A SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY A GENERALIZED S ˘ AL ˘ AGEAN AND RUSCHEWEYH OPERATOR

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1

A subclass of analytic functions defined by a generalized S˘al˘agean and

Ruscheweyh operator 1

Alina Alb Lupa¸s 2

Semireflexive subcategories 7

Dumitru Botnaru, Olga Cerbu 3

The Frattini theory for p - Lie algebras 21

Camelia Ciobanu 4

A characterization of n-tuples of commuting gramian subnormal operators 29 Loredana Ciurdariu

5

Darboux integrability in the cubic differential systems with three invariant

straight lines 45

Dumitru Cozma 6

Stability criteria for quasigeostrofic forced zonal flows; I. Asymptotically

vanishing linear perturbation energy 63

Adelina Georgescu, Lidia Palese 7

A Petri nets application in the mobile telephony 77

Monica Kere 8

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9

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Mariana Lut¸˘a, Emil M. Popa, Claudia Andra Dr˘agan, Constantin Bogdan Milian 10

Ekman currents on the Romanian Black Sea shore 97

Angela Muntean, Mihai Bejan

v

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vi 11

Transversality Conditions for infinite horizon optimization problems: three

additional assumptions 105

Ryuhei Okumura, Dapeng Cai, Takashi Gyoshin Nitta 12

A generalized mathematical cavitation erosion model 113 Constantin P˘atr˘a¸scoiu

13

Classification of the cubic differential systems with seven real invariant

straight lines 121

Vitalie Put¸untic˘a, Alexandru S¸ub˘a 14

Object Classification Methods with Application in Astronomy 123 Corina S˘araru

15

Universal regular autonomous asynchronous systems: fixed points, equiv-

alencies and dynamical bifurcations 131

S¸erban E. Vlad 16

Bogdanov-Takens singularities in an epidemic model 155 Mariana P. Trifan, George-Valentin Cˆırlig, Dan Lascu

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A SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY A GENERALIZED S ˘ AL ˘ AGEAN AND RUSCHEWEYH OPERATOR

Alina Alb Lupa¸s

Department of Mathematics and Computer Science, University of Oradea [email protected]

Abstract By means of a S˘al˘agean differential operator and Ruscheweyh derivative we define a new class BL(p, m, µ, α, λ) involving functionsf ∈ A(p, n). Paral- lel results, for some related classes including the class of starlike and convex functions respectively, are also obtained.

Keywords: analytic function, starlike function, convex function, S˘al˘agean operator, Ruscheweyh derivative.

2000 MSC:30C45.

1. INTRODUCTION AND DEFINITIONS

Denote byU the unit disc of the complex plane,U ={z∈C: |z|<1}and H(U) the space of holomorphic functions in U.

Let

A(p, n) ={f ∈H(U) : f(z) =z^p+ P^∞

j=p+n

a_jz^j, z∈U}, (1) withA(1, n) =A_n and

H[a, n] ={f ∈H(U) : f(z) =a+a_nzⁿ+a_n+1zⁿ⁺¹+. . . , z ∈U}, wherep, n∈N,a∈C.

LetS denote the subclass of functions that are univalent inU.

ByS^∗_n(p, α) we denote a subclass ofA(p, n) consisting of p-valently starlike functions of orderα, 0≤α < pthat satisfy

Re

µzf⁰(z) f(z)

¶

> α, z∈U. (2)

1

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2 Alina Alb Lupa¸s

Further, a functionf belonging toSis said to bep-valently convex of order α inU, if and only if

Re

µzf⁰⁰(z) f⁰(z) + 1

¶

> α, z∈U (3)

for some α, (0≤α < p). We denote by K_n(p, α), the class of functions in S which are p-valently convex of orderα in U and denote by R_n(p, α) the class of functions inA(p, n) which satisfy

Ref⁰(z)> α, z∈U. (4)

It is well-known that K_n(p, α)⊂S^∗_n(p, α)⊂S.

If f and g are analytic functions in U, we say that f is subordinate to g, writtenf ≺g, if there is a functionwanalytic inU, withw(0) = 0,|w(z)|<1, for all z ∈ U such that f(z) =g(w(z)) for all z ∈ U. If g is univalent, then f ≺g if and only iff(0) =g(0) andf(U)⊆g(U).

Let S^m be the S˘al˘agean differential operator [7], S^m :A(p, n) → A(p, n), p, n∈N, m∈N∪ {0} , defined as

S⁰f(z) = f(z),

S¹f(z) = Sf(z) =zf⁰(z),

S^mf(z) = S(S^m−1f(z)) =z(S^m−1f(z))⁰, z∈U.

In [6] Ruscheweyh has defined the operatorR^m :A(p, n)→A(p, n),p, n∈ N, m∈N∪ {0},

R⁰f(z) = f(z), R¹f(z) = zf⁰(z),

(m+ 1)R^m+1f(z) = z[R^mf(z)]⁰+mR^mf(z), z∈U.

LetD_λ^m be a generalized S˘al˘agean and Ruscheweyh operator introduced by A. Alb Lupa¸s in [1], D_λ^m :A(p, n)→ A(p, n),p, n ∈N, m∈N∪ {0}, defined as

D_λ^mf(z) = (1−λ)R^mf(z) +λS^mf(z), z∈U, λ≥0.

We note that iff ∈A(p, n), then D^m_λf(z) =z^p+

X∞

j=n+p

¡λj^m+ (1−λ)C_m+j−1^m ¢

a_jz^j, z∈U, λ≥0.

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Forλ= 1, we get the S˘al˘agean operator [7] and forλ= 0 we get the operator [6].

To prove our main theorem we shall need the following lemma.

Lemma 1.1. [5] Let u be analytic in U, withu(0) = 1,and suppose that Re

µ

1 +zu⁰(z) u(z)

¶

> 3α−1

2α , z∈U. (5)

Then Reu(z)> αfor z∈U and 1/2≤α <1.

2. MAIN RESULTS

Definition 2.1. We say that a functionf ∈A(p, n)is in the classBL(p, m, µ, α, λ), p, n∈N, m∈N∪ {0}, µ≥0, λ≥0, α∈[0,1) if

¯¯

¯

D^m+1_λ f(z) z^p

µ z^p D^m_λf(z)

¶µ

−p

¯¯

¯< p−α, z∈U. (6) Remark 2.1. The familyBL(p, m, µ, α, λ)is a new comprehensive class of an- alytic functions which includes various new classes of analytic univalent func- tions as well as some very well-known ones. For example, BL(1,0,1, α,1)≡

S^∗_n(1, α), BL(1,1,1, α,1)≡K_n(1, α) and BL(1,0,0, α,1)≡R_n(1, α). Another interesting subclasses are the special case BL(1,0,2, α,1)≡B(α) which has been introduced by Frasin and Darus [4], the class BL(1,0, µ, α,1)≡B(µ, α) introduced by Frasin and Jahangiri [5], the class

BL(1, m, µ, α,1) =BS(m, µ, α) introduced and studied by A.C˘ata¸s and A. Alb Lupa¸s[2]and the classBL(1, m, µ, α,0) =BR(m, µ, α)introduced and studied by A.C˘ata¸s and A. Alb Lupa¸s [1].

In this note we provide a sufficient condition for functions to be in the class BL(p, m, µ, α, λ). Consequently, as a special case, we show that convex functions of order 1/2 are also members of the above defined family.

Theorem 2.1. If for the function f ∈ A(p, n), p, n ∈ N, m ∈ N∪ {0}, µ≥0, λ≥0, 1/2≤α <1 we have

(m+ 2) (1−λ)R^m+2f(z)−(m+ 1) (1−λ)R^m+1f(z) +λS^m+2f(z) (1−λ)R^m+1f(z) +λS^m+1f(z) − (7) µ(m+ 1) (1−λ)R^m+1f(z)−m(1−λ)R^mf(z) +λS^m+1f(z)

(1−λ)R^mf(z) +λS^mf(z) +pµ−p+1≺1+βz,

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4 Alina Alb Lupa¸s z∈U, where

β = 3α−1 2α , thenf ∈BL(p, m, µ, α, λ).

Proof. If we consider

u(z) = D^m+1_λ f(z) z^p

µ z^p D^m_λf(z)

¶^µ

(8) thenu(z) is analytic in U with u(0) = 1. A simple differentiation yields

zu⁰(z)

u(z) = (m+ 2) (1−λ)R^m+2f(z)−(m+ 1) (1−λ)R^m+1f(z) +λS^m+2f(z) (1−λ)R^m+1f(z) +λS^m+1f(z) −

(9) µ(m+ 1) (1−λ)R^m+1f(z)−m(1−λ)R^mf(z) +λS^m+1f(z)

(1−λ)R^mf(z) +λS^mf(z) +pµ−p.

Using (7) we get

Re µ

1 +zu⁰(z) u(z)

¶

> 3α−1 2α . Thus, from Lemma 16, we deduce that

Re (

D^m+1_λ f(z) z^p

µ z^p D_λ^mf(z)

¶^µ)

> α.

Therefore, f ∈BL(p, m, µ, α, λ),by Definition 2.1.

As consequences of the above theorem we have the following corollaries.

Corolar 2.1. If f ∈A_n and Re

(9zf⁰⁰(z) + ⁷₂z²f⁰⁰⁰(z)

f⁰(z) +¹₂zf⁰⁰(z) −2zf⁰⁰(z) f⁰(z)

)

>−5

2, z∈U (10)

then

Re

½

1 +zf⁰⁰(z) f⁰(z)

¾

> 3

7, z∈U. (11)

That is, f is convex of order ³₇. Corolar 2.2. If f ∈A_n and

Re

½

1 +zf⁰⁰(z) f⁰(z)

¾

> 1

2, z∈U (12)

then

Ref⁰(z)> 1

2, z∈U. (13)

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In other words, if the functionf is convex of order ¹₂ thenf ∈BL(1,0,0,¹₂,1)≡ R_n¡

1,¹₂¢ .

Corolar 2.3. If f ∈A_n and Re

½zf⁰⁰(z)

f⁰(z) −zf⁰(z) f(z)

¾

>−3

2, z∈U (14)

then

Re

½zf⁰(z) f(z)

¾

> 1

2, z∈U. (15)

That is, f is a starlike function of order ¹₂. Corolar 2.4. If f ∈A_n and

Re

½2zf⁰⁰(z) +z²f⁰⁰⁰(z)

f⁰(z) +zf⁰⁰(z) −zf⁰⁰(z) f⁰(z)

¾

>−1

2, z∈U (16)

thenf ∈ BL(1,1,1,1/2,1)hence Re

½

1 +zf⁰⁰(z) f⁰(z)

¾

> 1

2, z∈U. (17)

That is, f is convex of order ¹₂.

References

[1] A. Alb Lupa¸s,Some diferential subordinations using S˘aj˘agean and Ruscheweyh oper- ators, Proceedings of International Conference on Fundamental Sciences, ICFS 2007, Oradea, 58-61.

[2] A. Alb Lupa¸s, A. C˘ata¸s,A note on a subclass of analytic functions defined by differential Ruscheweyh derivative, Journal of Mathematical Inequalities, (to appear).

[3] A. Alb Lupa¸s, A. C˘ata¸s,On a subclass of analytic functions defined by a generalized S˘al˘agean and Ruscheweyh operator, Journal of Approximation Theory and Applica- tions,4, 1-2(2008), 1-5.

[4] A. Alb Lupa¸s, A. C˘ata¸s,A note on a subclass of analytic functions defined by a gener- alized S˘al˘agean and Ruscheweyh operator, General Mathematics, (to appear).

[5] A. C˘ata¸s, A.Alb Lupa¸s,A note on a subclass of analytic functions defined by differential S˘al˘agean operator, submitted.

[6] B. A. Frasin, M. Darus, On certain analytic univalent functions, Int. J. Math. and Math. Sci.,25(5)(2001), 305-310.

[7] B. A. Frasin, J. M. Jahangiri,A new and comprehensive class of analytic functions, Analele Universit˘at¸ii din Oradea,XV, 2008.

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6 Alina Alb Lupa¸s

[8] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1975), 109-115.

[9] G. St. S˘al˘agean,Subclasses of univalent functions, Lecture Notes in Math., Springer, Berlin,1013(1983), 362-372.

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SEMIREFLEXIVE SUBCATEGORIES

Dumitru Botnaru, Olga Cerbu

Academy of Transport, Computer Science and Communications, Chi¸sin˘au, Republic of Moldova State University of Moldova, Chi¸sin˘au, Republic of Moldova

[email protected], [email protected]

Abstract In the topological locally convex Hausdorff vector spaces category, the semireflexive subcategories - a categorial notion which generates some well-known cases of semireflexivity, are examined.

Keywords: Lie triple system, homogeneous system, LT-algebra.

2000 MSC:17Dxx, 18G60.

1. INTRODUCTION

The results of the article are formulated and proved for the category C₂V of topological locally convex Hausdorff vector spaces. We denote by R the lattice of the non-null reflective subcategories of the categoryC₂V. Supposing thatR_m is the sublattice of the latticeRof thoseRelements that possess the property: R-replique of the categoryC₂Vobjects can be realized in two steps - first the topology is weakened, second it is completed somehow. The definded semireflexive spaces have such a property defined in different ways.

In the lattice Rtwo more complete sublattices are indicated.

R_b ={R∈R|R⊃S},R_p ={R∈R|R⊃Γ₀} where S(respectively Γ₀ ) is the subcategory of the weak topology spaces (respectively-complete).

In Section 2 some properties of the three latticesR_b,R_p and R_m are examined. In Section 3 the next issues are discussed (3.5 - 3.8).

1. Which elements of the latticeR_m can be realized as a semireflexive product of one element of the lattice R_b and of one element of the lattice R_p?

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8 Dumitru Botnaru, Olga Cerbu

2. Let L=R×_srA, where R∈R_b, and A∈R_p. The subcategory Ris compulsoryc- reflective, does S⊂Rand that mean the reflector functor r:C₂V−→Ris left exact?

3. LetL=R×_srA∈R_m. Are the factors Rand Adetermined in a unique way?

4. Let R₁,R₂ ∈R_b, Γ₁,Γ₂ ∈R_p and R₁×_srΓ₁ =R₂×_srΓ₂. What are the relations of inclusion between subcategories R₁ and R₂ or Γ₁ and Γ₂?

1.1. TERMINOLOGY AND NOTATIONS IN LOCALLY CONVEX SPACES THEORY

The c-reflective subcategories were studied in [9] and [7]. Left and right products were defined and studied in [6]. Other authors results concerning semireflexive subcategories can be found in [2].

In the category C₂Vwe consider the following bicategory structures:

(Epi,M_f)=(the class of epimorphisms, the class of strict monomorphisms);

(E_u,M_p)=(the class of universal epimorphisms, the class of precise monomorphisms)=(the class of surjective mappings, the class of topological embedings);

(E_p,M_u)=(the class of precise epimorphisms, the class of universal monomorphisms) [3], [7];

(E_f,Mono)=(the class of strict epimorphisms, the class of monomorphisms).

We will consider the following subcategories:

Π, the subcategory of complete spaces with weak topology [8];

S, the subcategory of spaces with weak topology [8];

sN, the subcategory of strict nuclear spaces [5];

N, the subcategory of nuclear spaces [10];

Sc, the subcategory of Schwartz spaces [8];

Γ₀, the subcategory of complete spaces [11];

qΓ₀, the subcategory of quasicomplete spaces [12];

sR, the subcategory of semireflexive spaces [8];

iR, the subcategory of inductive semireflexive spaces [4];

M, the subcategory of spaces with Mackey topology [11];

The last subcategory is coreflective and the others are reflective.

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Definition 1.1. Let Aand B be two classes of morphisms of the category C.

The class Ais B-hereditary, if f g∈Aand f ∈B, it follows that g∈A.

Dual notion: the class B-cohereditary.

2. THE FACTORIZATION OF THE REFLECTOR FUNCTORS

The results of this section can be found in [1] in Russian (see also [14]).

2.1. The lattice R of the non-null subcategories of the category C₂V is divided into three complete sublattices:

a) The sublatticeR_b of E_u-reflective subcategories. A subcategoryRisE_u- reflective if the R-replique of any object of the category C₂V is a bijection.

Moreover,

R_b ={R∈R|R⊃S}.

b) The sublattice R_p of M_p-reflective subcategories, the class of those reflective subcategoriesRforR-replique for any object of the categoryC₂Vis a topological embeding:

R_p ={R∈R|R⊃Γ₀}.

c)R_m= (R\(R_b∪R_p))∪ {C₂V}.

We mention thatR_m is a complete sublattice with the first element Π and the last element C₂V.

Figure 2.1

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2.2. LetLbe an element of latticeR_m. For any objectX of categoryC₂Vlet

Figure 2.2

l^X :X −→ lX be its L-replique, and l^X = p^Xb^X its (E_u,M_p)-factorization.

We denote by B = B(L) the full subcategory of the category C₂V consisting of allbX form objects and those isomorphic to these. We also can say thatB is the subcategory of all M_p-subobjects of the objects L. It is clear that B is a E_u-reflective subcategory, and b^X is B-replique of the objects X. Therefore B∈R_b.

2.3. Let Γ⁰⁰= Γ⁰⁰(L) be the full subcategory of all objectsY of the category C₂V,having the property:

Any morphism f :bX−→Y is extended through p^X: f =gp^X

for some morphismg. The subcategory Γ⁰⁰ is closed underM_f-subobjects and products. Further, Γ₀ ⊂ Γ⁰⁰, therefore Γ⁰⁰ ∈ R_p. It is obvious that p^X is Γ⁰⁰-replique of the objectbX.

We denote byG(L) the class of all theM_p-reflective subcategories for which p^X is the replique of the objectbX. The class G(L) has a minimal element

Γ⁰= Γ⁰(L) =∩{Γ|Γ∈G(L)}.

ThusG(L) is a complete lattice with first element Γ⁰(L) and the last element Γ⁰⁰(L).

We can write

G(L) ={Γ∈R_p |Γ⁰(L)⊂Γ⊂Γ⁰⁰(L)}.

2.4. For any element Γ∈G(L) the morphismp^X is Γ-replique of the object bX. Therefore if l : C₂V −→ L, b : C₂V −→ B and g : C₂V −→ Γ are the reflective functors, then

l=gb.

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Theorem. Let r :C₂V−→R and g :C₂V−→Γ be two reflective functors with R∈R_b and Γ∈R_p. The following affirmations are equivalent:

1. l=gr.

2. R =B andΓ∈G(L).

2.5. Example. Let us examine the caseL= Π. Then B(Π) =S

and

Γ⁰(Π) = Γ₀.

Theorem. The subcategory Γ⁰⁰(Π) contains all the normal spaces.

Proof. Let X be a weak topology space: X ∈| S |, and g₀^X : X −→ g₀X its Γ₀-replique. Theng₀^X is also the Π-replique of objectX. In this caseg₀X

∼K^τ, whereK is the field of numbers over which the vector spaces from the categoryC₂V:K=R orK=Care examined. Letf :X−→Y ,→Yb, where Y is a normal space, andYb is his completion.

Figure 1.3 Then

if =gg^X₀

for some morphismg, whereiis a canonical embedding. Sinceg₀X∼K^τ, we conclude that g(g₀X) is a finite dimensional subspace in Yb. Then subspace f(X) of the Y space as a finite dimensional space is complete and

f =g₁g₀^X for some morphism g₁. The theorem is proved.

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2.6. Let X and Y be two normal incomplete subspaces, the algebraical dimension of which is:

ℵ₀≤dimX < dimY

Let Γ₁ (respectively Γ₂ ) be the smallest reflective subcategory which contains the subcategory Γ₀ and X space (respectivelyY space). Then the subcategory Γ₁ is not contained in the subcategory Γ₂.

Theorem. Lattice G(Π) contains a proper class of elements.

2.7. Remark. On another side we have

B(C₂V) =C₂V, G(C₂V) ={C₂V}.

3. SEMIREFLEXIVE SUBCATEGORIES

3.1. Definition. Let R and A be two subcategories of the category C₂V, where R is a reflective subcategory. Object X of the category C₂V is called (R,A)-semireflexive, if hisR-replique belongs to the subcategoryA.

We denote by

L=R×_srA

the subcategory of all(R,A)-semireflexive objects. The subcategoryLis called the semireflexive product of the subcategories Rand A.

3.2 In the lattice R there are elements R such that their reflector functor r :C₂V−→ R is left exact. These kind of elements that belong to sublattice R_b are called thec-reflective subcategories.

The subcategoriesS,sNandScarec-reflective. The subcategoryNbelongs to classR_b but is notc-reflective.

We mention that also in lattices R_p and R_m there are elements of which reflector functor is left exact. For example, the functors

g₀ :C₂V−→Γ₀, π:C₂V−→Π have this property.

3.3. Theorem. Let RandAbe two reflective subcategories of the category C₂Vand the reflector functorr :C₂V−→Ris left exact. Then the subcategory

L=R×_srA

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is a reflective subcategory of the category C₂V.

Proof. It is easily to verify thatLis closed underM_f-subobjects and products (see [2]). So it is reflective.

3.4. From the definition we can deduce:

1. Let R,A∈R_b. ThenS⊂R×_srA.

2. Let R,A∈R_p. Then Γ₀⊂R×_srA.

3. Let R ∈R_b, A∈R_p and R×_srA be a reflective subcategory of the categoryC₂V. As a ruleR×_srA∈R_m.

3.5. Well known examples of semireflexive subcategories are represented by a semireflexive product of an element of the lattice R_b and of one element of the lattice R_p. Thus we formulate the following problem.

Problem. Which elements of the latticeR_m can be realized as a semireflex- ive product of one element of the lattice R_b and of one element of the lattice R_p?

3.6. Another problem concerning this topic is the following one.

Problem. Let L = R×_srA ∈ R_m, where R ∈ R_b, and A ∈ R_p. Is the subcategory R necessarilyc-reflective?

3.7. Problem. LetL=R×_srA∈R_m. Are the factorsRandAdetermined in a unique way?

3.8. Problem. Let R₁, R₂ ∈R_b, Γ₁, Γ₂ ∈R_p and R₁×_srΓ₁=R₂×_srΓ₂. What relations of inclusion are between the subcategoriesR₁ andR₂ or Γ₁ and Γ₂?

3.9. Let (E, t) be a locally convex Hausdorff space,m(t)- Mackey topology [11] compatible withttopology. Thus (E, m(t)) isM-coreplique of the object (E, t). For the elements of the latticeRwe will analyze the following condition (SR). Let (E, t) ∈|L|,L∈R. Then for any locally convex topology u on the vector spaces E

t≤u≤m(t),

the space (E, u) belongs to the subcategory L.

3.10 Categorial, the condition (SR) can be written this way (SR). Let X∈|L|, and b:Y −→X∈E_u∩M_u. Then Y ∈|L|.

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3.11. a) In the lattice R_b the elements S, sN, N, Sc do not satisfy the condition (SR). There are elements that satisfy this condition.

b) In the lattice R_m there are both elements that have the (SR) property and elements that do not have this property.

The subcategory Π has the (SR) property.

Indeed, let (E, t)∈|Π|. Then the topologytis a Mackey topology: t=m(t) [12].

3.12. Theorem. Any element of the lattice R_p has the property (SR).

Proof. Let Γ ∈ R_p, X ∈| Γ | and b : Y −→ X ∈ E_u ∩M_u. Further, let g^Y :Y −→gY be the Γ-replique of the objectY. Then

b=f g^Y for some morphism f.

Figure 2.1

Since b ∈ M_u, g^Y ∈Epi and the class M_u is Epi-cohereditary it follows that f ∈M_u. Also, from the above equality it follows that f ∈ E_u. Thus in this equality the mappings b and f are bijections. So g^Y also is a bijection, in particularyg^Y ∈E_u. Thereforeg^Y ∈M_p∩E_u =Iso. The theorem is proved.

3.13. Theorem. Given an elementL∈R_m,the following affirmations are equivalent

1. the subcategory L satisfies condition (SR);

2. L=B×_srΓ, where B=B(L) and Γ∈G(L);

3. there is an element R ∈ R_b and an element Γ ∈ R_p such that L=R×_srΓ.

Proof. We prove the following implications 1 =⇒2 =⇒3 =⇒1.

1 =⇒ 2. We verify the embedding L ⊂ B×_srΓ. Let X ∈| L |. Then in (E_u,M_p)-factorization of the morphisml^X =p^Xb^X bothb^X andp^X morphisms are isomorphisms.

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Figure 2.2 ThusbX ∈|Γ|. So X∈|B×_srΓ|.

Converse. We verify the embedding B×_srΓ ⊂L. Let (E, t) ∈|B×_srΓ |.

Then b(E, t) = (E, b(t))∈|Γ|, where the topologiestand b(t) are compatible with the same duality.

Figure 2.3

Thusp^E ∈Iso, and (E, b(t))∈|L|. By condition (SR),we have (E, t)∈|L|.

2 =⇒3.Obviously.

3 =⇒1. LetL=R×_srΓ, (E, t)∈|L|, and (E, u) be a locally convex space wheret≤u≤m(t)

Figure 2.4

Let (E, r(u)) be theR-replique of the object (E, u). Then r(t)≤r(u)≤m(t)

and Theorem 3.12 implies that the space (E, r(u)) belongs to the subcategory Γ. So (E, u)∈|L|. Theorem is proved.

3.14. Theorem. Assume R₁ ⊂ R₂ ⊂ R₃, where R_i ∈R_b, i= 1,2,3, and Γ∈R_p. Then

1. R₁×_srΓ⊂R₂×_srΓ;

2. if R₁×_srΓ =R₃×_srΓ, then R₁×_srΓ =R₂×_srΓ also.

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Proof. Let X be a object of the category C₂V. Since R₁ ⊂ R₂ ⊂ R₃, we deduce that between the respective repliques of the object X the following relations

Figure 2.5

r^X₂ =f r₃^X, (1)

r^X₁ =gr^X₂ =gf r^X₃ , (2) for some morphismsf and g, hold.

2. Assume X ∈| R₂×_srΓ |. Then r₂X ∈| Γ |, and from Theorem 3.12 and equality (1) it follows that r₃X∈| Γ|, soX∈|R₃×_srΓ|.

3.15. Theorem. For any reflective subcategory R with the property S⊂R⊂N, we have

R×_srqΓ₀ =sR, in particular,

S×_srqΓ₀ =sN×_srqΓ₀=N×_srqΓ₀ =sR.

Proof. Following the previous theorem, it is enough to prove that N×_srqΓ₀=sR, since, from the definition of the subcategorysRwe have

S×_srqΓ₀=sR.

LetX ∈|N×_srqΓ₀ |. ThenN-replique nX of the object X belongs to the subcategoryqΓ₀.

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Thus nX is a quasicomplete nuclear space. So it is semireflexive ([12] III 7.2. corollary 2, and also [12] IV 5.8 example 4).

3.16. Theorem. Assume that R∈R_b, Γ,Γ₁ ∈R_p, Γ⊂Γ₁, and g:C₂V−→Γ, g₁:C₂V−→Γ₁ are the reflector functors.

1. If R is a c-reflective subcategory, then g(R)⊂R.

2. If g(R)⊂R, then g₁(R)⊂R.

Proof. 1. Let X ∈| R|,g^X :X−→ gX be the Γ-replique of the objectX, and r^gX :gX −→rgX theR-replique of objectgX. Then

r^gXg^X =r(g^X)∈M_p,

since r(M_p) ⊂ M_p for a c-reflective subcategory ([2], theorem 2.8). In the above equality r(g^X) ∈ M_p, and the class M_p is the Epi-cohereditary. So r^gX ∈M_p∩E_u =Iso.

Figure 2.6

2. LetX∈|R|, and g^X :X−→gX and g^X₁ :X −→g₁X be the respective repliques of the objectX. Since Γ⊂Γ₁ it follows that

g^X =f g₁^X for some morphism f.

Figure 2.7

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Just like above we deduce thatf ∈M_p. The hypothesis implies thatgX ∈|R| and R is a E_u-reflective subcategory. So it is closed underM_p-subobjects. It follows g₁X∈|R|.

3.17. Theorem. Assume

L=R×_srΓ where R∈R_b and Γ∈R_p. If g(R)⊂R, then R⊂B=B(L).

Proof. Let X be an arbitrary object of the category C₂V, r^X : X −→ rX and g^rX :rX −→grX - Rand Γ-replique of the respective objects.

Figure 2.8 Since grX∈|R|we deduce thatgrX∈|L|.

Thus

f l^X =g^rXr^X (1)

for some morphism f. Supposing that the E_u,M_p-factorization of the morphisml^X,

l^X =p^Xb^X (3)

holds, we deduce

g^rXr^X =f p^Xb^X (4)

whereb^X ∈E_u, andg^rX ∈M_p, i.e. b^X ⊥g^rX. Thus

r^X =tb^X, (5)

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for some morphism t,

g^rXt=f p^X. (6)

The equality (4) indicates thatR⊂B.

3.18. Conclusions. Returning to problems 3.5-3.8 we can make the following assertions.

1. The L elements of the lattice R_m can be presented as a semireflexive product

L=R×_srΓ

withR∈R_b and Γ∈R_p having the property (SR) (Theorem 3.13).

2. sR=N×_srqΓ₀ and Nis not ac-reflective subcategory.

3. Let L=R×_srΓ. Then neither the first nor the second factor is determined in a unique way (Theorem 3.13 and 3.15).

4. A partially answer is given to question 3.8 by Theorem 3.17.

2.19. Examples. The right product of two subcategories and following examples are examined in more detail in the article [2].

1. Since (M,S) is a pair of conjugated subcategories in the category C₂V and Π =S∩Γ₀ we have ([2])

S×_srΓ₀=M×_dΠ.

2. Let qΓ₀ be a subcategory of the quasicomplete spaces, andsRthe subcategory of the semireflexive spaces [12]. Then

S×_sr(qΓ₀) =M×_d(S∩qΓ₀) =sR.

3. The subcategoryScof Schwartz spaces isc-reflective. LetKbe a coreflective subcategory of the category C₂V for which (K,Sc) is a pair of conjugated subcategories. Then

Sc×_srΓ₀ =iR=K×_d(Sc∩Γ₀);

iRis a subcategory of the inductive semireflexive spaces ([4], theorem 1.5).

References

[1] Botnaru D.,The factorization and commutativity and reflexive functors, Functionals Analysis, Ulianovsk,21(1983), 59-71. (in Russian)

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[2] Botnaru D., Cerbu O.,Semireflexive product of two subcategories (in print).

[3] Botnaru D., Gysin V. B.,Stable monomorphisms in the category of separating locally convex spaces, Bulletin. Acad. Sc. Moldova.,1(1973), 3-7. (in Russian)

[4] Berezansky I.A., Inductive reflexive locally convex spaces, DAN SSSR, v.182, Nr.1, 1968, p.20-20 (in Russian).

[5] Brudovsky B. S.,The additional nuclear topology, transformation of the types- reflexive spaces and the strict nuclear spaces, Dokl. Acad. Nauk, SSSR,178, 2(1968), 271-273.

[6] Botnaru D., T¸ urcanu A., Les produis de gauche et de droite de deux souscategories, Acta et Com., Chi¸sin˘au,VIII(2003), 57-73.

[7] Botnaru D., T¸ urcanu A., On Giraux subcategories in locally convex spaces, ROMAI Journal,1, 1(2005), 7-30.

[8] Grothendieck A.,Topological vector spaces, Gordon and Breach, New York, 1965.

[9] Geyler V.A., Gysin V.B., Generalized duality for locally convex spaces, Functionals Analysis, Ulianovsk,11(1978), 41-50 (in Russian).

[10] Pietsch R.,Nukleare lokal konvexe raume. Academie-Verlag, Berlin, 1965.

[11] Robertson A.P., Robertson W.J., Topological vector spaces, Cambridge University Press, England, 1964.

[12] Schaeffer H.H.,Topological vector spaces, The Macmillan Company, New York, 1966.

[13] Ra¨ıcov D.A.,Some properties of the bounded linear operators, The Sciences Bulletin, Pedagogical State College, Moscow ”V.I Lenin”,188(1962), p.171-191.

[14] T¸ urcanu A.,The factorization of reflexive functors, ”Al.I. Cuza” University, Ia¸si, 2007.

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THE FRATTINI THEORY FOR P - LIE ALGEBRAS

Camelia Ciobanu

“Mircea cel Batrˆan” Naval Academy, Constant¸a [email protected]

Abstract The aim of this paper is to presentFp(L)-the Frattini p-ideal ofLand Φp(L)- the Frattini p-subalgebra of L. The basic properties are presented in the second section. The third section is concerned with the main result F(L) ⊂ Fp(L)⊂CL(F(L)) and in the last section we present some results about p-c- supplemented subalgebras of p-Lie algebras.

Keywords: p-Lie algebra, Frattini p-ideal, p-subalgebra.

2000 MSC:17B05, 17B20, 17B66.

1. INTRODUCTION

In this paper we denote bya→a^p, p >0 the application which corresponds to a p- Lie algebra. Throughout the article L is a finite-dimensional p-Lie algebra over a fieldK of characteristic p >0. We denote by Φ(L) the Frattini subalgebra of L, that is the intersection of the maximal subalgebras ofLand byF(L) the Frattini ideal ofL, that is the largest ideal ofLwhich is contained in Φ(L). Analogously we denote by F_p(L) the Frattini p-ideal of L i.e. the largest p-ideal of L that is contained in Φ_p(L), where Φ_p(L) is the Frattini p-subalgebra of L, that is the intersection of the maximal p-subalgebras of L.

IfAis a p-subalgebra ofL, the p-core ofAis the largest ideal ofLcontained in Aand we denote that with_pA_L. We say thatAis p-core-free inLif_pA_L= 0.

A p-subalgebraAof Lis p-c-suplemented inL if there is a p-subalgebraB of Lsuch thatL=A+B andA∩B is a p-subalgebra of_pA_L. We say thatLis p-c-supplemented if every p-subalgebra ofL is p-c-supplemented inL.

2. BASIC PROPERTIES

We present some notions and results that we use in the sequel:

21

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22 Camelia Ciobanu

[x, y]is the product of x y in L;

L⁽¹⁾ is the derived algebra of L;

(A)_p = ({x^pⁿ |x∈A, n∈N})where x^pⁿ = (x^pⁿ⁻¹)^p; A^p = ({x^p|x∈ A}) where A is a subalgebra of L;

L₁ :=T_∞

i=1L^pⁱ;

L₀ :={x∈L¯¯ x^pⁿ = 0, n∈N};

C_L(M) :={x∈L|[x, M] = 0}, M ⊂L;

N_L(A) ={x∈L|[x, A]⊆A}, where A⊂L.

The first properties of the p-ideals and p-subalgebras of an p-algebra are stated by M. Lincoln and D.A. Towers [9] and we present them in the following.

Lemma 2.1. [9] Let A and B be p-subalgebras of L, such that A is an ideal of L . Then A+B is a p-subalgebra of L.

Lemma 2.2. [9] Let A be a subalgebra of L. Then (A)⁽¹⁾_p ⊆A⁽¹⁾.

Lemma 2.3. [9] If I is an ideal of L, then (I)_p ⊆ C_L(I). In particular, (I)_p is an ideal of L.

Lemma 2.4. [9] If A⊆L then N_L(A) is p-closed.

We give below some results relative toF_p(L) and Φ_p(L) inspired by results relative toF(L) and Φ(L) obtained by Stitzinger [8] for nilpotent Lie algebras.

Lemma 2.5. For any p-Lie algebra L, and A a p-subalgebra of L, the following statements are true

(i) if A+ Φ_p(L) =L then A=L;

(ii) if I is an ideal of L such that I ⊂Φ_p(A), then I ⊂Φ_p(L).

Proof. (i) We assume that A 6= L. Then there exists a maximal p- subalgebra M of L such that A ⊂ M. Now Φ_p(L) ⊂ M and so L = M, fact contradicting the maximality of M. The result follows.

(ii) BecauseAis a p-subalgebra ofLandI is an ideal ofL, Lemma 2.1 implies thatI+Ais a p-subalgebra ofLwithI ⊂Φ_p(A). IfI 6⊆Φp(L), then there is

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a maximal p-subalgebra M of L such thatI ⊂M,henceM ⊂Φ_p(A), which is in contradiction with the maximality of I, soI ⊂Φ_p(L).

Lemma 2.6. If I is a p-ideal of L, then (i) (Φ_p(L) +I)/I ⊂Φ_p(L/I);

(ii) (F_p(L) +I)/I ⊂F_p(L/I);

(iii) if I ⊂ Φ_p(L), then (i) and (ii) are true with equality; moreover, if F_p(L/I) = 0, then F_p(L)⊂I;

(iv) if A is a minimal p-subalgebra of L such thatL=I+AthenI∩A⊂F_p(A);

(v) if I∩F_p(L) = 0, then there is a p-subalgebra A of L such that L=I+A˙ (where +˙ denotes a vector space direct sum).

Proof. The assertions from (i), (ii) and (iii) are similar with those from Proposition 4.3 [9].

(iv) If I∩A6⊆F_p(A) then there is a maximal p-subalgebra M of A such that I ∩A+M = A. Hence L = I +M, contradicting the minimality of A. It follows thatI∩A⊂F_p(A).

(v) Let A be minimal with the propertyL=I+A. Then by (iv),I∩A⊂F_p(A).

But I∩A is a p-ideal of L hence, by Lemma 2.5(ii),I ∩A⊂F_p(L)∩I = 0.

We conclude that L=I+A.˙

Definition 2.1. (i) If L is a p-Lie algebra we denote by Sp(L) the sum of the minimal abelian p-ideals of L and we call it the abelian p-socle of L.

(ii) We say that L p-splits over an p-ideal I of p-Lie algebra L, if there is a p-subalgebra A of L such that L=I+A, where˙ +˙ represents the direct sum of vector spaces.

Lemma 2.7. The abelian p-socle Sp(L) is a p-ideal of L.

Proof. Letx∈Sp(L), l∈L. Thenx^pⁿ = 0, or£ l, x^pⁿ¤

= [l, x] (adx)^pⁿ⁻¹ = 0 and so (x^pⁿ) is a minimal abelian p-ideal of L. Hence x^pⁿ ∈Sp(L).

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3.

F_P

-FREE LIE ALGEBRA

In this section we will present the relationship between F(L) andF_p(L) . Definition 3.1. A p-Lie algebra is called F-free (respectively, F_p-free) if F(L) = 0 (respectively, F_p(L) = 0 ).

Theorem 3.1. If L is F_p-free, then L p-splits over its abelian p-socle.

Proof. If L is F_p-free, then F_p(L) = 0. Let I = Sp(L) be the abelian-p- socle of L which is an abelian p-ideal of L. Then I ∩F_p(L) = 0. In accord with Lemma 2.7(iv), there is a p-subalgebraA of Lsuch thatL=I+A, so˙ L p-splits over I.

Theorem 3.2. For any p-Lie algebra L, the relation F(L)⊂F_p(L) holds.

Proof. It is clear that L/F_p(L) is F_p(L)-free and hence the previous theorem implies thatL/F_p(L) splits over S(L/F_p(L)). So L/F_p(L) is F-free and it follows that F(L)⊂F_p(L).

In the following we introduce two further subalgebras of L.

Definition 3.2If L is a p-Lie algebra we note by T(L)-the intersection of all maximal subalgebras of L that are not ideals of L and correspondingly by T_p(L) the intersection of all maximal p-subalgebras ofLwhich are not p-ideals of L. We also defineτ(L) (respectively, τ_p(L)),to be the largest ideal (respec- tively, p-ideal) of L that is contained in T(L) (respectively, T_p(L)).

In these conditions the following statements hold.

Lemma 3.1. If I is a p-ideal of L, then:

(i) (T_p(L) +I)⊂T_p(L/I);

(ii) (τ_p(L) +I)/I ⊂τ_p(L/I);

(iii) if I ⊂T_p(L), then statements (i) and (ii) occur with equality; moreover, if τ_p(L/I) = 0 then τ_p(L)⊂I.

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Lemma 3.2. Let A be a maximal p-subalgebra of L. Then (i) if A is an ideal of L we have L⁽¹⁾ ⊂A;

(ii) if A is not an ideal of L, it is a p-subalgebra of L.

Proof. (i) Let x6∈A. ThenL =A+ (x)_p and according to Lemma 2.2 it follows thatL⁽¹⁾ ⊂A.

(ii) We assume that A is not p-close, then (A)_p =L, from where, according to Lemma 2.2 we deduce that L⁽¹⁾ = (A)⁽¹⁾_p ⊂A⁽¹⁾ ⊂A, hence A is an ideal of L, contradicting the hypothesis, so any maximal subalgebra of L which is not an ideal of Lis a p-subalgebra ofL.

Theorem 3.3 For any p-Lie algebra L, the following statements hold:

(i) τ_p(L) =C_L(F_p(L));

(ii) τ(L) =τ_p(L);

(iii) if N is the nilradical of L, then F_p(L) = Φ_p(L)∩N; (iv) if L is perfect (that is L=L⁽¹⁾), then Φ_p(L) = Φ(L).

Proof. (i) According to Lemma 3.2(i) we have [τ_p(L), L]⊂L⁽¹⁾∩τ_p(L)⊂ F_p(L), hence τ_p(L) ⊂C_L(F_p(L)). Now we assume that C_L(F_p(L)) 6⊆τ_p(L).

Then there is a maximal p-subalgebraA of L which is not an ideal ofLsuch that C_L(F_p(L))6⊆A. Now it is easy to prove that C_L(F_p(L)) is p-closed, so L =C_L(F_p(L)) +A. Then L⁽¹⁾ ⊂ F_p(L) +A ⊂ A and A is an ideal of L- a contradiction. ThereforeC_L(F_p(L))⊂τ_p(L).

(ii) According to Lemma 3.2(ii) it immediately follows that τ_p(L) ⊂ τ(L).

Consider now a x ∈ τ(L). Then, according with Theorem 2.8 [10] and with Theorem 3.2, [x, L]⊂F(L)⊂F_p(L). Therefore, according to Theorem 3.3(i) x∈C_L(F_p(L)) =τ_p(L) and hence we obtain the conclusion.

(iii) Let A = Φ_p(L)∩N. Then according to Theorem 3.2, N⁽¹⁾ ⊂ F(L) ⊂ F_p(L) and so N⁽¹⁾ ⊂ A. Let us assume that A is not an ideal of L. Then, since AL⊂ N L⊂N, we have that AL 6⊆Φ_p(L). Hence, there is a maximal p-subalgebra M of L such that AL 6⊆M. It follows that N 6⊆ M and hence L=N +M. So, AL =A(N +M)⊂N⁽¹⁾+M ⊂M, a contradiction. From these we can say thatAis a p-ideal of L which is contained in Φ_p(L) and thus A⊂F_p(L). The reverse assertion is immediate.

(iv) It is clear that any maximal p-subalgebra of L is a maximal subalgebra

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of L, hence Φ_p(L)⊂Φ(L).According to Lemma 3.2(ii) we have that Φ(L)⊂ Φ_p(L), and Φ_p(L) = Φ(L) follows. In accord with Theorem 3.2 and 3.3 we obtain F(L)⊂F_p(L)⊂C_L(F(L)) =C_L(F_p(L)).

4. P-C-SUPPLEMENTED SUBALGEBRAS OF P-LIE ALGEBRAS

In this section we present some results about p-c-Supplemented subalgebras of p-Lie algebras that take into account the results similar to those already presented by Ballester-Bolinches, Wang and Xiuyun in [1].

Lemma 4.1 Let L be a p-Lie algebra and A a p-subalgebra of L. The fol- lowing statements hold

(i) if B is a p-subalgebra of A and it is p-c-supplemented in L, then B is p-c- supplemented in A;

(ii) if I is a p-ideal of L and a p-subalgebra of A then A is p-c-supplemented in L if and only if A/I is p-c-supplemented in L/I.

Proof. (i) Assume thatAis a p-subalgebra ofLandBis p-c-supplemented inL. Then there is a p-subalgebra C ofL such that L=B+C andB∩Cis a p-subalgebra of _pB_L. It follows that A = (B+C)∩A and B ∩C ∩A is a p-subalgebra of _pB_L∩A which is a p-subalgebra of _pB_A, and so B is p-c- supplemented in A.

(ii) First we suppose that A/I is p-c-supplemented in L/I. Then there is a p-subalgebraB/I ofL/I such that L/I=A/I+B/I and (A/I)∩(B/I) is a p-subalgebra of _p(A/I)_L/I =_p A_L/I. It follows that L=A+B and A∩B is a p-subalgebra of _pA_L, whenceA is p-c-supplemented inL.

Now, conversely, we assume that I is an p-ideal of L, more than that, I is a p-subalgebra ofA and p-c-supplemented inL. In these circumstances there is a p-subalgebraB ofL such thatL=A+B and A∩B is a p-c-subalgebra of

pA_L. HenceL/I=A/I+(B+I)/I and (A/I)∩(B+I)/I = (A∩(B+I))/I = (I+A∩B)/I but (I+A∩B)/I is a p-subalgebra of _pA_L/I =_p (A/I)_L/I, and soA/I is p-c-supplemented inL/I.