aa r X i v : . [ m a t h . G R ] N ov Jordanable almost Abelian Lie algebras
Zhirayr AvetisyanDepartment of Mathematics, UCSBNovember 9, 2018
Abstract
We call a linear operator on a vector space over a field Jordanable if it has a Jordan canonical form.A Lie algebra will be called Jordanable if its adjoint representation is by Jordanable operators. Theexistence of canonical form allows to describe important structural properties of these Lie algebras inexplicit terms. Lie subalgebras, ideals, automorphisms and derivations, as well as quadratic Casimirelements are given explicitly in a suitable basis.
An almost Abelian Lie algebra is a non-Abelian Lie algebra L over a field F that contains a codimensionone Abelian Lie subalgebra. Such an algebra can always be written as a semidirect product F e ⋊ V of a one-dimensional Lie algebra F e with a codimension one Abelian ideal V . Among the mostprominent representatives of this class are the Heisenberg algebra H F , the algebra of generators of affinetransformations on the real line ax + b F , and the algebra of Killing vector fields on the plane E F (2).In a recent paper [1] we motivated the introduction of general almost Abelian Lie algebras, studiedtheir structure and obtained a classification up to isomorphism. More precisely, it was shown thatthese isomorphism classes correspond to the projective similarity classes of the linear operators ad e ∈ End( V ). In order to obtain more explicit formulae that are important in practice we need to fix a concreterepresentative from each class, and to work as far as possible with projective similarity invariants of linearoperators. This paper is devoted to those almost Abelian Lie algebras for which the operator ad e admitsa Jordan canonical form, i.e., is similar to a possibly infinite direct sum of ordinary finite dimensionalJordan blocks. We call such algebras Jordanable and use the convenient canonical forms to solve explicitlysome operator equations appearing in the more general results of [1]. Projective multiplicity functions areintroduced as invariant signatures of Jordanable almost Abelian Lie algebras, and all results are expressedin terms of them.The paper is structured as follows. In Section ?? we establish notation and remind the reader ofsome elementary facts about fields, polynomials and algebraic extensions. We introduce multiplicityfunctions and a few operations with them. Section 3 is devoted to Jordanable linear operators on agiven vector space. First we discuss conventional Jordan blocks corresponding to various roots x ∗ and lgebraic extensions F ( x ∗ ), and introduce flexible notation for switching between F -linear and F ( x ∗ )-linear objects. A Jordan canonical form J( ℵ ) corresponding to a multiplicity function ℵ is defined as thedirect sum of different Jordan blocks with multiplicities given by ℵ . Note that no restriction is posedon the cardinality of these multiplicities. A linear operator is defined to be Jordanable if it is similarto a Jordan canonical form. A Jordanable operator can be similar to many Jordan canonical forms,but all these forms differ only by the order of Jordan blocks and have the same multiplicity function,therefore a multiplicity function is a unique similarity invariant in the class of Jordanable operators.The operation of multiplication of a multiplicity function by a scalar introduced in Section ?? is usedto establish a bijective correspondence between projective classes of multiplicity functions and projectivesimilarity classes of Jordanable operators.In Section 4 the invariant vector subspaces of a given Jordanable operator are described in terms of abasis adapted to the structure of the multiplicity function. It is shown that the restriction to an invarianrsubspace of a Jordanable operator is again Jordanable. The Section 5 begins with certain technicalitiesregarding the passage from F -matrices to F ( x ∗ )-matrices. Then we proceed to explicitly describe thesolutions of the following operator equations: X T = λ T X and Y T − T Y = T for a given λ ∈ F ∗ = F \{ } and Jordanable operator T, as well as Z J( ℵ ) + J( ℵ ) ⊤ Z = 0 for a given nonzero multiplicity function ℵ inthe particular (Jordan) basis. Finally, in Section 6 we introduce Jordanable almost Abelian Lie algebrasas those for which the adjoint representation ad is by Jordanable operators. We show that isomorphismclasses of Jordanable almost Abelian Lie algebras correspond to projective classes of multiplicity functions,and for a given multiplicity function ℵ we choose a representative algebra a A ( ℵ ). The centre and the lowercentral series of a A ( ℵ ) are described, and it is shown that a A ( ℵ ) is nilpotent if and only if ℵ is supportedonly at the zero root x ∗ = 0. It is further shown that a Lie subalgebra of a Jordanable almost Abelian Liealgebra is again Jordanable, and all subalgebras and ideals of a A ( ℵ ) are given in terms of ad e -invariantsubspaces. At the end explicit matrix representations are given for automorphisms Aut( a A ( ℵ )), derivationsDer( a A ( ℵ )) and quadratic Casimir elements Q ( x ) ∈ Z (U( a A ( ℵ ))). The terminal Section 7 demonstratesthe utilization of the above results in studying three examples of concrete almost Abelian Lie algebras: theBianchi Lie algebra Bi(VI ), the Lie algebra of the Mautner group (the lowest dimensional non-type-I Liegroup), and the almost Abelian Lie algebra a A Q (1 × √ , q × ) over rationals. This choice of examplesdemonstrates a variety of different structures that trace different trajectories in the flow of argumentsleading to the final results. Let us start with establishing notations regarding fields and their spectra. Let F be a field of scalars,and F [ X ] the ring (actually F -algebra) of polynomials P ( X ) with coefficients in F . For each polynomial P = P ( X ) ∈ F [ X ] we denote by deg P ∈ N its degree. Let σ F ⊂ F [ X ] be the set of monic irreduciblepolynomials which is called the spectrum of F . It has a natural grading according to the degrees, σ F = G n ∈ N σ F ,n , σ F ,n . = { p ∈ σ F | deg p = n } , ∀ n ∈ N . or a p ∈ σ F there two possibilities. Either deg p = 1 and then p ( X ) = X − µ for some µ ∈ F , or deg p > p ( x ∗ ) = 0 has no roots in F . In the latter case the roots x , ..., x deg p are in the algebraicclosure F of F . In both cases for every n ∈ N the map σ F ,n ∋ p ( X ) = ( X − x ) ... ( X − x n )
7→ { x , ..., x n } ∈ F n / S n , n ∈ N is injective and can be used to embed σ F ,n into F n / S n (here S n is the symmetric group on n symbols).Let F ∗ . = F \ { } be the group of invertible elements of F . Define the following action of F ∗ on F [ X ],[ λ ⋆ P ]( X ) = λ deg P P ( Xλ ) , ∀ λ ∈ F ∗ . Remark 1
The following properties of this action are easily verified, P ( X ) = deg P X k =0 a k X k ⇔ λ ⋆ P ( X ) = deg P X k =0 λ deg P − k a k X k ,P ( x ∗ ) = 0 ⇔ λ ⋆ P ( λx ∗ ) = 0 ,λ ⋆ ( P ( X ) Q ( X )) = λ ⋆ P ( X ) λ ⋆ Q ( X ) , ∀ P ( X ) , Q ( X ) ∈ F [ X ] , ∀ λ ∈ F ∗ . Proposition 1
For every λ ∈ F ∗ and n ∈ N we have λ ⋆ σ F ,n ⊂ σ F ,n . More precisely, F ∗ acts on σ F ,n ⊂ F n / S n according to its natural action on F n by dilations, λ { x , ..., x n } = { λx , ..., λx n } . Proof:
That F ∗ acts by dilations of the set of roots { x , ..., x n } is already clear from Remark 1. We onlyneed to show that p ∈ σ F ,n entails λ ⋆ p ∈ σ F ,n . That deg λ ⋆ p = n and that λ ⋆ p is monic follows againfrom Remark 1. Suppose that λ ⋆ p ( X ) = Q ( X ) R ( X ) for some Q, R ∈ F [ X ] with deg Q, deg R >
0, i.e., λ ⋆ p is reducible. ⋆ -multiplying both sides by λ − we get p ( X ) = λ − ⋆ Q ( X ) λ − ⋆ R ( X ) by Remark 1which contradicts the irreducibility of p . (cid:3) For a subset σ ⊂ σ F the action of F ∗ is defined as pointwise, and the orbit F ∗ ⋆ σ . = (cid:8) σ ′ ⊂ σ F | σ ′ = λ ⋆ σ, λ ∈ F ∗ (cid:9) is the set of all possible subsets of the same cardinality related to σ by a uniform dilation.We proceed to introduce multiplicity functions. Let C be the class of cardinals as identified with theirrepresentative sets up to isomorphism. Definition 1 An N -graded multiplicity function is a map ℵ : σ F × N
7→ C . Denote by supp ℵ . = { p ∈ σ F | ℵ ( p, · ) } the support of ℵ and bydim F ℵ . = X p ∈ σ F ∞ X n =1 n ℵ ( p, n ) deg p = X p ∈ supp ℵ ∞ X n =1 n ℵ ( p, n ) deg p ts dimension . The action of F ∗ on multiplicity functions can be defined by the pushforward,[ λ ⋆ ℵ ]( λ ⋆ p, n ) = ℵ ( p, n ) , ∀ p ∈ σ F , ∀ n ∈ N , ∀ λ ∈ F ∗ . Proposition 2
The F ∗ -action on multiplicity functions ℵ has the following properties, supp λ ⋆ ℵ = λ ⋆ supp ℵ , dim F λ ⋆ ℵ = dim F ℵ , ∀ λ ∈ F ∗ . Proof:
To justify the first statement writesupp λ ⋆ ℵ = { p ∈ σ F | [ λ ⋆ ℵ ]( p, · ) } = { λ ⋆ p | [ λ ⋆ ℵ ]( λ ⋆ p, · ) , p ∈ σ F } = { λ ⋆ p | ℵ ( p, · ) , p ∈ σ F } = { λ ⋆ p | p ∈ supp ℵ} = λ ⋆ supp ℵ . For the second statement we observe thatdim F λ ⋆ ℵ = X p ∈ σ F ∞ X n =1 n [ λ ⋆ ℵ ]( p, n ) deg p = X p ∈ σ F ∞ X n =1 n [ λ ⋆ ℵ ]( λ ⋆ p, n ) deg λ ⋆ p = X p ∈ σ F ∞ X n =1 n ℵ ( p, n ) deg λ ⋆ p = dim F ℵ , where we used deg λ ⋆ p = deg p from Remark 1 in the last step. (cid:3) We denote the orbit of the F ∗ -action on ℵ by F ∗ ⋆ ℵ and call it a projective multiplicity function. Theaction of F ∗ on the class of all multiplicity functions C σ F × N is not regular in the sense that different orbitsmay have different ranks. In other words, the isotropy subgroups of F ∗ for different multiplicity functionsmay be different. Definition 2
A scalar λ ∈ F ∗ will be called a dilation symmetry of the multiplicity function ℵ if λ⋆ ℵ = ℵ . Remark 2
It is clear that the set
Dil( ℵ ) of all dilation symmetries of a given multiplicity function is theisotropy subgroup of F ∗ at ℵ . In low dimensions when there are finitely many nonzero entries {ℵ ( p i , n i ) } Ni =1 , N ∈ N in ℵ the followingnotation may be more convenient in practice, ℵ ↔ ( m × p n , ..., m N × p n N N ) , m = ℵ ( p , n ) , ..., m N = ℵ ( p N , n N ) . (1) In this section we are interested in linear operators T on F -vector space V which are Jordanable, i.e.,admit a Jordan canonical form. We refer to Chapter XIV of [5] for the relevant background and for thetheory of finite dimensional operators. In particular, Corollary 2.5 states that every finite dimensionallinear operator is Jordanable. The situation is not as clear in infinite dimensions, and below we will tryto recover some of the familiar facts in this new realm. or a given p ∈ σ F let x p be any one root of p . Denote by F /p . = F ( x p ) the extension field, whichregardless of the choice of the root x p is isomorphic to F [ X ] /p ( X ) F [ X ]. Then [ F /p : F ] = dim F F /p = deg p and F /p ≃ F deg p as an F -vector space. The left action of F /p on itself defines the embedding F /p ⊂ End F ( F /p ) ≃ End F ( F deg p ), and the element x p ∈ F /p regarded as an F -matrix takes the form x p = . . . − a . . . − a . . . − a . . . . . . . . . . . . . . . . . . − a deg p − , (2)which can be found on page 557 of [5]. However, in the special case F = R it is customary to adopt theform x p = a − bb a , p ( X ) = X + a X + a = ( X − a ) + b , b ≥ b ≥ ǫ = F = R and (3) is adopted. (4)In both cases, for deg p = 1 when p ( X ) = X + a we have x p = − a .Given an F /p -vector space V we can consider it as an F -vector space V ⊗ F F /p (tensor product of F -vector spaces), so that F /p -matrices T ∈ End F /p ( V ) become F -block-matrices T ∈ End F ( V ⊗ F F /p ) with F /p ⊂ End F ( F /p )-valued blocks. Hereafter we will freely switch between F /p -matrices and F -block matricesemphasizing the ground field whenever not obvious from the context. For instance, let us for n ∈ N denote by n the n × n identity matrix, where the ground field will be explicitly specified or clear fromthe context. If the field is F /p then as an F /p -matrix n is n × n , whereas as an F -matrix it is the n deg p × n deg p matrix n ⊗ deg p . More generally, by tensor product A ⊗ B we will mean the blockmatrix obtained by multiplying blocks B with the entries of the n × m matrix A,A ⊗ B = A B A B . . . A m B . . . . . . . . . . . .A n ;1 B A n ;2 B . . . A n ; m B . (5)For every p ∈ σ F and n ∈ N let J( p, n ) be the F /p -linear operator (Jordan block) on the vector space(block space) F /p n given by the matrix formJ( p, n ) = x p n + N n ∈ End F /p ( F /p n ) , (6) here x p ∈ F /p is as in (2) and N n is the nilpotent F /p -matrixN n = ...
00 0 1 ... ... ...
10 0 0 ... (note that our convention is the transpose of Theorem 2.4 in Chapter XIV of [5]). Each such block F /p n isan indecomposable F /p [J( p, n )]-module, i.e., it cannot be written as a direct sum of two J( p, n )-invariant F /p -vector subspaces.Let T i ∈ End F ( V i ) for i = 1 ,
2. The operators T and T are called similar (written T ∼ T ) if thereexists an invertible linear operator (intertwiner) S : V → V such that S T = T S. The two operatorsare called projectively similar if λ T ∼ T for some λ ∈ F ∗ . Proposition 3
For every p, q ∈ σ F , m, n ∈ N and λ ∈ F ∗ , λ J( p, m ) ∼ J( q, n ) as F -matrices if and onlyif q = λ ⋆ p and n = m . Proof:
First assume λ J( p, m ) ∼ J( q, n ) as F -matrices. Then[ λ ⋆ p ] m (J( q, n )) ∼ [ λ ⋆ p ] m ( λ J( p, m )) = p m (J( p, m )) = 0 . But also q n (J( q, n )) = 0. If λ ⋆ p = q then [ λ ⋆ p ] m and q n are mutually prime, and by Bezout theoremit follows that J( q, n ) = 0 which is not true. Therefore λ ⋆ p = q and by Remark 1 deg q = deg p .But similar matrices have the same dimensions, hence m deg p = n deg q and thus m = n . Conversely, F [ λ J( p, m )] = F [J( p, m )] acts indecomposably on F /p m , therefore the matrix λ J( p, m ) has only a singleJordan block, i.e., λ J( p, n ) ∼ J( q, n ). But then we have already shown that q = λ ⋆ p and n = m . (cid:3) Later in Lemma 5 we will find the corresponding intertwiner explicitly.
Definition 3
A canonical form J( ℵ ) ∈ End F ( F dim F ℵ ) associated with a multiplicity function ℵ is a linearoperator with matrix form J( ℵ ) = M p ∈ σ F ∞ M n =1 M ℵ ( p,n ) J( p, n ) . (7)Note that this definition is intrinsically ambiguous: with no a priori order on the set σ F , different ordersof direct sum will in general yield different matrix forms. One can fix this freedom by introducing anarbitrary order on σ F . But we will not need to do so as we will be interested in canonical forms only upto similarity. Different orders in the direct sum of blocks correspond to different choices of basis in thesame vector space. A change of basis is a similarity transformation, and all possible orders of summandsfall into the same similarity class. Thus, if we denote by [T] the similarity class containing the linearoperator T, the map ℵ 7→ [J]( ℵ ) from multiplicity functions to similarity classes is well defined. Definition 4
A linear operator T ∈ End F ( V ) on an F -vector space V will be called Jordanable if thereexists a multiplicity function ℵ T such that T ∼ J( ℵ T ) . n this case ℵ T is referred to as the multiplicity function of T, and J( ℵ T ) is called a Jordan canonicalform of T. Below we make sure that ℵ T is unique by showing how to extract it from T. Proposition 4
For a Jordanable operator T , ∀ p ∈ σ F , ∀ n ∈ N we have ℵ T ( p, n ) = dim F /p ker p n (T) /p (T) ker p n +1 (T) − dim F /p ker p n − (T) /p (T) ker p n (T) . Proof:
Let us first deal with the case T = J( ℵ ) (we drop the index T of ℵ T for brevity). For every fixed p ∈ σ F the subspace V p . = ∞ X n =1 ker p n (J( ℵ )) = ∞ M n =1 M ℵ ( p,n ) F /p n is actually an F /p -vector space. Choose an F /p -basis { ξ mα ( p, n ) } with α ∈ ℵ ( p, n ) and 1 ≤ m ≤ n such that V p = ∞ M n =1 M α ∈ℵ ( p,n ) n M m =1 F /p ξ mα ( p, n ) , p (J( ℵ )) ξ mα ( p, n ) = ξ m − α ( p, n ) , m > , m = 1 . Now it is not difficult to see thatker p k (J( ℵ )) = ∞ M n =1 M α ∈ℵ ( p,n ) min { k,n } M m =1 F /p ξ mα ( p, n ) ,p (J( ℵ )) ker p k +1 (J( ℵ )) = ∞ M n =1 M α ∈ℵ ( p,n ) min { k +1 ,n }− M m =1 F /p ξ mα ( p, n ) , whence ker p k (J( ℵ )) /p (J( ℵ )) ker p k +1 (J( ℵ )) ≃ k M n =1 M α ∈ℵ ( p,n ) F /p ξ nα ( p, n ) , and therefore dim F /p ker p k (J( ℵ )) /p (J( ℵ )) ker p k +1 (J( ℵ )) = k X n =1 ℵ ( p, n ) . Now come back to the general case with S being the intertwiner so that S T S − = J( ℵ ). Since S is anisometry, dim F /p ker p k (T) /p (T) ker p k +1 (T) = dim F /p S ker p k (T) / S p (T) ker p k +1 (T)= dim F /p ker p k (J( ℵ )) /p (J( ℵ )) ker p k +1 (J( ℵ )) = k X n =1 ℵ ( p, n ) . The result easily follows. (cid:3)
Corollary 1
The map ℵ 7→ [J( ℵ )] from multiplicity functions to similarity classes is injective. Finally we show how the multiplicity function captures projective similarity.
Proposition 5
For two multiplicity functions ℵ , ℵ ′ and a scalar λ ∈ F ∗ , J( ℵ ′ ) ∼ λ J( ℵ ) if and only if ℵ ′ = λ ⋆ ℵ . Proof:
First let ℵ ′ = λ ⋆ ℵ so thatJ( ℵ ′ ) = J( λ ⋆ ℵ ) = M p ∈ σ F ∞ M n =1 M λ⋆ ℵ ( p,n ) J( p, n ) = M p ∈ σ F ∞ M n =1 M ℵ ( p,n ) J( λ ⋆ p, n ) . y Proposition 3 there exists an invertible intertwiner S p,n ∈ End F ( F /p n ) such that S p,n J( λ ⋆ p, n ) = λ J( p, n ) S p,n . It follows that S J( λ ⋆ ℵ ) = λ J( ℵ ) S, whereS = M p ∈ σ F ∞ M n =1 M ℵ ( p,n ) S p,n . For the converse let us assume that J( ℵ ′ ) ∼ λ J( ℵ ) for some multiplicity functions ℵ , ℵ ′ and a scalar λ ∈ F ∗ . Above we have shown that λ J( ℵ ) ∼ J( λ ⋆ ℵ ), so that by transitivity J( ℵ ′ ) ∼ J( λ ⋆ ℵ ). Theassertion follows directly from Corollary 1. (cid:3) Corollary 2
The map F ∗ ℵ 7→ [ F ∗ J]( ℵ ) from projective multiplicity functions to projective similarityclasses of Jordanable operators is also injective. We end this section by putting the question of Jordanability into the context of general commutativealgebra, where it truly belongs.
Remark 3
The linear operator T on an F -vector space V is Jordanable if and only if the F [ X ] -module V is a direct sum of finite length submodules. Indeed, it is easy to see that a finite length F [ X ]-submodule of V is a finite direct sum of Jordan blocksspaces. In this section we will describe the invariant subspaces W ⊂ V of a Jordanable operator T on a vectorspace V over a field F . This will later be used to describe subalgebras and ideals of an almost AbelianLie algebra. Aside from that, invariant subspaces are of considerable importance in the context of lineardynamical systems. A very detailed exposition of invariant subspaces for linear operators on finite dimen-sional real and complex vector spaces can be found, for instance, in [4]. Here we will touch the subjectvery briefly, but in a larger context of arbitrary vector spaces.In order to do so let us give an adaptation to our setting of a theorem by Kulikov on commutativegroups. Theorem 1
For p ∈ σ F , an F /p -module V p is a direct sum of cyclic submodules if and only if V p is theunion of an ascending chain of submodules, V p ⊂ V p ⊂ . . . , V = ∞ [ n =1 V np , such that for every n ∈ N , the heights of nonzero elements in V np are bounded by natural numbers k n ∈ N . The original proof can be found in ([2], Theorem 17.1 on page 87), and it can be extended to modulesover PID with only trivial modifications.Let T be a Jordanable operator on the F -vector space V , and let W ⊂ V be a T-invariant vectorsubspace. Proposition 6
The restriction T | W of the Jordanable linear operator T to an invariant vector subspace W is again Jordanable. roof: Let V = M p ∈ σ F V p , W = M p ∈ σ F W p be the corresponding spectral decompositions. The F /p -module V p is obviously a direct sum of cyclicsubmodules (Jordan block spaces), so that by Theorem 1 we obtain a chain of submodules { V np } ∞ n =1 withthe above mentioned properties. Denote W np . = V np ∩ W p , which is an F /p -submodule of W p . The heightsof non-zero elements in W np ⊂ V np are obviously bounded by the same k n as above. Moreover, W p = W p ∩ V p = W p ∩ ∞ [ n =1 V np ! = ∞ [ n =1 (cid:0) W p ∩ V np (cid:1) = ∞ [ n =1 W np . Thus again by Theorem 1 we learn that each W p , and thereby also W is a direct sum of cyclic F [ X ]-submodules of V . Every such cyclic submodule is of finite length. Indeed, if v ∈ V is a cyclic vector thenthe submodule equals F [T] v . Write v = v + . . . v k , where v , . . . , v k are cyclic vectors in the original Jordandecomposition. Then our cyclic submodule F [T] v is a submodule of the direct sum F [T] v ⊕ . . . ⊕ F [T] v k ,which is of finite length by itself. Therefore by Remark 3 T | W is Jordanable. (cid:3) Thus the description of an invariant subspace of a Jordanable operator reduces to that of an irreducibleinvariant subspace, i.e., one with a single Jordan block space.
Corollary 3
An invariant subspace of a Jordanable operator on a vector space V with Jordan decompo-sition V = M p ∈ σ F ∞ M n =1 M α ∈ℵ ( p,n ) n M m =1 F /p ξ mα ( p, n ) is any subspace of the form W = M p ∈ σ F ∞ M n =1 M β ∈ i ( p,n ) n M m =1 F /p η mβ ( p, n ) , (8) where i is a multiplicity function and η mβ ( p, n ) = ∞ X k =1 X α ∈ℵ ( p,k ) min { k,m } X l =1 µ p ( n, β ; k, α, m − l ) ξ lα ( p, k ) , ∀ p ∈ σ F , ∀ n ∈ N , m = 1 , . . . , n. (9) For every β ∈ i ( p, n ) there exists an α ∈ ℵ ( p, ¯ n ) with ¯ n ≥ n such that the F -valued coefficient function µ p ( n, β ; ¯ n, α, = 0 . The support of µ p ( n, β ; ., ., . ) is finite. Proof:
In view of Proposition 6 it suffices to prove that a subspace of the form n M m =1 F /p η mβ ( p, n )is a Jordan block if and only if (9) holds. Being a Jordan block amounts to p (T) η mβ ( p, n ) = η m − β ( p, n )for 1 < m ≤ n and p (T) η β ( p, n ) = 0. The rest follows easily. (cid:3) Linear equations with Jordanable operators
In [1] the explicit representations of several properties of an almost Abelian Lie algebra (e.g., automor-phisms and derivations) were reduced to the solution of a few linear operator equations. These equationswill be solved in this section under the assumption that the coefficient operators are Jordanable. For thecase of finite dimensional matrices over F ∈ { R , C } the general solution of linear matrix equations canbe found in literature (e.g., [3]), but here we will obtain more explicit answers for the particular cases athand.Let us start by establishing the following simple fact. Lemma 1
For every p ∈ σ F , the subring F /p ⊂ End F ( F /p ) is self-normalizing, i.e., [T , F /p ] ⊂ F /p implies T ∈ F /p for every T ∈ End F ( F /p ) . Proof:
If deg p = 1 then F = F /p = End F ( F /p ) and the claim is trivial, so we will concentrate on the casedeg p >
1. Let us first note that F /p ⊂ End F ( F /p ) is self-centralizing, that is, [T , F /p ] = 0 implies T ∈ F /p forevery T ∈ End F ( F /p ). Indeed, since x = x x ∈ F /p , we haveT x = T( x
1) = (T x )1 = ( x T)1 = x (T 1) = x ((T 1)1) = ( x (T 1))1 = ((T 1) x )1 = (T 1) x, ∀ x ∈ F /p , which shows that T = T 1 ∈ F /p . Now let F /p = F ( x p ) and let by the assumption [T , x p ] = y for some y ∈ F /p . This can be written as T x p = x p T + y . One can prove inductively thatT x mp = x mp T + mx m − p y, which in its turn implies T P ( x p ) = P ( x p ) T + P ′ ( x p ) y, ∀ P ( X ) ∈ F [ X ] . Choosing P ( X ) = p ( X ) so that p ( x p ) = 0 gives p ′ ( x p ) y = 0. But p ( X ) is the minimal polynomial of x p and deg p ′ < deg p , therefore p ′ ( x p ) = 0. This shows that y = 0, and thus [T , F /p ] = [T , F ( x p )] = 0, whichby the self-centralization of F /p proves that T ∈ F /p . (cid:3) For every m, n ∈ N , p ∈ σ F and rectangular matrix T ∈ Hom F ( F /p m , F /p n ) denote[T , x p ] = T( m ⊗ x p ) − ( n ⊗ x p ) T , were the tensor product is understood in the same sense as in (5). Corollary 4
For every p ∈ σ F , m, n ∈ N , and for every rectangular matrix T ∈ Hom F ( F /p m , F /p n ) , thestatement [T , x p ] ∈ Hom F /p ( F /p m , F /p n ) implies T ∈ Hom F /p ( F /p m , F /p n ) . Proof:
Every matrix T ∈ Hom F ( F /p m , F /p n ) can be considered as a block matrix of dimension n × m withblocks being elements of End F ( F /p ). Therefore the commutatorR . = T( m ⊗ x p ) − ( n ⊗ x p ) T ∈ Hom F ( F /p m , F /p n ) s a block matrix with blocks being R k,l = [T k,l , x p ] ∈ End F ( F /p ), k = 1 , ..., n , l = 1 , ..., m . If we knowthat R ∈ Hom F /p ( F /p m , F /p n ), which means that each block entry satisfies R k,l = [T k,l , x p ] ∈ F /p , then byLemma 1 we get T k,l ∈ F /p , meaning that T ∈ Hom F /p ( F /p m , F /p n ). (cid:3) For two F -vector spaces V , V and two linear operators T ∈ End F ( V ) and T ∈ End F ( V ), an intertwiner of T and T is a linear operator ∆ ∈ Hom F ( V , V ) such that ∆ T = T ∆. We denote byC F (T , T ) the F -vector space of all intertwiners between T and T , and we set C F (T ) . = C F (T , T ).For a square matrix T ∈ End F ( F n ) we will denote by 0 x T ∈ Hom F ( F n + l , F n + m ) the rectangular matrixwith T occupying its top right corner and zeros elsewhere.First we find all intertwiners between two Jordan blocks. Lemma 2
For every m, n ∈ N , C F (N m , N n ) = (cid:8) x ∆ ∈ Hom F ( F m , F n ) ∆ ∈ F [N min { m,n } ] (cid:9) . Proof:
This can be proven by direct computation. The commutation relation ∆ N m = N n ∆ for a matrix∆ ∈ Hom F ( F m , F n ) literarily means ∆ k,l − = ∆ k +1 ,l for all 1 ≤ k ≤ n , 1 ≤ l ≤ m if we set for consistency∆ k, = ∆ n +1 ,l = 0. This forces ∆ to be constant on the diagonal and all superdiagonals of its top rightlargest square minor, and zero everywhere else, which is equivalent to the assertion. (cid:3) Lemma 3
For every ∆ ∈ C F (J( p, m ) , J( p, n )) and k ∈ N , the k -fold nested commutator satisfies ∆ k . = [ ... [∆ , x p ] , ..., x p ] = k X l =0 ( − l kl ! (N n ⊗ deg p ) k − l ∆(N m ⊗ deg p ) l . Proof:
The proof is based on induction by k . From ∆ J( p, m ) − J( p, n )∆ = 0 andJ( p, n ) = n ⊗ x p + N n ⊗ deg p , ∀ n ∈ N , we verify the statement for k = 1,∆ = [∆ , x p ] = ∆( m ⊗ x p ) − ( n ⊗ x p )∆ = (N n ⊗ deg p )∆ − ∆(N m ⊗ deg p ) . Now suppose that the statement is true for some k ∈ N . We establish that∆ k +1 = ∆ k ( m ⊗ x p ) − ( n ⊗ x p )∆ k = k X l =0 ( − l kl ! (N n ⊗ deg p ) k − l [∆ , x p ](N m ⊗ deg p ) l = k X l =0 ( − l kl ! (N n ⊗ deg p ) k +1 − l ∆(N m ⊗ deg p ) l + k +1 X l =1 ( − l kl ! (N n ⊗ deg p ) k +1 − l ∆(N m ⊗ deg p ) l = k +1 X l =0 ( − l k + 1 l ! (N n ⊗ deg p ) k +1 − l ∆(N m ⊗ deg p ) l , here in the last step we used the familiar properties of binomial coefficients, kl ! + kl − ! = k + 1 l ! , kk + 1 ! = k − ! = 0 . The proof is complete. (cid:3)
Proposition 7
For every p, q ∈ σ F and m, n ∈ N , C F (J( q, m ) , J( p, n )) = δ p,q C F /p (N m , N n ) . Proof:
Let ∆ J( q, m ) = J( p, n )∆. Then ∆ q m (J( q, m )) = 0 = q m (J( p, n ))∆. But if p = q then q m (J( p, n ))is invertible because it is upper block-triangular with all diagonal blocks being equal to 0 = q m ( x p ) ∈ F /p ,which forces ∆ = 0. This explains the factor δ p,q in the statement. Now assume p = q and let ∆ J( p, m ) =J( p, n )∆. Then by Lemma 3 we have∆ k . = [ ... [∆ , x p ] , ..., x p ] = k X l =0 ( − l kl ! (N n ⊗ deg p ) k − l ∆(N m ⊗ deg p ) l . (10)If we set k = m + n then the right hand side of (10) vanishes, because each summand contains a factorN kn with k ≥ n . This gives ∆ m + n = [∆ m + n − , x p ] = 0. Now if we apply Corollary 4 to ∆ k iterativelyfrom k = m + n − k = 1 we will eventually establish that ∆ ∈ Hom F /p ( F /p m , F /p n ). This will also implythat N n ∆ − ∆ N m = [∆ , x p ] = 0 ∈ End F /p ( F /p m , F /p n ) , so that ∆ ∈ C F /p (N m , N n ), as desired. Conversely, every ∆ ∈ C F /p (N m , N n ) satisfies ∆J( p, m ) = J( p, n )∆. (cid:3) Next we will describe the explicit solutions of certain linear equations involving Jordanable operators.Let us first introduce the set of invertible matricesV n ( λ ) . = diag( λ, . . . , λ n ) = λ . . . λ . . . . . . . . . . . . . . . . . . λ n , ∀ λ ∈ F ∗ , ∀ n ∈ N . These matrices have a few useful properties.
Lemma 4
For all p ∈ σ F and λ ∈ F ∗ we have V deg p ( λ | λ | ǫ − ) x λ⋆p = λx p V deg p ( λ | λ | ǫ − ) ∈ End F ( F /p ) , that is, V deg p ( λ | λ | ǫ − ) ∈ C F ( x λ⋆p , λx p ) , where ǫ is per (4). Proof:
Proof is by direct computation of matrix products. (cid:3)
Lemma 5
For all n ∈ N , p ∈ σ F and λ ∈ F ∗ we have λ J( p, n ) (cid:0) V n ( λ − ) ⊗ V deg p ( λ | λ | ǫ − ) (cid:1) = (cid:0) V n ( λ − ) ⊗ V deg p ( λ | λ | ǫ − ) (cid:1) J( λ ⋆ p, n ) , hat is, V n ( λ − ) ⊗ V deg p ( λ | λ | ǫ − ) ∈ C F (J( λ ⋆ p, n ) , λ J( p, n )) , where ǫ is per (4). Proof:
Proof is by direct computation of products of block matrices, λ J( p, n ) (cid:0) V n ( λ − ) ⊗ V deg p ( λ | λ | ǫ − ) (cid:1) = λx p λ deg p . . . λx p λ deg p . . . . . . . . . . . . . . . . . . . . . λx p λ − V deg p ( λ | λ | ǫ − ) 0 0 . . . λ − V deg p ( λ | λ | ǫ − ) 0 . . . . . . . . . . . . . . . . . . . . . λ − n V deg p ( λ | λ | ǫ − ) = x p V deg p ( λ | λ | ǫ − ) λ − V deg p ( λ | λ | ǫ − ) 0 . . . λ − x p V deg p ( λ | λ | ǫ − ) λ − V deg p ( λ | λ | ǫ − ) . . . . . . . . . . . . . . . . . . . . . λ − n +1 x p V deg p ( λ | λ | ǫ − ) = λ − V deg p ( λ | λ | ǫ − ) x λ⋆p λ − V deg p ( λ | λ | ǫ − ) 0 . . . λ − V deg p ( λ | λ | ǫ − ) x λ⋆p λ − V deg p ( λ | λ | ǫ − ) . . . . . . . . . . . . . . . . . . . . . λ − n V deg p ( λ | λ | ǫ − ) x λ⋆p = λ − V deg p ( λ | λ | ǫ − ) 0 0 . . . λ − V deg p ( λ | λ | ǫ − ) 0 . . . . . . . . . . . . . . . . . . . . . λ − n V deg p ( λ | λ | ǫ − ) x λ⋆p deg p . . . x λ⋆p deg p . . . . . . . . . . . . . . . . . . . . . x λ⋆p = (cid:0) V n ( λ − ) ⊗ V deg p ( λ | λ | ǫ − ) (cid:1) J( λ ⋆ p, n ) , where in the third step we used Lemma 4. (cid:3) We are now ready to solve the equation X T = λ T X for a Jordanable operator T and a nonzeronumber λ ∈ F ∗ . Proposition 8
For a Jordanable operator
T = S − J( ℵ ) S and number λ ∈ F ∗ it holds C F (T , λ T) = S − C F (J( ℵ ) , λ J( ℵ )) S , C F (J( ℵ ) , λ J( ℵ )) = V( λ ; ℵ ) · C F (J( ℵ ) , J( λ ⋆ ℵ )) , V( λ ; ℵ ) . = M p ∈ σ F ∞ M n =1 M ℵ ( p,n ) (cid:0) V n ( λ − ) ⊗ V deg p ( λ | λ | ǫ − ) (cid:1) , where ǫ is per (4). Every operator R ∈ C F (J( ℵ ) , J( λ ⋆ ℵ )) is a block operator with arbitrary blocks of theform R q,n,β ; p,m,α ∈ δ p,λ⋆q C F /p (N m , N n ) , ∀ p, q ∈ σ F , ∀ m, n ∈ N , ∀ α ∈ ℵ ( p, m ) , ∀ β ∈ ℵ ( q, n ) such that there are finitely many nenzero entries in every column of R . roof: The original equation X T = λ T X with the substitution T = S − J( ℵ ) S gives X = S − ∆ S wherethe operator ∆ satisfies ∆ J( ℵ ) = λ J( ℵ )∆. A blockwise application of Lemma 5 gives λ J( ℵ ) V( λ ; ℵ ) =V( λ ; ℵ ) J( λ ⋆ ℵ ) with V( λ ; ℵ ) as in the statement. This implies that ∆ = V( λ ; ℵ ) R where the operatorR satisfies R J( ℵ ) = J( λ ⋆ ℵ ) R. If we break down the operator R into blocks R q,n,β ; p,m,α then the latterequation reduces to R q,n,β ; p,m,α J( p, m ) = J( λ ⋆ q, n ) R q,n,β ; p,m,α , or R q,n,β ; p,m,α ∈ C F (J( p, m ) , J( λ ⋆ q, n )) = δ p,λ⋆q C F /p (N m , N n ) , were the second step follows from Proposition 7. (cid:3) Next we introduce for convenience the following sequence of matrices,U n . = ddλ (cid:2) λ n V n ( λ − ) (cid:3) (1) = n − . . . n − . . . . . . . . . . . . . . . , ∀ n ∈ N . These matrices are good for the following property.
Lemma 6
For all n ∈ N we have U n N n − N n U n = N n . Proof:
The proof is by direct computation of matrix products. (cid:3)
Let us proceed to the equation Y T − T Y = T for a Jordanable operator T. The monic irreduciblepolynomial p ( X ) = X will be denoted simply by X ∈ σ F . Proposition 9
For a nonzero Jordanable operator
T = S − J( ℵ ) S the equation Y T − T Y = T hassolutions Y if and only if supp ℵ = { X } , in which case the set of all solutions is S − (cid:0) U( ℵ ) + C F (J( ℵ )) (cid:1) S , U( ℵ ) . = ∞ M n =1 M ℵ ( X,n ) U n where C F (J( ℵ )) = C F (J( ℵ ) , J( ℵ )) can be found using Proposition 8 with λ = 1 . Proof:
The original equation Y T − T Y = T with the substitution T = S − J( ℵ ) S gives Y = S − ∆ Swhere the operator ∆ satisfies ∆ J( ℵ ) − J( ℵ )∆ = J( ℵ ). If we decompose the operator ∆ into blocks∆ q,n,β ; p,m,α then the latter equation reduces to∆ q,n,β ; p,m,α J( p, m ) − J( q, n )∆ q,n,β ; p,m,α = δ q,p δ n,m δ β,α J( p, m ) , ∀ p, q ∈ σ F , ∀ m, n ∈ N , ∀ α ∈ ℵ ( p, m ) , ∀ β ∈ ℵ ( q, n ) . n the diagonal q = p , n = m , β = α we get∆ p,m,α ; p,m,α J( p, m ) − J( p, m )∆ p,m,α ; p,m,α = J( p, m ) ∈ Hom F ( F /p m , F /p m ) , (11)which is equivalent to [∆ p,m,α ; p,m,α , x p ] = [(N m ⊗ deg p ) , ∆ p,m,α ; p,m,α ] + J( p, m ) . (12)From (11) one can show inductively that∆ p,m,α ; p,m,α J( p, m ) k − J( p, m ) k ∆ p,m,α ; p,m,α = k J( p, m ) k , ∀ k ∈ N , and thus∆ p,m,α ; p,m,α P (J( p, m )) − P (J( p, m ))∆ p,m,α ; p,m,α = P ′ (J( p, m )) J( p, m ) , ∀ P ( X ) ∈ F [ X ] . Choosing P ( X ) = p ( X ) we obtain∆ p,m,α ; p,m,α (N m ⊗ deg p ) − (N m ⊗ deg p )∆ p,m,α ; p,m,α = p ′ (J( p, m )) J( p, m ) , which combined with (12) yields[∆ p,m,α ; p,m,α , x p ] = − p ′ (J( p, m )) J( p, m ) + J( p, m ) ∈ Hom F /p ( F /p m , F /p m ) . Therefore by Corollary 4 we find that ∆ p,m,α ; p,m,α ∈ Hom F /p ( F /p m , F /p m ), i.e., it is F /p -linear. Thus we canview (11) as an F /p -linear equation and take the F /p -trace of both parts,tr F /p [∆ p,m,α ; p,m,α , J( p, m )] = 0 = tr F /p J( p, m ) = mx p , whence x p = 0 and p ( X ) = X . This proves that if a solution Y exists then necessarily supp ℵ = { X } .Now assume that supp ℵ = { X } . The equation [∆ , J( ℵ )] = J( ℵ ) is linear inhomogeneous, and itsgeneral solution is of the form ∆ = ∆ + X , where ∆ is a particular solution and X is an arbitrary solution of the homogeneous equation [X , J( ℵ )] = 0,i.e., ∆ ∈ ∆ + C F (J( ℵ )) , as desired. It remains to show that ∆ = U( ℵ ) = ∞ M n =1 M ℵ ( X,n ) U n s indeed a particular solution,[U( ℵ ) , J( ℵ )] = ∞ M n =1 M ℵ ( X,n ) [U n , N n ] = ∞ M n =1 M ℵ ( X,n ) N n = J( ℵ ) , where J( X, n ) = N n and Lemma 6 were used. (cid:3) Our final task in this section will be to solve the equation Z T + T ⊤ Z = 0 for a Jordanable operator T.Since the transpose of a linear operator does not generally make an invariant sence in infinite dimensions(but only in a basis where the operator has finitely many nonzero entries in every row), we restrict to thecase T = J( ℵ ). To this avail we first introduce the matricesP n = . . . . . . . . . . . . . . . . . . . . . ∈ End F ( F n ) , ∀ n ∈ N . For n = 1 we have P = 1. Lemma 7
For all n ∈ N we have P n N n = N ⊤ n P n . Proof:
The proof is by direct computation of matrix products. (cid:3)
For every p ∈ σ F let p ( X ) = X d + a d − X d − + . . . + a , d = deg p, and define d − µ , . . . , µ d − ∈ F recursively by µ n = − a d − n − n − X k =1 a d − n + k µ k , n = 1 , . . . , d − . (13)We introduce the symmetric matrixW ǫp = . . . . . . ǫµ . . . ǫµ µ . . . . . . . . . . . . . . . ǫµ . . . µ d − µ d − ∈ End F ( F /p ) . If deg p = 1 then x p ∈ F and W ǫp = 1. Lemma 8
For all p ∈ σ f we have W ǫp x p = x ⊤ p W ǫp ∈ End F ( F /p ) , where ǫ is per (4). roof: The proof is by direct computation of matrix products. If ǫ = 1 thenW p x p = . . . . . . µ . . . µ µ . . . . . . . . . . . . . . . . . . µ . . . µ d − µ d − µ d − . . . − a . . . − a . . . − a . . . . . . . . . . . . . . . . . . . . . − a d − = . . . − a d − . . . µ − a d − − µ a d − . . . µ µ − a d − − µ a d − − µ a d − . . . . . . . . . . . . . . . . . .µ µ . . . µ d − µ d − − a − µ a − . . . − µ d − a d − = . . . µ . . . µ µ . . . µ µ µ . . . . . . . . . . . . . . . . . .µ µ . . . µ d − µ d − − a − µ a − . . . − µ d − a d − = . . . . . . . . . . . . . . . . . . . . . . . . . . . − a − a − a . . . − a d − − a d − . . . . . . µ . . . µ µ . . . . . . . . . . . . . . . . . . µ . . . µ d − µ d − µ d − = x ⊤ p W p , where the defintion (13) of the numbers µ n was used in the third equality. If on the other hand ǫ = 0then W p x p = a − bb a = b aa − b = a b − b a = x ⊤ p W p , as desired. (cid:3) Proposition 10
For a nonzero multiplicity function ℵ , the solutions of theequation Z J( ℵ ) + J( ℵ ) ⊤ Z = 0 are
Z = W( ℵ ) A , where W( ℵ ) = M p ∈ σ F ∞ M n =1 M ℵ ( p,n ) P n ⊗ W ǫp ,ǫ is per (4) and A ∈ C F (J( ℵ ) , − J( ℵ )) as in Proposition 8 with λ = − . Proof:
Using Lemma 7 and Lemma 8 we establish for every n ∈ N and p ∈ σ F that(P n ⊗ W ǫp ) J( p, n ) = (P n ⊗ W ǫp )( n ⊗ x p + N n ⊗ deg p ) = P n ⊗ (W ǫp x p ) + (P n N n ) ⊗ W ǫp P n ⊗ ( x ⊤ p W ǫp ) + (N ⊤ n P n ) ⊗ W ǫp = ( n ⊗ x ⊤ p + N ⊤ n ⊗ deg p )(P n ⊗ W ǫp ) = J( p, n ) ⊤ (P n ⊗ W ǫp ) , which implies W( ℵ ) J( ℵ ) = J( ℵ ) ⊤ W( ℵ ) with operator W( ℵ ) as in the statement. The operator W( ℵ ) isinvertible, and we are free to make a substitution Z = W( ℵ ) A, which yields A J( ℵ ) + J( ℵ ) A = 0, i.e.,A ∈ C F (J( ℵ ) , − J( ℵ )) as desired. (cid:3) We finally come to the main subject of this paper, which is the study of Jordanable almost Abelian Liealgebras. An almost Abelian Lie algebra is a non-Abelian Lie algebra over a field F which contains acodimension 1 Abelian subalgebra. We refer the reader to [1] for all relevant definitions and facts aboutthis class of Lie algebras that are going to be used here. An almost Abelian Lie algebra can be writtenas the semidirect product L = F e ⋊ V of the 1-dimensional Abelian Lie algebra F e with a (dim F L − V . The Lie algebra structure is completely determined by the nonzerooperator ad e ∈ End F ( V ) defined by ad e v = [ e , v ] , ∀ v ∈ V . In Proposition 11 of [1] it was shown that isomorphism classes of almost Abelian Lie algebras are in abijective correspondence with projective similarity classes [ F ∗ ad e ]. Here we want to consider a subclassof almost Abelian Lie algebras which allows for a more explicit description of structure than in [1].In particular, several results in that paper are formulated in terms of solutions to operator (matrix)equations, which are hard to solve explicitly in general. In order to achieve explicit formulae we assumethat the operators (matrices) in question are Jordanable, which makes these matrix equations tractablealgebraically. Definition 5
We will say that an almost Abelian Lie algebra L = F e ⋊ V is Jordanable if the operator ad e over V is Jordanable. Remark 4
In particular, every finite dimensional almost Abelian Lie algebra is Jordanable. If L = F e ⋊ V is a Jordanable almost Abelian Lie algebra then by Proposition 11 of [1] and Corollary 2of the last section we see that the composite map[ L ] [ F ∗ ad e ] = [ F ∗ J]( ℵ ad e ) F ∗ ⋆ ℵ ad e (14)from isomorphism classes of Jordanable almost Abelian Lie algebras to projective multiplicity functionsis a bijection. Denote by a A F ( ℵ ) = a A ( ℵ ) . = F e ⋊ V , ad e = J( ℵ ) , V = F dim F ℵ the Jordanable almost Abelian Lie algebra associated to the multiplicity function ℵ , and by a A ( F ∗ ⋆ ℵ ) =[ a A ( ℵ )] its isomorphism class. Then the map F ∗ ⋆ ℵ 7→ a A ( F ∗ ⋆ ℵ ) is the inverse of the map in (14). Inthe sequel we will be interested only in almost Abelian Lie algebras up to isomorphism, and from every somorphism class a A ( F ∗ ⋆ ℵ ) we will always choose the convenient representative a A ( ℵ ) defined above.Every Jordanable almost Abelian Lie algebra L = F e ⋊ V can be brought to this canonical form bychoosing a suitable basis in V and rescaling e .Let us now fix a non-zero multiplicity function ℵ (remember that almost Abelian Lie algebras areassumed to be non-Abelian) and consider the associated canonical almost Abelian Lie algebra L = a A ( ℵ ) = F e ⋊ V . Let us for every p ∈ σ F , n ∈ N and α ∈ ℵ ( p, n ) choose standard F /p -basis { e mα ( p, n ) } , m = 1 , ..., n of the block space F /p n so that V = M p ∈ σ F ∞ M n =1 M α ∈ℵ ( p,n ) n M m =1 F /p e mα ( p, n ) , N n e mα ( p, n ) = e m − α ( p, n ) , m > , m = 1 . (15)According to Remark 2 from [1] we have for the centre Z ( L ) = ker ad e and for the lower central ceries L ( k ) = [ L , ..., [ L , L ] ... ] = ad ke V for all k ∈ N . Now ad e = J( ℵ ) allows us to explicitly write Z ( L ) = M p ∈ σ F ∞ M n =1 M ℵ ( p,n ) ker J( p, n ) = ∞ M n =1 M ℵ ( X,n ) ker N n , where we write X for the polynomial p ( X ) = X with x p = 0, and J( X, n ) = N n by formula (6). In thestandard basis this reads Z ( L ) = ∞ M n =1 M α ∈ℵ ( X,n ) F e α ( X, n ) (16)(we used the fact that F /p = F for p ( X ) = X ). Meanwhile L ( k ) = M p ∈ σ F ∞ M n =1 M ℵ ( p,n ) J( p, n ) k F /p n = M p ∈ σ F \{ X } ∞ M n =1 M ℵ ( p,n ) F /p n M ∞ M n =1 M ℵ ( X,n ) N kn F n , which in the standard basis becomes L ( k ) = M p ∈ σ F \{ X } ∞ M n =1 M ℵ ( p,n ) n M m =1 F /p e mα ( p, n ) M ∞ M n =1 M α ∈ℵ ( X,n ) n − k M m =1 F e mα ( X, n ) . (17)In particular, we arrive at the following. Remark 5 a A ( ℵ ) is nilpotent if and only if supp ℵ = { X } . Let us turn now to the decomposition L = L ⊕ W from Proposition 5 of [1]. Here only blocks with p ( X ) = X and n = 1 enter the direct summand W , W = M ℵ ( X, F = M α ∈ℵ ( X, F e α ( X, , (18)and the rest falls into L .Another important structural property of a Lie algebra is the family of its Lie subalgebras and ideals.Proposition 4 in [1] describes these for a given almost Abelian Lie algebra in terms of the kernel and theimage of ad e and subspaces of V invariant under ad e . For a Jordanable almost Abelian Lie algebraan explicit description is available for all three constituents. The kernel and the image of ad e were escribed above, and ad e -invariant subspaces were dealt with in Corollary 3. One important fact impliedby Proposition 6 is the following. Remark 6
Every almost Abelian Lie subalgebra of a Jordanable almost Abelian Lie algebra is Jordanable.
Next we will describe the automorphism group of a Jordanable almost Abelian Lie algebra other thanthe Heisenberg algebra with all details spelled out (for the Heisenberg algebra the answer is well knownand can be found, for instance, in [1]). The following is an easy corollary of Lemma 2, Proposition 8 andProposition 10 in [1].
Proposition 11
The automorphism group of the indecomposable Jordanable almost Abelian Lie algebra a A ( ℵ ) = F e ⋊ V with ad e = J( ℵ ) other than H F is Aut ( a A ( ℵ )) = ν γ ∆ ν ∈ Dil( ℵ ) , γ ∈ V , ∆ ∈ Aut( F dim F ℵ ) , ∆ q,n,β ; p,m,α = δ p,ν⋆q (cid:0) V n ( ν − ) ⊗ V d ( ν | ν | ǫ − ) (cid:1) R p ; n,β ; m,α ∈ Hom F ( F /p m , F /p n ) , (cid:0) V n ( ν − ) ⊗ V d ( ν | ν | ǫ − ) (cid:1) = . . . ν . . . . . . . . . . . . . . . . . . ν d − . . . ν − . . .
00 1 . . . . . . . . . . . . . . . . . . ν d − . . . . . . . . . . . . . . . . . . ν − n . . . ν − n . . . . . . . . . . . . . . . . . . ν d − n for ǫ = 1 and (cid:0) V n ( ν − ) ⊗ V d ( ν | ν | ǫ − ) (cid:1) = | ν | − ν | ν | − . . . ν − | ν | − | ν | − . . . . . . . . . . . . . . . . . . ν − n | ν | − ν − n | ν | − or ǫ = 0 , R p ; n,β ; m,α = . . . r r . . . r l . . . r . . . r l − . . . . . . . . . . . . . . . . . . . . . . . . . . . r ∈ End F /p ( F /p m , F /p n ) if m ≥ n, R p ; n,β ; m,α = r r . . . r l r . . . r l − . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . ∈ End F /p ( F /p m , F /p n ) if m < n,l = min { m, n } , d = deg p, r , . . . , r l ∈ F /p , and ǫ is per (4). For every p, m, α only finitely many q, n, β correspond to nonzero blocks. We can also describe the Lie algebra of derivations in a similar manner. Here we will distinguishbetween nilpotent and non-nilpotent Lie algebras a A ( ℵ ), see Remark 5. The below results are an immediateconsequence of Lemma 2, Proposition 8, Proposition 9 and Proposition 14 in [1]. Proposition 12
The algebra of derivations of the indecomposable Jordanable non-nilpotent almost AbelianLie algebra a A ( ℵ ) = F e ⋊ V with ad e = J( ℵ ) is Der ( a A ( ℵ )) = γ ∆ γ ∈ V , ∆ ∈ End F ( F dim F ℵ ) , ∆ q,n,β ; p,m,α = δ p,q R p ; n,β ; m,α ∈ Hom F ( F /p m , F /p n ) , R p ; n,β ; m,α = . . . r r . . . r l . . . r . . . r l − . . . . . . . . . . . . . . . . . . . . . . . . . . . r ∈ End F /p ( F /p m , F /p n ) if m ≥ n, R p ; n,β ; m,α = r r . . . r l r . . . r l − . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . ∈ End F /p ( F /p m , F /p n ) if m < n, = min { m, n } , d = deg p, r , . . . , r l ∈ F /p . For every p, m, α only finitely many q, n, β correspond to nonzero blocks.
Proposition 13
The algebra of derivations of the indecomposable Jordanable nilpotent almost AbelianLie algebra a A ( ℵ ) = F e ⋊ V with ad e = J( ℵ ) other than H F is Der ( a A ( ℵ )) = α γ ∆ α ∈ F , γ ∈ V , ∆ ∈ End F ( F dim F ℵ ) , ∆ X,n,β ; X,m,α = αδ n,m U n + R n,β ; m,α ∈ Hom F ( F m , F n ) , U n = n − . . . n − . . . . . . . . . . . . . . . . . . ∈ End F ( F n ) , R n,β ; m,α = . . . r r . . . r l . . . r . . . r l − . . . . . . . . . . . . . . . . . . . . . . . . . . . r ∈ End F ( F m , F n ) if m ≥ n, R n,β ; m,α = r r . . . r l r . . . r l − . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . ∈ End F ( F m , F n ) if m < n,l = min { m, n } , r , . . . , r l ∈ F . For every m, α only finitely many n, β correspond to nonzero blocks.
Again, for the Heisenberg algebra the answer is well known and can be found, for instance, in [1].We now proceed to the description of quadratic Casimir elements of the Jordanable almost AbelianLie algebra a A ( ℵ ) = F e ⋊ V . Following Section 6 of [1] we denote by h : a A ( ℵ ) → U( a A ( ℵ )) the embeddingof the Lie algebra into its universal enveloping algebra, and choose a basis { e i } i ∈ dim F ℵ in V . Note thedifference in the meaning of the symbol ℵ between the preset paper and [1]. What was ℵ in [1] now hasbecome dim F ℵ , where we identify the cardinality of a set with the set itself. One particular basis can beconstructed starting from the F /p -bases { e mα ( p, n ) } in each F /p n as above. In order to build an F -basis in F /p n we generate elements e m,kα ( p, n ) = x kp e mα ( p, n ), k = 1 , . . . , deg p −
1. This corresponds to the labelling i = ( p, n, α, m, k ). If we let x i = h ( e i ) and ad x i = h (ad e e i ) then since ad e = J( ℵ ) in the basis { e i } we ave that in the basis { x i } of h ( V ) the operator ad = J( ℵ ) as well.A quadratic form Q ( x ) on h ( V ) can be written as Q ( x ) = X i,j ∈ dim F ℵ A i ; j x i x j , where A ∈ End F ( h ( V )) is symmetric in the basis { e i } and has finitely many nonzero entries in every row(it always has finitely many non-zero entries in every column as long as it is a linear operator). Formallywe can also write Q ( x ) = x ⊤ A x . Proposition 14
Quadratic Casimir elements Q ( x ) ∈ Z (U( a A ( ℵ ))) of the Jordanable almost Abelian Liealgebra a A ( ℵ ) are x ⊤ A x , where the operator A ∈ End F ( h ( V )) is symmetric, has finitely many nonzeroentries in every column, and is of the form A q,n,β ; p,m,α = δ p, ( − ⋆q (P n ⊗ W ǫq )(V n ( − ⊗ V deg q ( − q,n,β ; p,m,α , (P n ⊗ W ǫq ) = . . . . . . . . . ǫµ . . . ǫµ µ . . . . . . . . . . . . . . . ǫµ . . . µ d − µ d − . . . . . . . . . ǫµ . . . ǫµ µ . . . . . . . . . . . . . . . ǫµ . . . µ d − µ d − . . . . . . . . . . . . . . . . . . ǫµ . . . ǫµ µ . . . . . . . . . . . . . . . ǫµ . . . µ d − µ d − . . . (V n ( − ⊗ V deg q ( − . . . − . . . . . . . . . . . . . . . . . . ( − d − . . . − . . .
00 1 . . . . . . . . . . . . . . . . . . ( − d − . . . . . . . . . . . . . . . . . . ( − − n . . .
00 ( − − n . . . . . . . . . . . . . . . . . . ( − d − n , R p ; n,β ; m,α = . . . r r . . . r l . . . r . . . r l − . . . . . . . . . . . . . . . . . . . . . . . . . . . r ∈ End F /p ( F /p m , F /p n ) if m ≥ n, R p ; n,β ; m,α = r r . . . r l r . . . r l − . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . ∈ End F /p ( F /p m , F /p n ) if m < n,l = min { m, n } , d = deg q, r , . . . , r l ∈ F /p , and ǫ is per (4). For every q ∈ σ F the numbers µ , . . . , µ d − are defined as in (13). Proof:
By Proposition 18 of [1] we know that a quadratic form Q ( x ) is a quadratic Casimir if and onlyif ad Q ( x ) = x ⊤ ad ⊤ A x + x ⊤ A ad x = x ⊤ (A ad + ad ⊤ A) x = 0 , that is, A ad + ad ⊤ A = A J( ℵ ) + J( ℵ ) ⊤ A = 0 . By Proposition 10 and Proposition 8 we know that this is equivalent toA = W( ℵ ) V( − ℵ ) C , here C ∈ C F (J( ℵ ) , J(( − ⋆ ℵ )). Together with Lemma 2 this yields the assertion. (cid:3) algebra Let F = R and p, q ∈ σ F , p ( X ) = X − , q ( X ) = X + 1 , i.e., x p = 1 and x q = −
1. Consider the multiplicity function ℵ = (1 × p , × q ) with notation from (1),or even ℵ = (1 × , × ( − ) if no confusion may arise. Bianchi VI is the Jordanable almost AbelianLie algebra Bi(VI ) = a A ( ℵ ) = a A (1 × p , × q ) = R e ⋊ R , with ad e = J( ℵ ) = J( p, ⊕ J( q,
1) = − . We set up the adapted basis as in (15), R = R e ( p, ⊕ R e ( q, , e ( p,
1) = (1 , ⊤ , e ( q,
1) = (0 , ⊤ . From (16) we see that the centre of the algebra is Z ( a A (1 × p , × q )) = ker ad e = 0 , while from (17) we read that the lower central series is a A (1 × p , × q ) ( k ) = R e ( p, ⊕ R e ( q,
1) = R . Formula (18) tells us that a A (1 × p , × q ) is indecomposable.Let us now investigate the invariant (proper) subspaces W of ad e which we do with the help ofCorollary 3. Since ℵ ( ., n ) = 0 for n > ℵ ( p,
1) = ℵ ( q,
1) = 1, in formula (9) we have k = 1, α = 1and l = 1, so that for all n ∈ N and m = 1 , . . . , n we get η mβ ( p, n ) = µ p ( n, β ; 1 , , m − e ( p, , η mβ ( q, n ) = µ q ( n, β ; 1 , , m − e ( q, . Corollary states that for every β ∈ i ( p, n ) there exists an α ∈ ℵ ( p, ¯ n ) such that ¯ n ≥ n and µ p ( n, β ; n, α, =0. Since α ∈ ℵ ( p, n ) and α ∈ ℵ ( q, n ) for n > i ( p, n ) = 0 and i ( q, n ) = 0 for n > n = 1 and m = 1 are attained, η β ( p,
1) = µ p (1 , β ; 1 , , e ( p, , η β ( q,
1) = µ q (1 , β ; 1 , , e ( q, . Further, since the sum in formula (8) must be direct, the vectors η β ( p,
1) must be linearly independentfor different β ∈ i ( p, i ( p,
1) = 1. Similarly, i ( q,
1) = 1. Thus we are left with only two ossibilites, µ q (1 ,
1; 1 , ,
0) = 0 and W = R e ( p, ⊕ µ p (1 ,
1; 1 , ,
0) = 0 and W = 0 ⊕ R e ( q, . This result was immediately obvious from the form of the matrix ad e , but our discussion above aimedat demonstrating how to use Corollary 3 rather than at the result as such.We now proceed to study the Lie subalgebras and ideals of a A (1 × p , × q ), which we do with thehelp of Proposition 4 in [1]. The (proper) subalgebras are of one of the following forms: • Abelian Lie subalgebras of the form L = W ( R • Abelian Lie subalgebras of the form L = R ( e + v ) for v ∈ R , since ker ad e = 0 • Almost Abelian Lie subalgebras of the form L = R ( e + v ) ⋊ W with v ∈ R and W = R e ( p, ⊕ W = 0 ⊕ R e ( q, e + v , e ( p, e ( p,
1) and[ e + v , e ( q, − e ( q, ax + b = a A (1 × )The proper ideals of a A (1 × p , × q ) are among the following: • Abelian ideals R e ( p, ⊕
0, 0 ⊕ R e ( q,
1) and R The other two possibilities in Proposition 4 in [1] offer L = R ( e + v ) ⋊ W with ad e R ⊂ W . But sincein our case ad e R = R we obtain L = R ( e + v ) ⋊ R , which is not a proper ideal.Proceeding to automorphisms we first note that Dil( ℵ ) = { , − } where( − ⋆ p = q, ( − ⋆ q = p. By Proposition 11 we establish thatAut( a A (1 × p , × q )) = γ ∆ p, , p, , γ q, , q, , , − γ p, , q, , γ ∆ q, , p, , ,γ , γ , ∆ p, , p, , , ∆ q, , q, , , ∆ p, , q, , , ∆ q, , p, , ∈ R , ∆ p, , p, , ∆ q, , q, , = 0 ∆ p, , q, , ∆ q, , p, , = 0 . For derivations we use Proposition 12,Der( a A (1 × p , × q )) = γ ∆ p, , p, , γ q, , q, , ,γ , γ , ∆ p, , p, , , ∆ q, , p, , ∈ R . Finally, quadratic Casimir elements are found from Proposition 14. It follows that the matrix A has the orm A = ab , a, b ∈ R , and the symmetry forces a = b . If we denote x = h ( e ( p, y = h ( e ( q, Q = axy, a ∈ R . Let F = R and p, q ∈ σ F , p ( X ) = X + 1 , q ( X ) = X + 4 π . On this occasion let us adopt the traditional representation (3), x p = −
11 0 , x q = − π π , so that ǫ = 0. Consider the multiplicity function ℵ = (1 × p , × q ). The Mautner algebra (the Liealgebra of the prominent Mautner group) is the Jordanable almost Abelian Lie algebra a A ( ℵ ) = a A (1 × p , × q ) = R e ⋊ R , with ad e = J( ℵ ) = J( p, ⊕ J( q,
1) = − − π − π . In both cases F /p = R ( x p ) = R ( x q ) = a − bb a a, b ∈ R ≃ C , and we will denote this representation of the field extension simply by C . Had we chosen (2) instead wewould need to distinguish between R ( x p ) = C p and R ( x q ) = C q which are both isomorphic to C but havedifferent representations as R -matrices. We set up the adapted basis as in (15), R = C e ( p, ⊕ C e ( q, , e ( p,
1) = (1 , , , ⊤ , e ( q,
1) = (0 , , , ⊤ . From (16) we see that the centre of the algebra is Z ( a A (1 × p , × q )) = ker ad e = 0 , while from (17) we read that the lower central series is a A (1 × p , × q ) ( k ) = C e ( p, ⊕ C e ( q,
1) = R . ormula (18) tells us that a A (1 × p , × q ) is indecomposable.Let us now investigate the invariant (proper) subspaces W of ad e which we do with the help ofCorollary 3. Since ℵ ( ., n ) = 0 for n > ℵ ( p,
1) = ℵ ( q,
1) = 1, in formula (9) we have k = 1, α = 1and l = 1, so that for all n ∈ N and m = 1 , . . . , n we get η mβ ( p, n ) = µ p ( n, β ; 1 , , m − e ( p, , η mβ ( q, n ) = µ q ( n, β ; 1 , , m − e ( q, . Corollary states that for every β ∈ i ( p, n ) there exists an α ∈ ℵ ( p, ¯ n ) such that ¯ n ≥ n and µ p ( n, β ; n, α, =0. Since α ∈ ℵ ( p, n ) and α ∈ ℵ ( q, n ) for n > i ( p, n ) = 0 and i ( q, n ) = 0 for n > n = 1 and m = 1 are attained, η β ( p,
1) = µ p (1 , β ; 1 , , e ( p, , η β ( q,
1) = µ q (1 , β ; 1 , , e ( q, . Further, since the sum in formula (8) must be direct, the vectors η β ( p,
1) must be linearly independentfor different β ∈ i ( p, i ( p,
1) = 1. Similarly, i ( q,
1) = 1. Thus we are left with only twopossibilites, µ q (1 ,
1; 1 , ,
0) = 0 and W = C e ( p, ⊕ µ p (1 ,
1; 1 , ,
0) = 0 and W = 0 ⊕ C e ( q, . We now proceed to study the Lie subalgebras and ideals of a A (1 × p , × q ), which we do with thehelp of Proposition 4 in [1]. The (proper) subalgebras are of one of the following forms: • Abelian Lie subalgebras of the form L = W ( R • Abelian Lie subalgebras of the form L = R ( e + v ) for v ∈ R , since ker ad e = 0 • Almost Abelian Lie subalgebras of the form L = R ( e + v ) ⋊ W with v ∈ R and W = C e ( p, ⊕ W = 0 ⊕ C e ( q, e + v , e ( p, x p e ( p,
1) and[ e + v , e ( q, x q e ( q, ) = E (2) = a A (1 × p ).The proper ideals of a A (1 × p , × q ) are among the following: • Abelian ideals C e ( p, ⊕
0, 0 ⊕ C e ( q,
1) and R The other two possibilities in Proposition 4 in [1] offer L = R ( e + v ) ⋊ W with ad e R ⊂ W . But sincein our case ad e R = R we obtain L = R ( e + v ) ⋊ R , which is not a proper ideal.Proceeding to automorphisms we first note that Dil( ℵ ) = { , − } where( − ⋆ p = p, ( − ⋆ q = q. y Proposition 11 we establish thatAut( a A (1 × p , × q )) = ± γ ∆ rp, , p, , − ∆ ip, , p, , γ ∆ ip, , p, , ∆ rp, , p, , γ rq, , q, , − ∆ iq, , q, , γ iq, , q, , ∆ rq, , q, , ,γ , γ , γ , γ ∈ R , ∆ rp, , p, , − ∆ ip, , p, , ∆ ip, , p, , ∆ rp, , p, , , ∆ rq, , p, , − ∆ iq, , p, , ∆ iq, , p, , ∆ rq, , p, , ∈ C , (∆ rp, , p, , ) + (∆ ip, , p, , ) + (∆ rq, , q, , ) + (∆ iq, , q, , ) > . For derivations we use Proposition 12,Der( a A (1 × p , × q )) = γ ∆ rp, , p, , − ∆ ip, , p, , γ ∆ ip, , p, , ∆ rp, , p, , γ rq, , q, , − ∆ iq, , q, , γ iq, , q, , ∆ rq, , q, , , ∆ rp, , p, , − ∆ ip, , p, , ∆ ip, , p, , ∆ rp, , p, , , ∆ rq, , p, , − ∆ iq, , p, , ∆ iq, , p, , ∆ rq, , p, , ∈ C . Finally, quadratic Casimir elements are found from Proposition 14. It follows that(P ⊗ W p )(V ( − ⊗ V ( − ⊗ W q )(V ( − ⊗ V ( − −
11 0 , and the matrix A has the form A = a − b b a c − d d c , a, b, c, d ∈ R , while symmetry requires b = d = 0. If we denote x = h ( e ( p, , y = h ( x p e ( p, , z = h ( e ( q, , w = h ( x q e ( q, , hen the quadratic Casimir is Q = a ( x + y ) + c ( z + w ) , a, c ∈ R . a A Q (1 × √ ) ⊕ Q Let F = Q and p, q ∈ σ F , p ( X ) = X − , q ( X ) = X. According to (2), x p = , x q = 0 , and ǫ = 1. Consider the multiplicity function ℵ = (1 × p , × q ) and the Jordanable almost Abelian Liealgebra a A Q ( ℵ ) = a A ( ℵ ) = a A (1 × p , × q ) = Q e ⋊ Q , with ad e = J( ℵ ) = J( p, ⊕ J( q,
1) = . In this case we have F /p = Q ( x p ) = a c bb a cc b a a, b, c ∈ Q ≃ Q ( √ , whereas Q ( x q ) = Q . We set up the adapted basis as in (15), Q = F /p e ( p, ⊕ F /p e ( p, ⊕ Q e ( q, ,e ( p,
2) = (1 , , , , , , ⊤ , e ( p,
2) = (0 , , , , , , ⊤ , e ( q,
1) = (0 , , , , , , ⊤ . From (16) we see that the centre of the algebra is Z ( a A (1 × p , × q )) = ker ad e = Q e ( q, , while from (17) we read that the lower central series is a A (1 × p , × q ) ( k ) = F /p e ( p, ⊕ F /p e ( p, ⊕ . ormula (18) tells us that a A (1 × p , × q ) = a A (1 × p ) ⊕ Q , hence the title of this subsection.Let us now investigate the invariant (proper) subspaces W of ad e which we do with the help ofCorollary 3. First consider p . Since ℵ ( p,
2) = 1 and ℵ ( p, n ) = 0 for n = 2 we have k = 2 and α = 1 onlyin formula (9). Moreover, the corollary states that for every β ∈ i ( p, n ) there exists an α ∈ ℵ ( p, ¯ n ) suchthat ¯ n ≥ n and µ p ( n, β ; n, α, = 0. Since α ∈ ℵ ( p, n ) for n = 2 we find that n ∈ { , } . For n = 1 weget η β ( p,
1) = µ p (1 , β ; 2 , , e ( p, , ∀ β ∈ i ( p, . For n = 2 we obtain for ∀ β ∈ i ( p, η β ( p,
2) = µ p (2 , β ; 2 , , e ( p, , η β ( p,
2) = µ p (2 , β ; 2 , , e ( p,
2) + µ p (2 , β ; 2 , , e ( p, . Now since the sum in formula (8) must be direct, we see that either µ p (1 , β ; 2 , ,
0) = 0 and i ( p,
1) = 0or µ p (2 , β ; 2 , ,
0) = 0 and i ( p,
2) = 0. Following similar reasoning, from ℵ ( q,
1) = 1 and ℵ ( q, n ) = 0 for n > k = 1, α = 1 and n = 1 in (9). Therefore η β ( q,
1) = µ p (1 , β ; 1 , , e ( q, , ∀ β ∈ i ( q, . Again arguments of linear independence show that i ( q, ≤
1. To conclude, we have the followingpossibilities for an invariant proper subspace: • i = (1 × p ) and W = F /p e ( p, ⊕ F /p e ( p, • i = (1 × p ) and W = F /p e ( p, • i = (1 × q ) and W = Q e ( q, • i = (1 × p , × q ) and W = F /p e ( p, ⊕ Q e ( q, a A (1 × p , × q ), which we do with thehelp of Proposition 4 in [1]. The (proper) subalgebras are of one of the following forms: • Abelian Lie subalgebras of the form L = W ( Q • Abelian Lie subalgebras of the form L = Q ( e + v ) or L = Q ( e + v ) ⊕ Q e ( q,
1) for v ∈ Q • Almost Abelian Lie subalgebras of the form L = Q ( e + v ) ⋊ W with v ∈ Q and either ofthe following: a) W = F /p e ( p,
2) so that L ≃ a A (1 × p ), b) W = F /p e ( p, ⊕ F /p e ( p,
2) so that L ≃ a A (1 × p ), c) W = F /p e ( p, ⊕ Q e ( q,
1) so that L ≃ a A (1 × p ) ⊕ Q .The proper ideals of a A (1 × p , × q ) are among the following: • Abelian ideals L = W ⊆ Q that are invariant subspaces, proper as above or improper • Almost Abelian ideal L = Q ⋊ ( F /p e ( p, ⊕ F /p e ( p, ≃ a A (1 × p )The remaining possibility in Proposition 4 in [1] requires ad e Q ⊂ ker ad e which does not hold (seeRemark 3 in [1]). et us now proceed to the automorphisms. Our algebra has a decomposition a A (1 × p , × q ) = L ⊕ W , L = Q e ⋊ ( F /p e ( p, ⊕ F /p e ( p, a A (1 × p ) , W = Q e ( q, . Let us deal with L first. Since Dil( ℵ ) = { } , by Proposition 11 we establish thatAut( a A (1 × p )) = γ ∆ a c b ∆ d f e γ ∆ b ∆ a c ∆ e ∆ d f γ ∆ c ∆ b ∆ a ∆ f ∆ e ∆ d γ a c b γ b ∆ a c γ c ∆ b ∆ a ,γ , γ , γ , γ , γ , γ ∈ Q , ∆ a c b ∆ b ∆ a c ∆ c ∆ b ∆ a , ∆ d f e ∆ e ∆ d f ∆ f ∆ e ∆ d ∈ F /p , (∆ a ) + (∆ b ) + (∆ c ) > . Coming back to the direct sum a A (1 × p , × q ) = a A (1 × p ) ⊕ Q , Proposition 7 and Proposition 8 in[1] yield Aut( a A (1 × p , × q )) = φ φ φ φ ,φ ∈ Aut( L ) , φ ( W ) ∈ Z ( L ) = 0 , ( L ) (1) = F /p e ( p, ⊕ F /p e ( p, ⊂ ker φ , ker φ = 0 . Thus Aut( a A (1 × p , × q )) = γ ∆ a c b ∆ d f e γ ∆ b ∆ a c ∆ e ∆ d f γ ∆ c ∆ b ∆ a ∆ f ∆ e ∆ d γ a c b γ b ∆ a c γ c ∆ b ∆ a γ δ ,γ , δ ∈ Q , δ = 0 . n dealing with derivations let us again start with L = a A (1 × p ). We use Proposition 12,Der( a A (1 × p )) = γ ∆ a c b ∆ d f e γ ∆ b ∆ a c ∆ e ∆ d f γ ∆ c ∆ b ∆ a ∆ f ∆ e ∆ d γ a c b γ b ∆ a c γ c ∆ b ∆ a ,γ , γ , γ , γ , γ , γ ∈ Q , ∆ a c b ∆ b ∆ a c ∆ c ∆ b ∆ a , ∆ d f e ∆ e ∆ d f ∆ f ∆ e ∆ d ∈ F /p . For the direct sum a A (1 × p , × q ) = a A (1 × p ) ⊕ Q we apply Proposition 12 in [1], which tells us thatDer( a A (1 × p , × q )) = φ φ φ φ ,φ ∈ Der( L ) , φ ( W ) ∈ Z ( L ) = 0 , ( L ) (1) = F /p e ( p, ⊕ F /p e ( p, ⊂ ker φ . Thus Der( a A (1 × p , × q )) = γ ∆ a c b ∆ d f e γ ∆ b ∆ a c ∆ e ∆ d f γ ∆ c ∆ b ∆ a ∆ f ∆ e ∆ d γ a c b γ b ∆ a c γ c ∆ b ∆ a γ δ ,γ , δ ∈ Q . Finally, quadratic Casimir elements are found from Proposition 14. Since ( − ⋆ q = q and ℵ (( − ⋆ p, . ) = 0 , e see that only the q ; q -block contributes to a Casimir element,A = a , a ∈ Q . If we denote w = h ( e ( q, Q = aw . References [1] Z. Avetisyan. Structure of almost Abelian Lie agebras.
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