INTERNATIONAL JOURNAL OF MULTIDISCIPLINARY SCIENCES AND ENGINEERING, VOL. 12, NO. 2, MARCH 2021 Creating a New Nest Algebra Structure for Nilpotent Lie Algebras
Atefeh Hasan-Zadeh Fouman Faculty of Engineering, College of Engineering, University of Tehran, Iran
Abstract—In this paper a functional analysis approach to solve such that the main concepts of Lie derivations and the results the rigidity problems of the Riemannian nilmanifolds using obtained from it in nilmanifolds are consistent with the information from their geodetic flows is considered. The corresponding results in nest algebras. For this reason, we proposed method combines analytical and geometric materials to could prove one proposition of Gordon, which can be used in construct a structure of nest algebra for nilpotent Lie algebras. In this way, the main concepts of Lie derivations and the results solving the rigidity problem for 2-step nilmanifolds, in a new obtained from it in nilmanifolds are consistent with the structure, Corollary 3.1. It should be noted that this corresponding results in nest algebras. Due to this adaptation of proposition was proved once again in [4] with the use of the the main concepts, one proposition of Gordon, which can be used concepts of Lie groupoids. Lastly, in expressing the in solving the rigidity problem for 2-step nilmanifolds, has been importance of the approach used, an extension of Corollary proved in the new structure. Also, the resulting nesting algebra 3.1 to quasi-nilpotent operators has been mentioned in structure leads to an extension of that proposition to quasi- nilpotent operators. Corollary 3.2.
Keywords— Nilpotent Lie Algebra, Inner Derivations, Trace II. PRELIMINARIES Class Operators and Quasi-nilpotent Operators
A. Ingredients of operator algebra
I. INTRODUCTION Let H be a complex Hilbert space and B()H is the algebra of all linear bounded operators on H . A nest N of N recent years, considerable attention has been paid to projections on is a chain of orthogonal projections on I solving the rigidity problems for nilmanifolds via some information about their geodesic flows: which contains 0 , I and is closed in the strong operator topology. The nest algebra TN() corresponding to the nest M Problem. ([2]) Whether two compact 2-step nilmanifolds N is the set of all operators A in such that AP= PA for all PN . Clearly, is a weak* closed operator algebra. M and are isometric or not, if they have conjugated geodesic For more details refer to [7-9]. flows? Theorem 2.1. ([7]) Let be a nest of projections on a complex Hilbert space H and B be a weak* closed Among which one can be mentioned through the works of Eberlein, Gordon, Mao and Schueth in [1-3]. This problem has subalgebra of which contains . Then a linear been already considered in [4] by an algebraic-geometric map :()TNB→ is a Lie triple derivation if and only if approach, especially in the category of Lie groupoids, and in there are an element S B and a linear functional f on [5] in notions of von Neumann algebras. Also, the symplectic with fD( )= 0 for every second commutator version of it, [3], has been extended to Poisson notions, [6]. In our ongoing research on this issue, we have come up with DXYZ= [[ , ], ] such that ()[,]()A=+ S A f A I , A TN(). an analytical approach. The method presented in this paper Corollary 2.1. ([7]) Let be a nest of infinite multiplicity uses the structure of nest algebras corresponding to the algebra on a complex Hilbert space H and let B be a weak* closed of projections on complex Hilbert spaces. In this way, by subalgebra of which contains . Let be a Lie some ingredients of operator algebra and geometry, we triple derivation from into . Then there is an construct a structure of nest algebra for an 푚-dimensional nilpotent Lie algebra, Theorem 3.1. The structure obtained is element S B such that ()[,]ASA= , A TN(). In fact, Lie derivations are special cases of Lie triple derivations.
A. Hasan-Zadeh is a faculty member of Fouman Faculty of Engineering, College of Engineering, University of Tehran, Iran, P.O..Box: 43581-39115, (e-mail: [email protected]).
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B. Ingredients of geometry e e, it can be defined the layer MMB ={ :J = 0} . By As the notions of [1-3], a Riemannian nilmanifold is a construction, each layer is a M -invariant subset of M and M quotient of a simply connected nilpotent Lie group M by M is a disjoint finite union of layers MM= e . Also, B e a discrete subgroup ; together with a Riemannian metric g all the orbits contained in a given layer have the same e whose lift to is left invariant. A nilmanifold has step dimension ( card e ). Let the orbits contained in M B are 2r - dimensional. Then, there exists on size k if is -step nilpotent. M Especially, a Lie group is said to be two-step nilpotent if i) mr− 2 polynomial functions zz12,..., mr− ; its Lie algebra M satisfies [[MMM , ], ]= 0 , equivalently ii) 2r rational functions p11,..., prr , q ,..., q which are regular [,] is central in . If be a left-invariant MM M e Riemannian metric on 2-step nilpotent Lie group , then it on M B such that the polynomial functions zz12,..., mr− defines an inner product .,. on the Lie algebra of . separate the orbits contained in ; Let ZMM= [,] and V denote the orthogonal complement The first layer has additional properties: it is a Zariski dense of Z in M relative to . For zZ , a skew symmetric open subset of M , it contains only orbits of maximal linear transformation Jz():VV→ can be defined by dimension 2d , and the polynomial functions zz12,..., md− J( z ) x= ( ad ( x )) z for x V , where (ad ( x )) denotes the separating the orbits of are M -invariant. adjoint of ad() x . Equivalently, Moreover, if we identify the symmetric algebra SM() of J( z ) x , y= [ x , y ], z , x , y VZ , z (2.1) with the space of polynomial functions on M and and this process is reversible. denote by SM()M the subring of of the 푀invariant An authomorphism of M is said to be -almost inner if polynomial functions, then the quotient field of () is conjugate to for all . It is said to be almost coincides exactly with the field R( z ,..., z ) of rational inner if ()x is conjugate to x for all xM . A derivation 12md− e of the Lie algebra is said to be -almost inner functions in the zk variables. The open set M B is usually (respectively, almost inner) if ()[,]XX M for all X log called the generic set associated to the basis B , the orbits e (respectively, for all xM . The spaces of -almost inner contained in M B are the generic orbits and the automorphisms is denoted by AIA() and similarly, AID() is corresponding polynomial functions zz12,..., md− are the its infinitesimal counterpart, namely -almost inner generic invariants. derivations. Let M is any open subset of M such that the III. MAIN STRUCTURE polynomial functions separate the orbits contained in and the vectors dz( ),..., dz ( ) are linearly Theorem 3.1. For an m-dimensional nilpotent Lie algebra 12md− , it can be constructed a structure of nest algebra TM() independent for all in . Also, since the generic corresponded to it which the notions of Lie derivations in invariants associated to any basis B generate the these two apparently different structures produce the same quotient field of SM()M , then satisfies the above meanings and results (especially Theorem 2.1 and Corollary conditions for any Jordan-Holder basis. 2.1 in this structure will also be satisfied). Corollary 3.1. ([2]) Let be a 2-step nilpotent Lie algebra Proof. Suppose that is an m -dimensional nilpotent Lie with an inner product .,. and be an almost inner algebra. Denote by M the dual space of and M by the connected and simply connected Lie group with Lie algebra derivation of continuous type on say (x )= [ ( x ), x ] with . Let MMMM01... m = be a flag of continuous on M \{0} . Let z Z() M and yker( J ( z )) .
( dimMi = i ) such that []M,Mii M −1 for all im{1,..., } . Let Then (x ), z= [ ( x ), y ], z , x M , where Jz():VV→ , also BXX= ( 1,..., m ) be a Jordan-Holder basis adapted to is a skew symmetric linear transformation defined by equation (2.1). (Mi ) that Mii=RRXX1 ... , for all i . Then, M has a i In special case, if the center of properly contains the natural layering which can be summarized as follows. derived algebra, then every almost inner derivation of For in , it can be defined the set of indexes continuous type on M is inner. Proof. From Theorem 3.1 applied to Theorem 2.1 and J= j: X ji (MM−1 + ) , where Corollary 2.1, it can be concluded that: MMM =XXYY : ,[ , ] = 0, . If J = j12 ... j r A Lie derivation (especially, almost inner derivation) then = RRXX ... . Let ={;}J . For (especially, , on the nest algebra where MM jj12r M
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S is a weak* closed subalgebra which contains TM() REFERENCES (especially, (x ), z= [ ( x ), y ], z , x M ) and vice versa. [1] P. Eberlein, “Geometry of two-step nilpotent groups with a left As a result, it has been archived more than we expected. invariant metric”, Ann. Sci. Ecole Norm. Sup., 27, 1994, pp. Corollary 3.2. Corollary 3.1 can equally be extended to 611-660. quasi-nilpotent operators rather than almost inner derivations. [2] C. S. Gordon, and Y. Mao, “Geodesic conjugacy of two-step Proof. A bounded linear operator T over a (separable) Hilbert nilmanifolds”, Mich. Math. J., 45, 1998, pp. 451-481. [3] C. S. Gordon, Y. Mao, D. Schueth, “Symplectic rigidity of space H is said to be in the trace class if for some (and geodesic flows on two-step nilmanifolds”, Annh. Sci. École. hence all) orthonormal bases {}ekk of H , the sum of Norm. Sup., 30, 1998, pp. 417427. 1 [4] H. R. Fanai, A. Hasan-Zadeh, “An application of Lie groupoids positive terms A== Tr A:(), A A2 e e is finite. Then to a rigidity problem of 2-step nilmanifolds”, Mathematica 1 kk Bohemica, 144, 2019, pp. 1-12. k [5] A. Hasan-Zadeh, H. R. Fanai, “Applications of von Neumann Algebras to Rigidity Problems of (2-step) Riemannian (Nil- it can be defined the trace of T , Tr(), T= Aekk e which is k )Manifolds”, Australian Journal of Mathematical Analysis and independent of the choice of the orthonormal basis. Applications, 17(1) 2020, pp. 1-5. [6] H. R. Fanai, A. Hasan-Zadeh, “A symplectic rigidity problem On the other hand, as a corollary of Erdos Density for 2-step nilmanifolds”, Houston Journal of Mathematics, 2, Theorem, ([7]), every trace class operator on a Hilbert space 2017, pp. 363-374. ℋ is the limit in the trace norm of finite rank operators in [7] K. R. Davidson, Nest Algebras: triangular forms for operator TN(). Then every trace class quasi-nilpotent operator on a algebras on Hilbert space, Longman Scientific and Technical, Hilbert space ℋ is the limit in the trace norm of finite rank 1988. [8] M. Takesaki, Theory of Operator Algebras II, Springer-Verlag, nilpotent operators. Then, Corollary 3.1 can bring the same 2003. result for the quasi-nilpotent operators instead of almost inner [9] H. Li, Y. Wang, “Generalized Lie triple derivations”, Linear derivations. and Multilinear Algebra, 3, 2011, pp. 237-247.
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