poisson algebras satisfying polynomial identities. Remark that if a Leibniz-Poisson algebra A satisfies the identity {x,x} = 0 then A be a Poisson algebra. Poisson algebras arise naturally in different areas of algebra, topology, theoretical physics. We study varieties of Leibniz-Poisson algebras, whose ideals of identities contain the identity {x, y} ■ {z,t} = 0. We show that the properties of such Leibniz-Poisson algebras are close to
* RatseevSM@rambler.ru © Siberian Federal University. All rights reserved
the properties of Leibniz algebras. We show that Leibniz-Poisson algebra can have overexponen-tial growth of the codimensions and construct a variety of Leibniz-Poisson algebras with almost exponential growth.
1. Preliminaries
Let A(+, •, {, }, K) be a K-algebra with two binary multiplications • and {, }. Let the algebra A(+,-,K) with multiplication • be a commutative associative algebra with unit and let the algebra A(+, {, }, K) be a Leibniz algebra under the multiplication {, }. The latter means that A(+, {, }, K) satisfies the Leibniz identity
{{x, y}, z} = {{x, z}, y} + {x, {y, z}}.
Assume that these two operations are connected by the relations (a, b, c G A)
{a • b, c} = a • {b, c} + {a, c} • b,
{c, a • b} = a • {c, b} + {c, a} • b.
Then the algebra A(+, •, {, }, K) is called a Leibniz-Poisson algebra.
We make the convention that brackets in left-normed form arrangements will be omitted:
{... {{xi,x2}, X3}, ...,i„} = {xi,x2, ...,x„}.
Let L(X) be a free Leibniz algebra with multiplication [, ] freely generated by the countable set X = {x1,x2,...}. Let also F(X) be a free Leibniz-Poisson algebra. Denote by PL and Pn the vector spaces in L(X) and F(X) accordingly, consisting of the multilinear elements of degree n in the variables xi , . . . , xn.
Proposition 1 ( [7]). A basis of the vector space Pn consists of the following elements:
xki • • • • • xkr • {xii, • • • , xis } • • • • • {xji , • • • , xjt }, (1)
where:
(i) r > 0, ki < .. . < kr;
(ii) all elements are multilinear in the variebles x1,...,xn;
(iii) each factor {xi1,..., xis},..., {xj-1,..., xjt} in (1) is left normed and has lenngth ^ 2;
(iv) in each product (1) the shorter factor precede the longer: s ^ ... ^ t;
(v) if two consecutive factors in (1) are brackets {...} of equal length
• • • • {xpi, • • • , xPs } • {xqi, • • • , xqs } • • • • ,
then pi < q1.
Denote by rn the subspace of Pn spanned by the elements (1) with r = 0. Denote by L^2(X) the subspace of the free Leibniz algebra L(X) spanned by the elements [xii, ...,xin ] with n > 2. Also denote by PL^2(X ) the subspace of F (X ) spanned by the elements {xii,..., xin} with n > 2. Obviously, L^2(X) = PL^2(X) as Leibniz algebras. We will use only the notation L^2(X) everywhere as L^2(X) = PL^2(X) up to isomorphism of Leibniz algebras.
Let V be a variety of Leibniz-Poisson algebras (pertinent information on varieties of PI-algebras can be found, for instance, in [8], [9]). Let Id(V) be the ideal of identities of V. Denote
Pn(V) = Pn/(Pn n Id(V)), cn(V) = dim Pn(V).
For a variety of Leibniz algebras VL denote
PL(Vl) = PL/(PL n Id(VL)), cL(Vl) = dim PL(Vl).
Let Id,(A) be the ideal of the free algebra F(X) of polynomial identities of A.
The next proposition shows how from every Leibniz algebra one can construct a Leibniz-Poisson algebra with some conditions of the source Lebniz algebra.
Proposition 2 ( [7]). Let AL be a nonzero Leibniz algebra with multiplication [, ] over an infinite field K and let
A = Al © K
be a vector space with multiplications • and {, } defined as
(a + a) • (b + fi) = (fia + ab) + afi, {a + a, b + fi} = [a, b], a, b G AL, a,fi G K.
Then the algebra (A, +, •, {}, K) is a Leibniz-Poisson algebra and the following conditions are true:
(i) Id(AL) = Id(A) n L-^2(X) and the algebra A satisfies the identity {x\,x2} • {x3,x4} = 0;
(ii) for any n ^ 2
rn(A) = PL (A) = PL(Al)
up to isomorphism of vector spaces;
(iii) for any n the following equality holds:
Cn(A) = l + £ (?) • dim PL(Al).
2. Leibniz-Poisson Algebras with Identity
{xl,X2} • {X3,X4} = 0
Denote by Id({xi,x2} • {x3,x4}) the ideal of identities of the free Leibniz-Poisson algebra F(X) generated by the element {xi,x2} • {x3,x4}.
Theorem 1. Let VL be a variety of Leibniz algebras over an infinite field K defined by a system of identities
{fi = 0 | fi G L^(X), i G I} (3)
and let {gj G Id({x1, x2} • {x3,x4}) | j G J}, where | J| > 0, be a set of elements in the ideal Id({x1, x2} • {x3,x4}). Let V be a variety of Leibniz-Poisson algebras defined by the system of identities
{fi = 0, gj =0 | i G I, j G J}.
Then:
(i) MVl) = Id(v) n L^(X);
(ii) PL(V) = P„l(Vl);
n
(iii) Cn(V) > 1+£ (n) • cL(Vl);
k=2
(iv) if |11 = 0 then cn(V) ^ [n! • e] — n, where e = 2.71..., [ ] is «n integer part of a number.
Proof. (i) Let f G Zd(VL). Then f follows from the system of identities (3). Therefore, f G 1d(V)nL^(X) and Jd(VL) Ç 1d(V)nL^(X). We will show that 1d(V)nL^(X) Ç /d(VL).
Let W be a Leibniz-Poisson variety defined by the system of identities (3) and the identity (xi,x2| • {x3,x4} = 0. Since the element |xi,x2} • {x3,x4} generates the ideal /d({xi,x2} • {x3,x4}) and |J | > 0 then WÇV, 1d(V ) Ç 1d(W ).
Let L(X, VL) be the relatively free algebra of the variety VL of countable rank. Theorem of Birkhoff implies that the algebra L(X, VL) generates the variety VL. Hence 1d(VL) = 1d(L(X, VL)). Let A = L(X, VL) © K be a Leibniz-Poisson algebra with the multiplications (2). Proposition 2 implies that A G W, hence 1d(W) Ç Id(A). Proposition 2 also implies the equality
«(Vl) = Jd(L(X, Vl)) = Id(A) n L^(X).
Since 1d(V) Ç 1d(W) Ç Id(A), it follows
1d(V) n L^(X) Ç 1d(W) n L^(X) Ç Id(A) n L^(X) = 1"d(VL).
(ii) Condition (i) implies that 1d(V) n P^ = MVL) n Pjf for any n ^ 2. Therefore,
PL(Vl) = P„L/(/d(VL) n PL) = PL/(1d(V) n P„L) = P„L(V).
(iii) follows from (ii) and [7, Proposition 4].
(iv) Applying the formula
1
n! • > — = [n! • e], ^ k! L J
k=0
inequality from (iii) and P^ = n!, we obtain that
cn(V) > 1 + Ê (n) • k! = 1 + Ê (n—ï)ï =
k=2 k=2
n—2 ! n i
= /t = n — k/ = 1 + t! = n! t! — n = [n! • e] — n. t=0 ! t=0 !
□
Define the lower and upper exponents for the codimension sequence {cn(V)}nas follows: EXP(V) = lim nyCn(V), EXP(V) = US VCn(V).
If the lower and the upper limits coincide, we use the notation Exp(V).
Theorem 2. Let VL be a variety of Leibniz algebras over an infinite field K defined by the system of identities (3) and let V be a variety of Leibniz-Poisson algebras defined by the system of identities (3) and the identity {x1,x2} • {x3,x4} = 0. Then: 1 ) For any n ^ 2
Tn(V ) = PnL(V )= PL(Vl)
up to isomorphism of vector spaces.
2) Let
), s = 1,...,cL(VL), (4)
be a basis of the vector space P^CVl), n ^ 2. Then Pn(V) has a basis
x 1 • • • • • xn ? /
ki \ (5)
x«i ••• us (x3i T--Th3k
k = 2, ...,n, s = 1,... , ^(Vl), ii <...<in-k, ji <...<jk;
3) For any n
n
n
k=2 ^
4) If exponent EXP(VL) exists, then EXP(V) = EXP(VL) + 1, in particular if there exist constants d ^ 0, a and ft such that for all sufficiently large n the double inequality holds
cn(V) = 1 + ¿ Q •dim PL(VL).
nadn < cL(VL) < n3dn,
then there exist constants y and 5 such that for all sufficient large n the following double inequality holds
nY(d + 1)n < cn(V) < ns(d + 1)n.
5) If some Leibniz algebra AL generate the variety VL, then the Leibniz-Poisson algebra A = Al © K with multiplications (2) generates the variety V.
6) If |I| < and the variety VL has the Specht property (i.e. all subvarieties of VL, including VL itself, are finite based), then the variety V has the Specht property.
7) Let W be a proper subvariety of V. Then the ideal of identities Id(W )n L^2(X) determines the proper subvariety of VL.
8) The variety VL is nilpotent if and only if the variety V has a polynomial growth.
Proof. 1) The equality P^(Vl) = P« (V) follows from Theorem 1. Since for any n holds equality
rn = PL © Id({xi, X2} • {X3,£4}) n Tn,
then
Tn(V) =rn/(Id(V) n Tn) =
= PL © Id({xi,x2} • {x3,X4}) n fn = = Id(V) n (PL © Id({xi, x2} • {x3, x4}) n Tn) =
= PL © Id({xi,x2} • {x3,x4}) n fn = = (Id(V) n PL) © (Id({xi, x2} • {x3, x4}) n Tn) =
- pL/(Id(V) n pL) = PL (V).
2) Follows from 1) and [7, Proposition 4].
3) Follows from 2).
n
4) Follows from 3) and the equality (t + 1)n = £ (£) • tk.
k=0
5) Let some Leibniz algebra AL generates the variety VL. Define the Leibniz-Poisson algebra A = Al © K with multiplications (2). Then Proposition 2 and Theorem 1 imply such equalities
Id(A) n L^(X) = Id(AL) = Id(VL) = Id(V) n L^(X), (6)
with 1d(V) C Id(A). We will show that Id(A) C 1d(V).
Denote by B the subspace of the free Leibniz-Poisson algebra F(X) spanned by the elements
{x«i, • • •, xis} • • • • • {xji, • • •, xjt}, s ^ 2,..., t ^ 2.
In particular rn = B n Pn, n = 1,2,... Note that
B = L^(X) © B n Id({xi, X2} • {X3, X4}). (7)
From [7] it follows that the ideal of identities Id(A) is generated by the set of identities B n Id(A). Let f e B n Id(A). Since
Id({xi, X2} • {x3,X4}) C Id(A)
and (7) then
B n Id(A) = L^2(X) n Id(A) © B n Id({x1,x2} • {x3,x4}). Hence there exist unique
g e L^2(X) n Id(A), h e B n Id({x1, x2} • {x3, x4}),
such that f = g + h. (6) implies that g e 1d(V). Obviously, h e 1d(V), hence f = g + h e 1d(V). Thus Id(A) = 1d(V).
6) Let |11 < and the variety of Leibniz algebras VL has the Specht property. Let W be a subvariety of the variety V. Obviously, 1d(W) n L^2(X) is an ideal of identities of the free Leibniz algebra L(X). Theorem 1 implies that
MVl) C 1d(W) n L^2(X).
Hence the ideal of identities 1d(W) n L^2(X) is generated by a finite number of elements
fi, • • •, fk e ¿^2(X).
Using the notations of 5), we have
B n 1d(W) = L^2(X) n 1d(W) © B n Id({x1, x2} • {x3, x4}). (8)
Since 1d(W) is generated by B n 1d(W) (see [7]) then the variety W is generated by the elements fi,..., ffc and {xi, X2} • {X3, X4}.
7) Let W be a proper subvariety of V. Then the strict inclusion 1d(V) £ 1d(W) holds. We will show that
MVl) £ 1d(W) n L^(X),
where 1d(W) n L^2(X) is an ideal of identities of L(X).
Since 1d(W) is generated by the set B n 1d(W) (see [7]) and 1d(V) £ 1d(W), there is such element f e B n 1d(W) that f e 1d(V). Equality (8) implies that there exist unique
g e L^2(X) n 1d(W), h e B n Id({xi, x2} • {x3, x4})
such that f = g + h. Since h e 1d(V) and f e 1d(V), we obtain that
g e l^2(x ) n id(v ) = /d(vL).
Therefore, Jd(VL) £ 1d(W) n L^(X).
8) Follows from 1), 3) and [7, Theorem 1] □
Corollary. Let L(X) be a free Leibniz algebra over infinite field K and let L(X) © K be a Leibniz-Poisson algebra with multiplications (2). Then:
(i) 1d(L(X) © K) n L(X) = {0}.
(ii) 1d(L(X) ©K) = Id({x1, x2} • {x3, x4}), i.e. the ideal of identities of the algebra L(X) ©K is generated by the identity {x1,x2} • {x3,x4} = 0.
Denote by the variety of Leibniz-Poisson algebras defined by the identity {x1,x2} • {x3, x4} = 0. Theorems 1 and 2 imply that the codimension growth of f is overexponential.
Proposition 3. For any n ^ 1 the codimension of the identities of V1 satisfy
cn (V1) = [n! • e] — n.
Proposition 4. Let 3Nf be a Leibniz-Poisson variety, defined by the identity
{X1, {X2, {X3,X4}}} = 0.
Then the variety V1 n 3Nf over a field K of characteristic 0 has almost exponential growth of the codimension sequence.
Proof. [11] and [10] implies that the variety of Leibniz algebras 3N, defined by the identity
[x1, [x2, [X3,X4]]] = 0,
has almost exponential codimension growth. Therefore, by Theorem 1, the variety of Leibniz-Poisson algebras V1 n 3Nf has overexponential codimension growth.
Let W be a proper subvariety of f 1 n 3Nf. Condition 7) of Theorem 2 implies that the ideal of identities 1d(W) n (X) defines the proper subvariety of 3N, which has exponentially bounded codimension growth. By condition 4) of Theorem 2, the sequence of codimensions of W is exponentially bounded. □
Denote by A/*sA the variety of Leibniz-Poisson algebras, defined by the identity
{{X1, X2}, . . . , {x2s+1, X2s+2}} = 0.
Proposition 5. Variety f1 n A/"s A over a field K of characteristic 0 has the Specht property. Proof. [12] implies that the variety of Leibniz algebras A/"sA, defined by the identity
[[X1, X2], ..., [X2s + 1, X2s+2]] = 0.
has the Specht property. Therefore, by 6) of Theorem 2, f 1 n A/"sA has the Specht property. □
References
[1] A.Regev, Existence of identities in A <g> B. Israel J. Math. 11(1972), 131-152.
[2] A.R.Kemer, T-ideals with power growth of the codimensions are Specht. Sibirsk. Mat. Zh. 19(1978), 54-69 (Russian).
[3] A.Giambruno, M.V.Zaicev, On codimension growth of finitely generated associative algebras. Adv. Math. 140(1998), 145-155.
[4] A.Giambruno, M.V.Zaicev, Exponential codimension growth of P.I. algebras: an exact estimate. Adv. Math. 142(1999), 221-243.
[5] I.B.Volichenko, Varieties of Lie algebras with identity [[Xi, X2, X3], [X4, X5, X6]] = 0 over a field of characteristic zero. Sibirsk. Mat. Zh. 25(1984), no. 3, 40-54 (Russian).
[6] V.M.Petrogradsky, Growth of polynilpotent varieties of Lie algebras, and rapidly increasing entire functions, Mat. Sb., 188(1997), no. 6, 119-138 (in Russian).
[7] S.M.Ratseev, Commutative Leibniz-Poisson algebras of polynomial growth, Vestn. Samar. Gos. Univ. Estestvennonauchn. Ser., 94(2012), no. 3/1, 54-65 (in Russian).
[8] Yu.A.Bahturin, Identical relations in Lie algebras, Nauka, Moscow, 1985 (in Russian); English transl. in VNU Science Press, Utrecht, 1987.
[9] V.Drensky, Free algebras and Pi-algebras, Graduate Course in Algebra, Singapore, SpringerVerlag, 2000.
[10] S.P.Mishchenko, T.V.Skoraya, Yu.Yu.Frolova, New properties of the variety of Lebniz algebras 3N, defined by the identity x(y(zt)) = 0, Algebra and number theory: Modern Problems and Applications. Abstracts of VIII International conference, dedicated to the 190th anniversary of P.L.Chebyshev and 120th anniversary of I.M. Vinogradov, Saratov Gos. Univ., Saratov. 2011, 49-51 (in Russian).
[11] L.E.Abanina, S.P.Mishchenko, The variety of Leibniz algebras defined by the identity x(y(zt)) = 0, Serdica Math. J. 29(2003), 291-300.
[12] S.M.Ratseev, On the property of having a finate basis of some varieties of Lebniz algebras, Izvestiya Vysshih uchebnyh zavedenii: Povolzhskii region, 33,(2007), no. 6, 12-16 (in Russian).
О многообразиях алгебр Лейбница-Пуассона с тождеством {x,y} • {z,t} = 0
Сергей М. Рацеев
В данной 'работе исследуются многообразия алгебр Лейбница-Пуассона, идеалы тождеств которых содержат тождество {x,y} ■ {z,t} = 0, исследуется взаимосвязь таких многообразий с многообразиями алгебр Лейбница. Показано, что из любой алгебры Лейбница можно построить алгебру Лейбница-Пуассона с похожими свойствами исходной алгебры. Показано, что если идеал тождеств многообразия алгебр Лейбница-Пуассона V не содержит ни одного тождества из свободной алгебры Лейбница, то рост многообразия V является сверхэкспоненциальным. Приводится многообразие алгебр Лейбница-Пуассона почти экспоненциального роста.
Ключевые слова: алгебра Пуассона, алгебра Лейбница-Пуассона, многообразие алгебр, рост многообразия.
