SKEW-ADJOINT MAPS AND QUADRATIC LIE ALGEBRAS

. The procedure of double extension of vector spaces endowed with non-degenerate bilinear forms allows us to introduce the class of generalized K -oscillator algebras over any arbitrary ﬁeld K . Starting from basic structural properties of such algebras and the canonical forms of skew-adjoint endomorphisms, we will proceed to classify the subclass of quadratic nilpotent algebras and characterize those algebras in the class with quadratic dimension 2. This will enable us to recover the well-known classiﬁcation of real oscillator algebras, also known as Lorentzian algebras, given by Alberto Medina in 1985.


Introduction
In linear algebra, considerable work has been done on isometric, selfadjoint, and skew-adjoint endomorphisms with respect to a bilinear form.Canonical matrices for operators of this type on a complex inner product vector space have been provided in [Horn and Merino, 1999].Similar results for more general fields can be found in [Sergeichuk, 1988] and, more recently, in [Caalim et al., 2020].In this paper, we will focus on skew-adjoint endomorphisms and their significance as a fundamental tool in constructing quadratic Lie algebras.
The algebra L is said to be quadraticif it is endowed with a non-degenerate symmetric bilinear form, ϕ, which is invariant with respect to the Lie bracket: (2) ϕ([x, y], z) + ϕ(y, [x, z]) = 0.
If (L, ϕ) is quadratic, the invariance of the equation( 2) is equivalent to the left multiplication operators, called adjoint or inner derivations denoted by ad x, being ϕ-skew-adjoint endomorphisms.In fact, the set of inner derivations of L, Inner(L), is an ideal of the whole algebra of derivations of L, Der(L).
Semisimple Lie algebras under their Killing-Cartan form, which is defined as κ(x, y) = Tr(ad x ad y), are nice examples of quadratic algebras.On the opposite structural side, we find abelian Lie algebras; all of them are quadratic by using any non-degenerate symmetric form.The orthogonal sum (as ideals) of semisimples and abelian algebras allows us to assert that reductive Lie algebras are also quadratic.According to [Medina and Revoy, 1985], any non-simple, non-abelian, and indecomposable quadratic Lie algebra (i.e. the algebra does not break as an orthogonal sum of two regular ideals) is a double extension either by a one-dimensional or by a simple Lie algebra.The main tool enabling this classical procedure is the existence of skewadjoint derivations.We point out that the class of quadratic Lie algebras is quite large and contains reductive Lie algebras and also infinitely many non-semisimple examples.Most of the examples, structures, and constructions on quadratic algebras have been set over fields of characteristic zero (see [Ovando, 2016] for a survey-guide).In positive characteristic, they have not been as extensively studied.
Over the reals, the double extension of any Euclidean vector space by a skew-adjoint automorphism yields to the class of real oscillator algebras.They are quadratic and solvable Lie algebras of dimension 2n + 2 and a bilinear invariant form of Lorentzian type (inner product with metric signature (2n + 1, 1)).The real oscillator class was firstly introduced and easily described, thanks to the Spectral Theorem on real skew-adjoint operators, by ALberto Medina in [Medina, 1985, Section 4].The name oscillator comes from quantum mechanics because they describe a system of a harmonic oscillator n-dimensional Euclidean space.At the same time, J. Hilgert and K. H. Hofmann arrive at oscillator algebras in their characterization of Lorentzian cones in real Lie algebras [Hilgert and Hofmann, 1985].In [Hilgert et al., 1989, Definition II.3.6], the authors term them as (solvable) Lorentzian algebras.For n ≥ 2, the Levi subalgebra of the algebra of skew-derivations of a (2n + 2)-dimensional oscillator algebra is the special unitary real Lie algebra su n (R) (see [Benito and Roldán-López, 2023, Theorem 3.2]).So, oscillator algebras can be doubly extended to a countable series of non-semisimple and non-solvable quadratic Lie algebras.Moreover, the study of other non-associative structures on oscillator algebras provides information on connections and metrics on oscillator Lie groups [Albuquerque et al., 2021, Section 5].
Throughout this paper, we will extend the notion of real oscillator algebras to any arbitrary field K of characteristic different from 2. Under the name of generalized K-oscillator algebras, we encode the double extensions of any abelian quadratic Lie algebra by any skew-adjoint endomorphisms.In the particular case K = R, extensions through skew-adjoint automorphisms allow us to recover the class of real oscillator algebras [Medina, 1985, Lemme 4.2].
The paper is organized as follows.In Section 2, we assemble some basic properties, orthogonal decompositions, and canonical forms of skew-adjoint endomorphisms.Definition 3.1 in Section 3 establishes the concept of generalized oscillator algebras over arbitrary fields of characteristic not 2, and Lemma 3.2 reviews some of structural properties of this class of quadratic algebras.In Section 3, we also classify those algebras in the class that are nilpotent in Theorem 3.5, and provide a characterization of those with quadratic dimension two.In Section 4, we prove that indecomposable quadratic algebras with Witt index 1 are simple or solvable.The solvable ones are just the subclass of generalized oscillator algebras which are constructed as a double extension of skew-adjoint automorphisms.The proof of the assertion is based on the concept of isomaximal ideal introduced in [Kath and Olbrich, 2004].This result enables us to recover the well-established classification of real oscillator algebras given by Medina in 1985.

Skew-maps on orthogonal subspaces
Let V be a K-vector space and f : , with π i (x) monic and k i ≥ 1, induces a direct sum vector decomposition of V , referred to as primary decomposition: where . Each primary component is an finvariant subspace.In the particular case π 1 (x) = x − λ, the scalar λ is an eigenvalue of f , and V x−λ is usually denoted as V λ .In this case, the subspace is called a generalized λ-eigenspace.Thus, V 0 is the generalized 0-eigenspace, and V 0 will be null if and only if f is a bijective map.Let J n (λ) denote the Jordan n-by-n canonical block and C(π(x)) the companion matrix of a given monic polynomial π Now, let ϕ be a symmetric bilinear form, and assume that f is ϕ-skewadjoint (henceforth, we will use ϕ-skew to abbreviate this term), which means, ϕ(f (x), y) = −ϕ(x, f (y)).Then, for any s ≥ 0, we have: This implies that, ( 6) From now on, the pair (V, ϕ), where ϕ is a symmetric bilinear form, will be called an orthogonal K-vector space.
Proposition 2.1.Let (V, ϕ) be an orthogonal K-vector space, and , and primary decomposition V = r i=1 V π i .The orthogonal subspaces V ⊥ π i are f -invariant, and for any π i (x), we have two possibilities: a) for any direct summand of W with s ≥ 1. Suppose W = V , which is equivalent to f not being one-to-one.Then, the polynomial x appears in the factorization of m f (x).Reordering, if necessary, we assume Thus, V 0 ⊥ W , and any assertion regarding the primary component V π 1 follows.This is a particular case of item b).Without loss of generality, we can assume f is one-to-one.Since π i (x) is irreducible, so is π i (−x), thus either gcd(π i (x), π j (−x)) = 1 or gcd(π i (x), π j (−x)) = π i (x).The second case happens if and only if π j (−x) = ±π i (x).Firstly, assume π i (−x) = ±π j (x) for any 1 ≤ j ≤ r.Then, gcd(π v, w).This implies 0 = V π i ⊂ V ⊥ , and a) follows.Otherwise, the uniqueness of the decomposition into irreducibles of the minimal polynomial ensures that there exists a unique 1 ≤ j i ≤ r such that π i (−x) = (−1) deg π i π j i (x).This implies that π i (−x) and π k (x) are coprime for any k = j i .Using the Bezout's identity as in the previous reasoning, V π i ⊥ V π k , and the direct sum decomposition of V ⊥ π i follows from the f -invariance of V ⊥ π i .This decomposition also shows that V π i ⊆ V ⊥ π i when i = j i .Finally, we will prove the equality ) n a 0 is the unique monic polynomial fulfilling p(−x) = (−1) deg p q(x).The set of roots of q(x) is just {−λ : p(λ) = 0}.In particular, q(x) = p(x) if and only if the monomials of p(x) have even degree, and the roots of p(x) are of the form ±λ 1 , . . ., ±λ n .
From the previous Proposition 2.1, we arrive at the following general orthogonal decomposition of (V, ϕ) through a ϕ-skew map.
Corollary 2.3.Let (V, ϕ) be an orthogonal K-vector space over a field of characteristic not 2, and f : V → V be a ϕ-skew linear map.Then, up to permutation, the factorization of into irreducible monic polynomials splits as m f (x) = x α p(x)q(x)s(x)n(x) where α ≥ 0, and if the degree of some of the p, q, s, n factors is ≥ 1: This yields to the orthogonal sum, V π j ⊕ V π j+l with V π j and V π j+l totally isotropic subspaces, and Corollary 2.4.Let (V, ϕ) be an orthogonal subspace over a field K of characteristic not 2 such that ϕ is non-degenerate, and denote by and a 0 = 0.In particular, f = 0 is the only possibility if the base field is algebraically closed.b) There exists a permutation σ ∈ S r , σ 2 = 1, such that π , and either m f (x) = x α or m f (x) = x α p(x), p(0) = 0, and the monomials of p(x) are of even degree.So, the nonzero roots of m f (x) are of the form ±λ 1 , . . ., ±λ n .
Proof.Suppose that there are no nonzero isotropic vectors.Thus with a 0 = 0. Denote by E 1,2n i the 2n i × 2n i elemental matrix (a r,s = 0 for (r, s) = (1, 2n i ) and a 1,2n i = 1), and let C(π i (x)) be the companion matrix described in equation ( 4).If k i ≥ 2, there is a set T of 4n i linearly independent vectors such that U = span T is f -invariant, and the matrix of Then we can find v, w ∈ T such that . Using previous equalities, we have that: And thus, ϕ(w, w) = 0, a contradiction.This proves assertion a).From V ⊥ = 0, the decomposition in c) and items b) and d) follow from Proposition 2.1 and Remark 2.2.Finally, to get dim From previous equalities, we conclude the equidimensionality.
Example 2.1.Applying previous Corollary 2.4 we get the well-know Spectral Theorem (skew-Hermitian case).Let (V, ϕ) be a real orthogonal vector space without isotropic vectors, and and there are 0 and U i (v) ⊥ are f -invariant subspaces, just reducing dimension by orthogonal decompositions and rescaling, we get an orthonormal basis of V such that the pair (f, ϕ) is represented by the pair of matrices (A f , Id dim V ) and A f is the matrix that decomposes as diagonal blocks Here is the number of 2 × 2 matrix λ i 's blocks and d 0 = dim ker f denotes the number of 1 × 1 matrix 0's blocks.For f | V 0 , we use any orthonormalization process.This is just the Spectral Theorem for real matrices.
For any arbitrary field, assuming no isotropic vectors and the irreducible semisimple decomposition m f (x) = x(x 2 + µ 1 ) • • • (x 2 + µ t ), the pair (f, ϕ) is represented by the pair of matrices (A f , B ϕ ) where B ϕ is a diagonal matrix and ( 9) Next, let us return to assertion c) in Corollary 2.4, and consider the orthogonal decomposition The subspace V 0 and the subspaces of V 2 of the form V λ ⊕ V −λ (associated with the primary components V π(x) where π(x) = x ± λ) admit canonical forms determined by the Jordan blocks J n (λ) described in equation ( 4).
Theorem 2.5.Let (V, ϕ) an orthogonal vector space over an arbitrary field of characteristic not two such that ϕ is non-degenerate.Suppose that f : are equidimensional and totally isotropic subspaces.Even more, V λ ⊕ V −λ decomposes as orthogonal sum of a finite number of f -invariant subspaces (W i , ϕ| W i ) that have a basis in which the pair ) of the following type: And, for the zero eigenvalue, V 0 is an orthogonal sum of f -invariant subspaces U j that have a basis in which the pair ) as in equation ( 10) with λ = 0 and n j = 2k j or (here Proof.From Corollaries 2.3 and 2.4, we establish that and then ϕ(v, (f +λ Id) k (w)) = (−1) k α = 0, so (f +λ Id) k (w) = 0. Rescaling w if necessary, we can suppose α = 1.
Next, define v := v 0 , w := w 0 , v r := (f − λ Id) r (v 0 ) and w s := (f + λ Id) s (w 0 ) for 1 ≤ r, s ≤ k.We point out that ϕ(v r , w s ) = (−1) s ϕ(v r+s , w 0 ) by applying equation (6).We now set a new w ′ = w ′ 0 obtained in the linear span of {w 0 = w, w 1 , . . ., w k }, i.e., Those α i coefficients are obtained as solutions from the system of equations for j = 0, . . ., k: ϕ(v j , w ′ ) = δ jk , where δ jk is the Kronecker delta.This is a triangular system whose diagonal coefficients are ±1.Therefore there is a unique solution w ′ , and, as ) is expressed by a matrix pair as described in expression (10) with 1 is f -invariant and regular, so recursively we get the desired ortogonal decomposition for the nonzero eigenvalue λ.
In the sequel we assume that 0 = V 0 , equivalent to 0 being an eigenvalue of f .Let k 0 be the multiplicity of this eigenvalue in the minimum polynomial, and k := k 0 − 1.Hence, V 0 = ker f k+1 and there exists a nonzero v ∈ V 0 such that f k (v) = 0. We will consider two different cases depending on the parity of k and, in both cases, we will start with the previous v.Let us first suppose that k = 2n − 1 is odd and k 0 = 2n.Then, Since 2 = 0, ϕ(v, f k (v)) = 0 and, we can find an element w ∈ V 0 with ϕ(w, f k (v)) = 0 by means of the non-degenerancy of ϕ and the equality In this context, the previous procedure for the nonzero eigenvalue λ remains valid, allowing us to identify an f -invariant and regular vector space U 1 with a basis in which (f | U 1 , ϕ| U 1 ) is expressed by a matrix pair as described in equation ( 10) with n 1 = k 0 = 2n and λ = 0.
The other case arises when k = 2n is even, so Moreover, using the previous assumptions and k = 2n, we have We can then replace v with v ′ that fulfills the required condition.Starting next from v ∈ V 0 such that 0 = µ = ϕ(v, f k (v)), we define recursively w 0 = v and for 1 ≤ j ≤ n: A straightforward computation shows that w n satisfies ϕ(w n , f 2n (w n )) = µ = 0, and ϕ(w n , f t (w n )) = 0 for any 1 ≤ t < 2n (for t odd numbers, apply equation ( 5)).Even more, Then, the subspace is f -invariant and regular, and the ordered set {v 0 , . . ., v 2n } where v j := f j (w n ) is a basis in which (f | U 1 , ϕ| U 1 ) is expressed by a matrix pair as described in equation ( 12) with n 1 = n, so k 0 = 2n 1 + 1.By reescaling and reordering in the following way we arrive at the matrix pair described in equation ( 11).
Concluding the proof, note that U 1 is a regular subspace, thus V 0 = U 1 ⊕ (U 1 ) ⊥ .Since (U 1 ) ⊥ is f -invariant and regular, recursively accounting for the parity in each step yields the desired orthogonal decomposition.
Remark 2.6.Over the complex field, the previous result is established in [Horn and Merino, 1999, Theorem 2] as a classical characterization of the Jordan canonical forms of complex orthogonal and skew-symmetric matrices.The analogous result over algebraically closed fields of characteristic not 2 appears in [Caalim et al., 2020, Corollary 1.2].Here, we provide an alternative proof that highlights a recursive constructive method based on straightforward linear arguments.The canonical form (A f | U j , B ϕ| U j ) proposed in [Caalim et al., 2020] is:

Generalized oscillator algebras
Along this section, we assume the base field K is of characteristic not 2 unless otherwise stated.
In any quadratic solvable and non-abelian n-dimensional Lie algebra (L, ϕ), such that Z(L) ⊆ L 2 = Z(L) ⊥ the centre is a totally isotropic ideal.According to [Favre and Santharoubane, 1987, Proposition 2.9], any solvable quadratic Lie algebra (characteristic zero) contains a central isotropic element z, and the algebra L can be obtained as a double extension of the (n − 2)-dimensional quadratic and solvable Lie algebra (K•z) ⊥ K•z .Applying this one-step process iteratively, we get the class of solvable quadratic algebras from the class of quadratic abelian Lie algebras.Let's start this section with this construction.
Example 3.1 (One-dimensional-by-abelian double extension).Let (V, ϕ) be an orthogonal K-vector space with ϕ non-degenerate, and δ be any ϕ-skew map.Denote by δ * the dual 1-form of δ, so δ * : K • δ → K and λδ → λ.On the vector space d(V, ϕ, δ) for t, t ′ , s, s ′ ∈ K, x, y ∈ V .It is easily checked that this product is skew and satisfies the Jacobi identity, Lie algebra, and the bilinear form is symmetric, invariant, non-degenerate, and it extends ϕ by the hyperbolic space span K δ, δ * .The method of constructing this algebra is known as double extension.Over fields of characteristic zero, this procedure was introduced simultaneously by several authors in the 80s (see [Bordemann, 1997] and references therein), but starting from any quadratic Lie algebra (L, ϕ) and any δ ∈ Der ϕ L. In this case, the bracket [x, y] L must be added in the binary product given by equation ( 13).To obtain new indecomposable quadratic algebras, it is important to take a non-inner δ ϕ-skew derivation [Figueroa-O'Farrill and Stanciu, 1996, Proposition 5.1].This onedimensional construction is also valid in characteristic other than 2 (see [Bordemann, 1997, Theorem 2.2] for a more detailed explanation).
The Lie bracket and the invariant bilinear form ϕ δ are given in equation ( 13) and ( 14).Real oscillator algebras are quadratic and solvable.The Witt index of ϕ δ is 1, so ϕ δ is a Lorentzian form.This characteristic leads to these algebras also being referred to in the literature as (real) Lorentzian algebras (see [Hilgert et al., 1989, Definition II.3.16]).The one-dimensional-byabelian construction of this class of algebras appears in [Hilgert et al., 1989, Proposition II.3.11].The algebras obtained through this procedure are called standard solvable Lorentzian algebras of dimension 2n + 2, and they are denoted as A 2n+2 .
Previous examples serve as the introduction to the definition of oscillator algebras over arbitrary fields of characteristic not 2.The definition first appears in [Benito and Roldán-López, 2023].Definition 3.1.Let (V, ϕ) be a K-vector space of dimension greater than or equal to 2, endowed with a symmetric and non-degenerate bilinear form ϕ. For any ϕ-skew map δ : V → V and its dual 1-form linear map δ * : K•δ → K, described as λδ → λ, the one-dimensional-by-abelian double extension Lie algebra d(V, ϕ, δ) := K • δ ⊕ V ⊕ K • δ * with product as in equation ( 13) and bilinear form as in expression ( 14) will be referred to in this paper as a generalized K-oscillator algebra and as a K-oscillator algebra in the particular case where δ is automorphism.
In the sequel, we use the following terminology.A Lie algebra L is reduced if Z(L) ⊆ L 2 and local if L has only one maximal ideal [Bajo and Benayadi, 2007, Definition 3.1].The derived series of L is recursively defining as The algebra L is nilpotent if there exists k ≥ 2 such that L k = 0.The smallest k such that L k = 0 and L k+1=0 is the nilpotency index of L. A quadratic algebra is said to be decomposable if it contains a proper ideal I that is non-degenerate (I ∩ I ⊥ = 0, the ideal is also called a regular subspace), and indecomposable otherwise.Equivalently, L is decomposable if and only if L = I ⊕ I ⊥ .In the literature, the terms reducible and irreducible are also used as synonyms for decomposable and indecomposable.
Remark 3.1.In general, a non-reduced quadratic Lie algebra splits as an orthogonal sum, as ideals, of an abelian quadratic algebra and another reduced quadratic algebra.In characteristic zero, this assertion is just given in [Tsou and Walker, 1957, Theorem 6.2].Its proof also works in characteristics other than 2. Therefore, any non-reduced quadratic Lie algebra is decomposable.
Consider now the class of Lie algebras h that satisfy: Since, for any x, y ∈ h, [x, y] = λ x,y z and λ x,y ∈ K, the structure constants λ x,y allow us to define in h the skew-symmetric bilinear form ϕ h (x, y) := λ x,y .Then, if V is a complement vector space of the centre, i.e., h = V ⊕ K • z, the vector space is regular.The non-degeneracy of the skew-symmetric form ϕ h | V implies that there is a basis of V , {v 1 , . . ., v n , w 1 , . . ., w n , } such that ϕ h (v i , w j ) = δ ij and ϕ h (v i , v j ) = ϕ h (w i , w j ) = 0. Therefore, the algebras that satisfy equation ( 16) have odd dimension and exhibit a basis B = {x 1 , . . ., x n , y 1 , . . ., y n , z} such that [v i , w j ] = δ ij z and all other products are zero.Thus, for any n ≥ 1, there is only one algebra of dimension 2n + 1 that satisfies equation ( 16).In characteristic zero, these algebras are known as Heisenberg algebras.Throughout the paper, we will refer to them as generalized K-Heisenberg algebras.
Lemma 3.2.Let A = d(V, ϕ, δ) be a generalized K-oscillator Lie algebra.Then the following holds: a) For any k ≥ 1, A is nilpotent if and only if δ is a nilpotent map.And the nilpotency index is the degree of the minimum polynomial of δ.
In addition, if δ = 0, g) If δ is an automorphism, the dimension of V is even, and the derived algebra From equation ( 13), t 0 δ + x 0 + s 0 δ * ∈ Z(A) is equivalently to: (17) t 0 δ(y) − t ′ δ(x 0 ) + ϕ(δ(x 0 ), y)δ * = 0, for all t ′ ∈ K and y ∈ V .If δ = 0, A = Z(A) and then A 2 = 0. Otherwise, the centre turns out to be Z , the last equality is a consequence of the non-degeneracy of ϕ.This proves a) and b) when k = 1.Assume For the (k + 1)-term of the descending series, we have: If δ k = 0, that is, δ is a nilpotent map, A k+1 = 0 = A m for m ≥ k + 1, and A is a nilpotent algebra.Otherwise, δ k = 0 and we can take x ∈ V such that δ k (x) = 0.By the non-degenerancy of ϕ, there exists y ∈ V such that ϕ(δ k (x), y) = 0. Then A k+1 = Im δ k ⊕ K • δ * .This proves item a) and the assertion from right to left in item d).Suppose now that A is nilpotent and k is its nilpotency index, i.e., A k = 0 and A k+1 = 0.The assumption implies δ k = 0 = δ k−1 , so k is just the degree of the minimum polynomial m δ (x) and d) follows.
Next, for the upper central series, we assume Equivalently, the two following elements are in Z k (A): That is δ(x 0 ), t 0 δ(v) ∈ ker δ k for all v ∈ V .Since δ k+1 = 0, the only possibility is t 0 = 0 and x 0 ∈ ker δ k+1 .Therefore, Z k+1 (A) = ker δ k+1 ⊕K•δ * and b) is proven.The statement c) regarding the orthogonality of the terms of the descending and upper central series follows by using ϕ(δ k (x), y) = (−1) k ϕ(x, δ k (y)), so we have (20) , and the description of the terms in both series given in items a) and b).
, and items e) and f) follow easily.In the particular case that Im δ = V , we have A 2 = V ⊕ K • δ * , and looking for x ∈ V such that [y, x] = ϕ(δ(y), x)δ * = 0 for all y ∈ V , we get ϕ(V, x) = 0.This implies x = 0 from the non-degenerancy of ϕ.
Remark 3.3.Statement c) in Lemma 3.2, that is, the orthogonal terms, (A i+1 ) ⊥ , of the descending central series give us the upper central terms Z i (A) and vice versa, is well known for quadratic algebras in characteristic zero [Medina and Revoy, 1985].According to assertion e), generalized Koscillator algebras are reduced if and only if δ is an automorphism or the length n of any Jordan block J n (0) of δ is greater than or equal to 2. Remark 3.4.Indecomposable real quadratic Lie algebras satisfying that their quotient Lie algebra A 2 Z(A) is abelian have been treated in [Kath and Olbrich, 2004].
In this paper, the authors give the classification of real Lie algebras of maximal isotropic centre of dimension less or equal to 2. The method they use is the two-fold extension.In the next section, we will relate K-oscillator algebras and quadratic algebras with maximal isotropic centre of dimension 1.
Our previous Lemma 3.2 ensures the existence of quadratic nilpotent Lie algebras over fields of characteristic different from 2, with arbitrary nilpotence index, using one-dimensional-by-abelian double extensions and skewnilpotent maps.As a corollary of the results in Section 2, we can give the complete description of this type of algebras.
Proof.The result follows from Theorem 2.5 and Lemma 3.2.
The algebras in Example 3.3 admit other non-degenerate and invariant bilinear forms.In the case of n 3,2 all of their quadratic structures are isometrically isomorphic.This is clear from Theorem 2.5.The same is true for n 2,3 if K is algebraically closed, but not so clear from the previous theorem.If K = R, we have two non-isometrically isomorphic quadratic structures: the one given through µ 1 = −1 and that corresponding to µ 1 = 1.This comment leads us to the concept of quadratic dimension.
The quadratic dimension [Bajo and Benayadi, 1997] of a Lie algebra L is defined as d q (L) = dim B s inv (L) where B s inv (L) is the subspace of symmetric invariant bilinear forms of L. Note that L is quadratic if and only if d q (L) ≥ 1.For the quadratic free-nilpotent algebras in Example 3.3, d q (n 2,3 ) = 4 and d q (n 3,2 ) = 7.In characteristic zero, any quadratic Lie algebra such that d q (L) = 1 is simple [Bajo and Benayadi, 1997, Theorem 3.1], and the converse is also true over algebraically closed fields.The paper [Bajo and Benayadi, 2007] is devoted to the structure of quadratic Lie algebras with quadratic dimension 2. According to [Bajo and Benayadi, 2007, Lemma 3.1] (characteristic zero), any indecomposable Lie algebra whose quadratic dimension is 2 is a local Lie algebra.
For each pair (i, j), define the symmetric bilinear form on W as T i,j (w i , w j ) = 1 = T i,j (w j , w i ) and T i,j (w k , w s ) = 0 for (k, s) = (i, j).We extend T i,j to a symmetric form in A by defining T i,j (A 2 , A) = 0. We have that T i,j is invariant because A 2 ⊂ A ⊥ T i,j .So, the vector space span T i,j , ϕ : and, as the generator forms are linearly independent, the result follows.
We recall that a Lie algebra is local if it has a unique maximal ideal.We note that, in the case of quadratic Lie algebras, I is a maximal ideal if and only if I ⊥ is minimal.Therefore, having a unique maximal ideal is equivalent to having a unique minimal ideal in the class of quadratic algebras.
Lemma 3.7.Let A = d(V, ϕ, δ) be a generalized K-oscillator Lie algebra.The following assertions are equivalent: The quadratic dimension of A is 2. In this case, A 2 is a generalized K-Heisenberg algebra.And the set of invariant symmetric forms is the vector space B s inv (A) = span ϕ 1,0 , ϕ δ where ϕ 1,0 (A 2 , A) = 0 and ϕ 1,0 (d, d) = 1.
Proof.Any subspace U containing A 2 is an ideal and any subspace of Z(A) is an ideal.Assume firstly A is local.As dim A ≥ 4, A is not abelian, so δ = 0 and Z(A) = ker δ ⊕ K • δ * .Since there is only one minimal ideal, ker δ = 0, so Im δ = V , A 2 = V ⊕ K • δ * and b) follows.From b) and A 2 = (Z(A)) ⊥ , we have dim A = dim A 2 + dim Z(A) and we get c).If Z(A) = K • δ * = A, using Lemma 3.2 we have δ = 0 and Z(A) = ker δ ⊕ Kδ * , so ker δ = 0 and δ is an automorphism.Then c) implies d).Now we will prove the final comment and d) ⇒ e).From Lemma 3.6, So, ψ(δ * , δ * ) = 0. Now, for any x, y ∈ A and y = 0, Then, ϕ(x, y) = ϕ(y, x)ψ(δ * , δ) = βϕ δ (x, y) and the equality also holds for y = 0.In this way, we have that ψ = αϕ 1,0 + βϕ δ .Therefore, d q (A) = 2 and assertion e) follows.Assume finally e).From Lemma 3.6, r = dim Z(A) = 1 so Z(A) is a minimal ideal and therefore δ is an automorphism and A 2 is a maximal ideal.Let I be a minimal ideal different from Z(A), then I ∩Z(A) = 0 and A = I ⊥ + A 2 .As I ⊥ is an ideal, applying the non-degenerancy of ϕ δ , we have [I, I ⊥ ] = 0 and therefore 0 Our assumption implies ϕ(δ(v), w) = 0 for all w ∈ V , a contradiction because ϕ is non-degenerate and δ is an automorphism.This proves that Z(A) is the unique minimal ideal and therefore A is a local Lie algebra.
Remark 3.8.In characteristic zero, [Bajo and Benayadi, 2007, Theorem 3.1] offers a characterization of local algebras that includes the class of K-oscillators.The main goal in this paper is to provide examples and characterizations of algebras whose quadratic dimension is 2. In fact, our proof of d) ⇒ e) is the one presented in Proposition 4.1.This proposition asserts that the result is true for a double extension of any quadratic Lie algebra by any skew-symmetric derivation.
In the sequel, we will tackle the case of isomorphisms and isometric isomorphisms in the subclass of K-oscillator algebras.A similar result appears in [Favre and Santharoubane, 1987, Proposition 2.11].According to Definition 3.1, K-oscillator algebras are one-dimensional-by-abelian double extensions of orthogonal subspaces through skew-automorphisms.Theorem 3.9.Let A i = d(V i , ϕ i , δ i ) be two K-oscillator algebras.Then, A 1 and A 2 are isomorphic if and only if there exists an isomorphism f : V 1 → V 2 and scalars λ, µ ∈ K with λµ = 0 such that: a) . Moreover, they are isometrically isomorphic if and only if the previous conditions a) and b) stand with λµ = 1.
For the converse implication, assuming that a) and b) hold, take any z ∈ V 2 and ν ∈ K, and define F : and extend it by linearity.This way, the map is linear and bijective.To prove that F is an isomorphism, we just have to check If we also assume λµ = 1 and then we have that: This proves that, in this case, F is also an isometry.

Isomaximality and Lorentzian algebras
Along this section K is a field of characteristic zero.According to [Hilgert et al., 1989, Definition II.3.16], a Lorentzian Lie algebra is a pair (L, ϕ) with a real Lie algebra L and ϕ an invariant and non-degenerate Lorentzian form, i.e., a real invariant symmetric bilinear form with signature (p, 1) where p is the number of positive eigenvalues and q is the number of negative eigenvalues.Section 6 of Chapter II in [Hilgert et al., 1989] focuses on these algebras and explores the study of Lie semialgebras in them.The section points out that the complete classification of Lorentzian algebras is reduced to the indecomposable (named as irreducible by the authors) subclass, see [Hilgert et al., 1989, Remark II.6.1].And the classification of this subclass is fully covered by Theorem II.6.14.Throughout this final section, we will prove that, over any arbitrary field K of characteristic zero, the class of solvable quadratic irreducible Lie algebras with quadratic form of Witt index 1 (only one hyperbolic plane) is just the class of one-dimensional-by-abelian double extension through skewautomorphisms of orthogonal subspaces without isotropic vectors (i.e., the class of K-oscillator algebras according to Definition 3.1).The Witt index 1 condition generalizes the (p, 1) or the (1, q) signature condition in the real case.To achieve our result, we will use the concept of isomaximal ideal introduced in [Kath and Olbrich, 2004, Definition 2.3] and some basic facts on quadratic Lie algebras.Definition 4.1.Let I be a totally isotropic ideal in a quadratic Lie algebra (L, ϕ).The ideal I is called isomaximal if it is not contained in any other totally isotropic ideal.Proof.This is Lemma 2.2 and Lemma 2.3 in [Kath and Olbrich, 2004].The proofs of both lemmas hold over fields of characteristic zero.
Theorem 4.2.The following assertions are equivalent: a) (L, ψ) is an non-semisimple indecomposable quadratic Lie algebra such that ψ has Witt index 1.b) L = d(V, ϕ, δ) is a K-oscillator algebra, and ϕ(v, v) = 0 for all v ∈ V .Lie algebras that satisfy one of the previous equivalent conditions are double extensions of an abelian quadratic algebra (V, ϕ) by a ϕ-skew semisimple automorphism δ.Even more, any irreducible polynomial of the factorization of the minimum polynomial m Proof.Assume condition a) and note that the maximal dimension of any totally isotropic subspace is one.From Lemma 4.1, R(L) ⊥ ⊆ R(L), so R(L) ⊥ is a totally isotropic ideal, and therefore d = dim R(L) ⊥ ≤ 1.Since d = dim L−dim R(L) is just the dimension of any Levi factor of L, d ≥ 3 if d = 0. Thus, d = 0 and L = R(L) is a solvable Lie algebra.This implies L 2 = L, and then Z(L) = (L 2 ) ⊥ is a nonzero ideal.From Remark 3.1, 0 = Z(L) ⊂ L 2 = Z(L) ⊥ by indecomposibility; therefore, Z(L) is a totally isotropic ideal.So, Z(L) = K • z is a minimal ideal, ψ(z, z) = 0, and (Z(L)) ⊥ = L 2 is a maximal ideal of codimension one.Then L = K • x ⊕ L 2 , and from the non-degeneracy of ψ, we can assume without loss of generality ϕ(x, z) = 1, ϕ(x, x) = 0, i.e., x, z is a hyperbolic plane.From [Favre and Santharoubane, 1987, Lemma 2.7], we get that L is isometrically isomorphic to the double extension of the quadratic algebra (V = L 2 Z(L) , ϕ), ϕ(x + Z(L), y + Z(L)) := ψ(x, y).But the centre is an isomaximal ideal; thus, (V = L 2 Z(L) , ϕ) is a quadratic abelian Lie algebra following b) in Lemma 4.1.Since the Witt index of ψ is one, from the decomposition L = x, z ⊕ x, z ⊥ and L 2 = K • z ⊕ x, z ⊥ , it is easy to check that ϕ has no isotropic vectors.Note that, as L ∼ = d(V, ϕ, δ) and Z(L) is one dimensional, δ is an automorphism according to Lemma 3.7.Finally, as L 2 is the only maximal ideal of L, if I is any proper ideal, I ⊥ ⊆ L 2 and therefore Z(L) ⊆ (I ⊥ ) ⊥ = I.This implies that L is indecomposable.The final assertion on δ semisimple and irreducible factors of m δ (x) follows from the Corollary 2.4.
Remark 4.3.The class of K-oscillator algebras described in Theorem 4.2 is broad.According to Lemma 3.7, the quadratic dimension of any algebra in this class is 2.Moreover, Theorem 3.9 provides criteria for isomorphism (isometric isomorphism) between two algebras of this class.In the real case, the algebraic structure of oscillator algebras provides information on the geometry of oscillator groups [Baum and Kath, 2003, Theorem 5.1].
Remark 4.4.Any indecomposable quadratic Lie algebra with trivial solvable radical is simple.For any simple Lie algebra S, the Killing form κ(x, y) = Tr(ad x ad y) is invariant and, from Cartan's Criteria, κ is non-degenerate.For the special simple Lie algebra sl(n, K) of zero-trace matrices, the bilinear form b(A, B) = 1 2 Tr AB allows the recovery of the Killing form as κ = 8b.Over the reals, (sl(2, R), λκ) with λ > 0 are the unique simple Lorentzian algebras.All of them are isometrically isomorphic to (sl(2, R), κ).
Lemma 4.1(Kath, Olbrich, 2003).Let (L, ϕ) be a quadratic indecomposable Lie algebra over a field of characteristic zero, and denote by R(L) the solvable radical of L. Then: a) L has no proper simple ideals and R(L) ⊥ ⊆ R(L).b) If I is an isomaximal ideal, then I = L, I ⊥ ⊆ R(L), and the quotient Lie algebra I ⊥ I is abelian.