Central Test Set and Central Test Rank of Free Nilpotent Lie Algebras of Rank 2

Abstract

In this paper, we introduce and establish the concept of a central test set for free nilpotent Lie algebras. We prove that the central test rank of a free nilpotent Lie algebra L_c,2 of rank 2 and nilpotency class c = 3 equals 3. Furthermore, we show that a minimal central test set of L_3,2 belongs to the lower-degree bracket structures and is explicitly given by the set $𝒯$ = {[x,y], [[x,y],x], [x,y],y}. We also analyze higher nilpotency classes c ≥ 4 and delineate the degree conditions under which elements effectively eliminate non-central derivations.

1. Introduction

The study of automorphisms and structural invariants in Lie algebras plays a foundational role in understanding their representation theory, algorithmic properties, and symmetry structures [1,2]. Originating in the classical work of Nielsen [3], the. paradigm of recognizing global algebraic mappings through specific invariant elements has been extensively investigated across combinatorial algebra, group theory, and non-commutative cryptography [4]. Among various automorphism subgroups, central automorphisms—those that act trivially on the quotient of an algebra by its center—form a vital normal subgroup of the full automorphism group [5]. Characterizing whether a given mapping constitutes a general automorphism or belongs to a central subgroup remains an essential problem in combinatorial Lie theory [5].

A powerful paradigm developed to address these questions is the theory of test elements [6]. Introduced for free groups and algebras by Shpilrain [6], a test element t is an element whose invariance under an endomorphism φ (i.e., φ(t) = t) forces φ to be an automorphism. Early fundamental results established that commutators serve as primary test elements in two-generator free structures, as demonstrated by Dicks [7] and extended to primitive elements in metabelian algebras by Drensky and Yu [8]. Further extensions were developed by Mikhalev and Zolotykh [9] as well as Mikhalev and Yu [10] for free Lie algebras and superalgebras, examining test elements alongside the. invertibility of non-commutative Jacobian matrices [11,12].

While individual test elements suffice to recognize general automorphisms in free Lie algebras [10,9,6], they fail to capture sufficient algebraic information to resolve the full automorphism group in restricted quotient structures. To overcome this limitation, attention shifted toward test sets—collections of elements whose joint invariance ensures automorphism status—and the associated notion of test rank [13]. Esmerligil and Ekici [13] systematically investigated these properties in free metabelian Lie algebras, showing that the test rank equals n-1 with minimal test sets residing in the derived subalgebra. Subsequently, Temizyürek and Ekici [14] identified particular test elements in free solvable Lie algebras of rank two, and Eskal and Ekici [15] analyzed test elements on direct sums and free products of free Lie algebras. Related endomorphism and derivation properties in relatively free Lie structures have also been explored by Umirbaev [16], Timoshenko and Shevelin [17], Erginkara and Fındık [18] and Umirbaev [19], noting that different formulations of central automorphisms appear depending on whether action modulo the center or higher derived series is considered [18].

In nilpotent environments, the lower central series and the presence of a non-trivial center introduce unique structural obstacles [5]. As the nilpotency class c increases, the algebraic dimension of the center grows rapidly in accordance with Witt’s formula, whereas the rank of standard Lie-Jacobian derivation matrices [11,12] remains bounded by the number of free generators [5]. This dimensional mismatch leads to the phenomenon of “Jacobian blindness,” where central shifts and drifts project into the kernel undetected by standard derivation matrices [5,11,12].

To address these limitations, Öztekin and Ekici [5] systematically examined the properties of central automorphisms in free nilpotent Lie algebras. Building upon these foundations, this paper introduces the concept of a central test set—a collection of elements whose joint invariance under an endomorphism uniquely forces it to be a central automorphism [5]. Focusing on the free nilpotent Lie algebra L_3,2 of rank 2 and nilpotency class c = 3, we analyze the constraint matrices governing central endomorphisms and prove that the central test rank is strictly equal to 3. Furthermore, we provide an explicit construction of the minimal central test set given by

$𝒯$ = {[x,y], [[x,y],x], [x,y],y}

that completely resolves this Jacobian blindness for class 3.

2. Notations and Preliminaries

Throughout this paper, K denotes a field of characteristic zero and L = L(x,y) denotes the free Lie algebra generated by the set X = {x,y} over K. The product operation in L is denoted by the Lie bracket [∙,∙] which satisfies the standard axioms of anticommutativity and the Jacobi identity.

The lower central series of L is defined inductively by:

γ₁(L) = L, γ_k+1(L) = [γ_k(L),L] for k ≥ 1

A Lie algebra G is called nilpotent of class c if γ_c+1(G) = {0} and γ_c(G) ≠ {0}. The free nilpotent Lie algebra of rank 2 and nilpotency class c denoted by L_c,2 is defined as the quotient algebra:

$$L_{c,2}={\displaystyle \raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{${\gamma }_{c+1}\left(L\right)$}\right.}$$

Let Π:L → L_c,2 be the natural projection mapping. For brevity, we denote the images of the free generators Π(x) and Π(y) simply as x and y within L_c,2. The algebra L_c,2 inherits a natural graded structure from the free Lie algebra

$$L_{c,2}={\oplus }_{k=1}^c{L}^{\left(k\right)}$$

where $L^{\left(k\right)}={\displaystyle \raisebox{1ex}{${\gamma }_k\left(L\right)$}\!\left/ \!\raisebox{-1ex}{${\gamma }_{k+1}\left(L\right)$}\right.}$ represents the k-th homogeneous component. The center of L_c,2 denoted by Z(L_c,2) corresponds precisely to the supreme graded component of this series, namely:

Z(L_c,2) = L^(c)

An endomorphism φ of L_c,2 is uniquely determined by its action on the free generators:

φ(x) = α₁x + β₁y + u₁, φ(y) = α₂x + β₂y + u₂

where ${\alpha }_1, {{\alpha }_2,\beta }_1, {\beta }_2\in K$ and $u_1, u_2\in {\oplus }_{k=2}^c{L}^{\left(k\right)}.$

Furthermore, φ is an automorphism if and only if the linear component matrix (the Jacobian matrix) modulo γ₂(L_c,2) is invertible:

$$M_{\phi }=\left(\begin{array}{cc}{\alpha }_1& {\beta }_1\\ {\alpha }_2& {\beta }_2\end{array}\right)\in {\mathit{GL}}_2\left(K\right)$$

Definition 1.

An automorphism φ ∈ Aut(L_c,2) is called a central automorphism if it acts as the identity mapping modulo the center of the algebra. That is,

φ(v) ≡ v(modZ(L_2,c))

for all v ∈ L_2,c. Consequently, φ is central if and only if the shift elements u₁ and u₂ satisfy u₁, u₂ ∈ Z(L_2,c) = L^(c) and the linear part matrix M_φ is the identity matrix I₂.

Definition 2.

A subset $𝒯$ = {𝓉₁, 𝓉₂,…,𝓉_m} is called a central test set if for any endomorphism φ of L_2,c, the condition φ(𝓉_j) = 𝓉_j for all 1 ≤ j ≤ m implies that φ is a central automorphism. The central test rank of L_2,c is the minimal cardinality among all central test sets of L_2,c.

3. Main Results and Theorems

In this section, we establish the main results of the paper. We begin by proving that any candidate central test set of cardinality m < 3 is insufficient, thereby establishing the lower bound.

Theorem 1

(Lower Bound on Central Test Cardinality). Let L_3,2 be a free nilpotent Lie algebra of rank 2 and nilpotency class c = 3. The central test rank of L_3,2 required to uniquely characterize central automorphisms is at least 3.

Proof of Theorem 1.

Let T = {u₁ … u_m} be a subset of L_3,2 with m < 3. Any endomorphism φ is defined by its action on the generators x,y:

φ(x) = α₁x + β₁y + w₁, φ(y) = α₂x + β₂y + w₂

where ${\alpha }_1, {{\alpha }_2,\beta }_1, {\beta }_2\in K$ and w₁, w₂ ∈ γ₂(L_c,2). The linear part of φ is represented by the matrix $M_{\phi }=\left(\begin{array}{cc}{\alpha }_1& {\beta }_1\\ {\alpha }_2& {\beta }_2\end{array}\right)$.

For φ to fix an element u ∈ L^(k) (where k ≥ 2), φ(u) − u must vanish. The action φ on L⁽²⁾ = span_K{[x,y]} yields:

$$\phi \left(\left[x,y\right]\right)=\left[\phi \left(x\right),\phi \left(y\right)\right]\equiv \left({\alpha }_1{\beta }_2-{{\alpha }_2\beta }_1\right)\left[x,y\right] \left(\mathit{mod}{\gamma }_3\left(L_{c,2}\right)\right)$$

Thus, fixing [x,y] imposes the constraint $\mathit{det}\left(M_{\phi }\right)={\alpha }_1{\beta }_2-{{\alpha }_2\beta }_1=1$.

Now consider the homogeneous components L⁽³⁾ = span_K{[[x,y],x],[[x,y],y]}, which has dimension d₃ = 2. The action of φ on L⁽³⁾ modulo modγ₄(L_c,2) is governed by the matrix product of det(M_φ) and M_φ:

φ([[x,y],x]) ≡ α₁[[x,y],x] + β₁[[x,y],y] (modγ₄(L_c,2))

φ([[x,y],y]) ≡ α₂[[x,y],x] + β₂[[x,y],y] (modγ₄(L_c,2))

If m = 2, say T = {u₁,u₂} the requirement φ(u_i) = u_i (i = 1,2) induces a linear system on the parameters $\left({\alpha }_1,{{\alpha }_2,{\beta }_1,\beta }_2\right)$. Since M_φ ∈ GL₂(K) has four entries subject to det(M_φ) = 1 (leaving three degrees of freedom), two test conditions impose at most two independent constraints. Consequently, the linear system for the non-central components has a non-trivial solution space containing matrices M_φ ≠ I₂.

Explicitly, setting T^′ = {[x,y],[[x,y],x]}, we can define an endomorphism φ by φ(x) = x and φ(y) = y + [x,y]. Then φ([x,y]) = [φ(x),φ(y)] = [x,y + [x,y]] = [x,y] + +[x,[x,y]] ≠ [x,y]. Alternatively, choosing linear combinations shows that m = 2 elements leave an undetermined component in L⁽¹⁾. Thus, m ≥ 3 is necessary. □

Theorem 2

(Explicit Minimal Central Test Set). The lower bound established in Theorem 1 is sharp. For the free nilpotent Lie algebra L_3,2 of rank 2 and nilpotency class c = 3, the set:

$𝒯$ = {[x,y], [[x,y],x], [x,y],y}

is a minimal central test set of cardinality exactly 3 that uniquely characterizes central automorphisms in L_3,2. Thus, the central test rank of L_3,2 is exactly 3.

Proof of Theorem 2.

Let φ ∈ End(L_c,2) such that:

• φ([x,y]) = [x,y]
• φ([[x,y],x]) = [[x,y],x]
• φ([[x,y],y]) = [[x,y],y]

The action of the endomorphism φ on the generators x and y can be expressed in general form as:

φ(x) = α₁x + β₁y + u₁, φ(y) = α₂x + β₂y + u₂

where ${\alpha }_1, {{\alpha }_2,\beta }_1, {\beta }_2\in K$ and $u_1, u_2\in {\oplus }_{k=2}^3{L}^{\left(k\right)}.$ The goal is to establish that α₁ = 1, α₂ = 0, β₁ = 0, β₂ = 1 and u₁, u₂ ∈ Z(L_3,2).

• Step 1: Determination of the linear part

Applying the endomorphism φ to [x,y] yields:

φ([x,y]) = [φ(x),φ(y)] = [α₁x + β₁y + u₁,α₂x + β₂y + u₂]

By projecting this equality onto the homogeneous component of degree 2, we obtain

$$\left[{\alpha }_{1}x+{\beta }_{1}y,{\alpha }_{2}x+{\beta }_{2}y\right]=\left({\alpha }_1{\beta }_2-{{\alpha }_2\beta }_1\right)\left[x,y\right]$$

Since φ([x,y]) = [x,y], comparing degree 2 terms implies that the determinant of the linear transformation matrix is 1:

$$\det \left(A\right)={\alpha }_1{\beta }_2-{{\alpha }_2\beta }_1=1$$

Next, we apply φ to the degree 3 elements in $𝒯$:

φ([[x,y],x]) = [φ([x,y]),φ(x)] = [[x,y],α₁x + β₁y + u₁]

Comparing the degree 3 components on both sides gives:

α₁[[x,y],x] + β₁[[x,y],y] = 1∙[[x,y],x] + 0∙[[x,y],y]

By the linear independence of the basic commutators of degree 3, we deduce α₁ = 1, and β₁ = 0. Analogously, evaluating φ([[x,y],y]) yields α₂ = 0 and β₂ = 1.

Hence, the linear part of φ is the identity matrix (which automatically implies φ is surjective and thus an automorphism), reducing the action to:

φ(x) = x + u₁, φ(y) = y + u₂

• Step 2: Elimination of Non-central Degree 2 Shifts

Decompose u₁ and u₂ into their homogeneous degree components:

$$u_1={\displaystyle {\sum }_{k=2}^3}{u}_1^{\left(k\right)},{ u}_2={\displaystyle {\sum }_{k=2}^3}{u}_2^{\left(k\right)} \mathrm{where} u_1^{\left(k\right)},u_2^{\left(k\right)}\in L^{\left(k\right)}$$

Since L⁽²⁾ is one-dimensional and spanned by [x,y], let $u_1^{\left(2\right)}=k_1\left[x,y\right]$ and $u_2^{\left(2\right)}=k_2\left[x,y\right]$ for k₁, k₂ ∈ K. The degree 3 projection of φ([x,y]) gives:

$$\phi {\left(\left[x,y\right]\right)}^{\left(3\right)}=\left[x,u_2^{\left(2\right)} \right]+\left[u_1^{\left(2\right)},y\right]=-k_2\left[\left[x,y\right],x\right]+k_1\left[\left[x,y\right],y\right]$$

Since φ([x,y]) has no component of degree 3, we must have. −k₂[[x,y],x] + k₁[[x,y],y] = 0, which forces k₁ = 0 and k₂ = 0. Thus $u_1^{\left(2\right)}=0$ and $u_2^{\left(2\right)}=0$.

Consequently, $u_1=u_1^{\left(3\right)}\in L_3=Z\left(L_{3,2}\right)$ and u₂ = $u_2^{\left(3\right)}\in L_3=Z\left(L_{3,2}\right)$. Therefore, φ(x) ≅ x(modZ(L_3,2)) and φ(y) ≅ y(modZ(L_3,2)) proving that any endomorphism fixing $𝒯$ pointwise is necessarily an automorphism and specifically a central automorphism. □

Remark 1

(The Degenerate Case c = 2). For a free nilpotent Lie algebra L_c,2 of rank 2, the case c = 2 is degenerate as the notion of a central test set fails to exist. When c = 2, the center coincides entirely with the derived subalgebra, i.e.,

Z(L_2,2) = L⁽²⁾ = span_K{[x,y]}

Consequently, higher-degree bracket structures such as [[x,y],x] and [[x,y],y] vanish identically to zero. Furthermore, any central shift u₁, u₂ ∈ Z(L_2,2) trivially commutes with all elements. This implies that testing elements within L⁽²⁾ fails to isolate central automorphisms from general non-central endomorphisms.

To illustrate this breakdown, consider the following concrete counterexample in L_2,2:

Example 1

(Counterexample for c = 2). Let L_2,2 = span_K{x, y, [x,y]} be the. three-dimensional Heisenberg Lie algebra. Consider the non-central endomorphism φ defined on the generators by:

$$\phi \left(x\right)=2x,\hspace{1em}\phi \left(y\right)={\displaystyle \frac{1}{2}}y+3\left[x,y\right]$$

Notice that φ is clearly not a central automorphism, as φ(x) = 2x ≇ x(modZ(L_2,2). However, evaluating φ on the candidate test element 𝓉 = [x,y] ∈ L⁽²⁾ yields:

$$\phi \left(\left[x,y\right]\right)=\left[\phi \left(x\right),\phi \left(y\right)\right]=\left[2x,{\displaystyle \frac{1}{2}}y+3\left[x,y\right]\right]=2·{\displaystyle \frac{1}{2}}\left[x,y\right]+6\left[x,\left[x,y\right]\right]=\left[x,y\right]$$

since φ([x,y]) = [x,y], the non-central endomorphism φ passes the invariance condition under [x,y]. Because no higher-degree non-zero elements exist in L_2,2 to impose additional constraints, no subset of L_2,2 can serve as a central test set.

Remark 2

(Higher Nilpotency Classes c ≥ 4). For nilpotency classes c ≥ 4, test sets constructed exclusively from degree 2 and degree 3 elements (such as $𝒯$) are insufficient to guarantee central automorphism status. As demonstrated by the derivation D(a) = [[x,y],a], the mapping φ = exp(D) = Id + D defines an automorphism under which all elements of $𝒯$ remain invariant because D([x,y]) = 0 and higher bracket applications vanish modulo γ_c+1(L_c,2). However, φ(x) − x = [[x,y],x] has degree 3, whereas the center Z(L_c,2) = L^(c) resides at degree c ≥ 4. Thus φ is non-central. To form a complete central test set for general c ≥ 4 higher-degree bracket structures, up to L_c−1 must be included, resulting in central test ranks strictly exceeding 3.

4. Classification and Interdisciplinary Discussion

4.1. Comparison Across Lie Algebra Classes

To contextualize our main result, Table 1 summarizes the test ranks and central test rank across fundamental classes of free and nilpotent Lie algebras.

Table 1. Test ranks and central test ranks of free nilpotent Lie algebras of rank 2.

Lie Algebra Type	Nilpotency Class	Central Test Rank	Test Rank
Free nilpotent of rank 2	2	Not exist	1
Free nilpotent of rank 2	3	3	1
Free nilpotent of rank 2	c ≥ 3	≥4	1

4.2. Interdisciplinary Applications

While our results are combinatorial and algebraic, structural invariants of Lie algebras provide algebraic foundations for operator theory. In quantum field theory and Fock space calculus, creation and annihilation operators satisfy canonical commutation relations that form nilpotent Lie algebraic structures. Minimal central test sets guarantee that field transformations preserve canonical bracket structures and energy state symmetries without collapsing into deeper central ideals.

5. Conclusions and Future Outlook

In this paper, we introduced and established the concept of a central test set for the free nilpotent Lie algebra L_3,2. We proved that the central test rank of L_3,2 is strictly equal to 3 and explicitly identified the minimal central test set

$𝒯$ = {[x,y], [[x,y],x], [x,y],y}

Future research directions include:

Extending the determination of central test ranks to free nilpotent Lie algebras of rank n > 2.

Analyzing the algorithmic implementations and matrix representations of automorphism group actions on second cohomology Grassmannians and invariant spaces.

Investigating analogous notions in Lie superalgebras and quantum group structures.