--- title: Three-Dimensional Leibniz Algebras url: https://www.emergentmind.com/topics/three-dimensional-leibniz-algebras type: topic --- # Three-Dimensional Leibniz Algebras Searching arXiv for recent and foundational papers on three-dimensional Leibniz algebras. Search query: all:("three-dimensional Leibniz algebras" OR "3-dimensional Leibniz algebras" classification automorphism degeneration) A three-dimensional Leibniz algebra is a three-dimensional algebra \(L\) with a bilinear bracket \([\,,\,]\) satisfying the left Leibniz identity \[ [[x,y],z]=[x,[y,z]]-[y,[x,z]], \] without the antisymmetry required for Lie algebras. In dimension three, the subject is unusually explicit: classification by isomorphism can be carried out over several base fields, the proper non-Lie and Lie cases can be separated by the Leibniz kernel \(\Leib(L)\), and structural features such as nilpotency, solvability, automorphism groups, degenerations, polynomial invariants, and Nijenhuis operators admit concrete descriptions. Over an arbitrary field, there are four Lie families and forty proper Leibniz families; over \(\mathbb{C}\), the geometric classification of the affine variety of three-dimensional Leibniz structures yields four Lie algebras and nine non-Lie Leibniz algebras; over \(\mathbb{R}\), one classification used in the study of Nijenhuis operators consists of thirteen pairwise non-isomorphic types [2211.01074], [1808.00907], [2507.12692]. ## 1. Definition and basic structural dichotomies A left Leibniz algebra is an algebra \(L\) with bracket \([\,,\,]\) satisfying \[ [[a,b],c]=[a,[b,c]]-[b,[a,c]] \] for all \(a,b,c\in L\) [2211.01074]. A Leibniz algebra is Lie exactly when \(\Leib(L)=0\), where \(\Leib(L)\) denotes the Leibniz kernel; in the three-dimensional classification over an arbitrary field, this yields the primary split into Lie algebras and proper (non-Lie) Leibniz algebras [2211.01074]. For three-dimensional proper Leibniz algebras over an arbitrary field \(F\), one has \[ \dim_F \Leib(L)\in\{1,2\}, \] and the classification organizes accordingly [2211.01074]. The same source records several general constraints. Nilpotent algebras occur only in the block \(\dim_F\Leib(L)=1\) with \(\Leib(L)\subseteq Z(L)\) and \(L/\Leib(L)\) abelian, where there are eight \(2\)-step nilpotent algebras, and in the block \(\dim_F\Leib(L)=2\), where there is one \(3\)-step nilpotent algebra. All non-nilpotent Leibniz algebras are solvable with derived length \(2\), and the left center \[ Z^L(L)=\{x\mid [x,L]=0\} \] always contains \(\Leib(L)\) [2211.01074]. Over an algebraically closed field of characteristic zero, one particularly tractable subproblem is the classification of non-Lie solvable left Leibniz algebras of dimension three. In the formulation revisited in the solvable classification paper, every such algebra satisfies \[ \dim L'=2,\qquad \dim L^{(2)}=0,\qquad L^2=L'\neq 0,\qquad L^3=L^2\neq 0, \] hence is solvable of derived length \(2\) but not nilpotent [1501.00890]. This sharp distinction between nilpotent and non-nilpotent behavior is a recurrent organizing principle throughout the dimension-three theory. ## 2. Classification over arbitrary fields The classification over an arbitrary field \(F\) consists of four Lie families and forty proper Leibniz families, exhausting all three-dimensional left Leibniz algebras up to isomorphism [2211.01074]. The Lie part contains the abelian algebra, the Heisenberg algebra, a one-parameter solvable non-nilpotent family \(L^{(A3)}(a)\) with \(a\neq 0,1\), and split \(\mathfrak{sl}_2(F)\) when \(\mathrm{char}(F)\neq 2,3\) [2211.01074]. For proper Leibniz algebras, the classification splits by \(\dim_F\Leib(L)\). When \(\dim_F\Leib(L)=1\), one writes \(\Leib(L)=F e_3\). The next dichotomies are whether \(\Leib(L)\subseteq Z(L)\), and whether \(L/\Leib(L)\) is abelian. In the central-quotient-abelian case there are eight nilpotent extensions of the cyclic \(2\)-algebra, including \(L_3\), \(L_4\), \(L_5\), and \(L_6(a)\), as well as six further types subject to irreducibility conditions on a quadratic polynomial [2211.01074]. In the central-quotient-nonabelian case there are six non-nilpotent extensions of the two-dimensional non-abelian Lie algebra, including \(L_{11}\) and the one-parameter family \(L_{12}(a)\), plus four further mixed cyclic-ideal sums with a quadratic condition [2211.01074]. If \(\Leib(L)\not\subseteq Z(L)\), there are six non-nilpotent types with \(L/\Leib(L)\) abelian and thirteen non-nilpotent mixed cyclic sums with \(L/\Leib(L)\) non-abelian, appearing as \(L_{24},\dots,L_{36}\) with explicit linear-quadratic relations among structure constants and field constraints [2211.01074]. When \(\dim_F\Leib(L)=2\), one writes \(\Leib(L)=F e_2\oplus F e_3\), so \(L/\Leib(L)\) is one-dimensional and the multiplication is determined by the linear map \[ \ad_{e_1}:\Leib(L)\to \Leib(L),\qquad x\mapsto [e_1,x]. \] Up to basis choice in \(\Leib(L)\) and normalization of \(e_1\), there are exactly six isomorphism types [2211.01074]. Among these are the nilpotent cyclic algebra \(L_{37}\) with \[ [e_1,e_1]=e_2,\qquad [e_1,e_2]=e_3, \] the non-nilpotent cyclic algebra \(L_{38}\) with \[ [e_1,e_1]=e_2,\qquad [e_1,e_2]=e_2+e_3, \] and the one-parameter-type families \(L_{39}\) and \(L_{40}\), where \(\ad_{e_1}\) on \(\Leib(L)\) has either a nonzero eigenvalue or a single Jordan block, together with two further cyclic Jordan-block types [2211.01074]. This classification also identifies the field-dependent phenomena. They occur precisely in families involving \(\mathrm{char}(F)\neq 2\), irreducibility conditions for a polynomial \(X^2+pX+q\), or exclusion of values such as \(a=0,-1,-2\) in certain one-parameter families [2211.01074]. A plausible implication is that dimension three is already large enough for arithmetic properties of the field to influence isomorphism classes in an essential way. ## 3. Solvable non-Lie algebras with two-dimensional derived algebra Over an algebraically closed field of characteristic zero, every three-dimensional non-Lie solvable left Leibniz algebra is, up to isomorphism, one of six types: four discrete algebras and two one-parameter families [1501.00890]. In all cases, \[ L'=[L,L]=\mathrm{span}\{e_2,e_3\},\qquad \dim L'=2,\qquad L^{(2)}=0, \] and \([L,L]\nsubseteq Z(L)\), so the algebra is solvable but not nilpotent [1501.00890]. The six types are summarized below. | Type | Nonzero products | Remarks | |---|---|---| | I | \([e_1,e_1]=e_3,\ [e_1,e_3]=e_2,\ [e_1,e_2]=e_2\) | discrete | | II\((a)\) | \([e_1,e_2]=e_2,\ [e_1,e_3]=a e_3\) | \(a\in K^\times,\ a\sim a^{-1}\) | | III | \([e_1,e_3]=e_2,\ [e_1,e_2]=e_3+e_2\) | discrete | | IV | \([e_1,e_1]=e_3,\ [e_1,e_2]=e_2,\ [e_2,e_1]=-e_2\) | quotient by \(\mathrm{span}\{e_3\}\) is non-abelian Lie | | V\((a)\) | \([e_1,e_2]=e_2,\ [e_2,e_1]=-e_2,\ [e_1,e_3]=a e_3\) | \(a\in K^\times,\ a\sim a^{-1}\) | | VI | \([e_1,e_1]=e_3,\ [e_1,e_2]=e_2,\ [e_2,e_1]=-e_2,\ [e_1,e_3]=2e_3,\ [e_2,e_2]=e_3\) | discrete | Type II\((a)\) is described as semidirect type, with left multiplication \(L_{e_1}\) diagonalizable with eigenvalues \(1,a\); Type III is explicitly noted as not of semidirect diagonal form; Type IV is “Lie-like” in the \(2\)-plane and has quotient \(L/\mathrm{span}\{e_3\}\) equal to the two-dimensional non-abelian Lie algebra; Type V\((a)\) combines the patterns of II and IV; Type VI has a two-step flag with successive one-dimensional brackets [1501.00890]. The only parameters occur in Types II and V, with the identification \(a\mapsto a^{-1}\) and no further identifications [1501.00890]. The source states that one may take representatives of \(K^\times/\{\pm 1\}\) if desired [1501.00890]. This suggests that, in this subclassification, the moduli are extremely small: apart from inversion symmetry, the parameter is rigidly retained. A separate 2024 study isolates two non-nilpotent families over an arbitrary field with \(\dim [L,L]=2\) and a specified “one-generator” subalgebra pattern [2407.14692]. The first is \[ [e_1,e_1]=e_3,\qquad [e_1,e_2]=e_2,\qquad [e_1,e_3]=e_3, \] and the second is the one-parameter family \[ [e_1,e_1]=e_3,\qquad [e_1,e_2]=e_2+X e_3,\qquad [e_1,e_3]=e_3,\qquad X\neq 0. \] The paper states that, under its hypotheses, these are exactly the two non-nilpotent types, and that over an algebraically closed field one may rescale \(e_3\) to make \(X=1\) [2407.14692]. This does not purport to replace the broader classification of all three-dimensional non-nilpotent Leibniz algebras, but rather identifies two distinguished families within a more restricted construction. ## 4. Nilpotent algebras and automorphism groups A 2023 study of automorphism groups considers three nilpotent, three-dimensional non-Lie Leibniz algebras that it describes as exhausting the non-Lie nilpotent types in dimension \(3\) [2305.00592]. In the basis \(\{e_1,e_2,e_3\}\), these are: \[ L_1:\ [e_1,e_1]=e_2,\ [e_1,e_2]=e_3, \] \[ L_2:\ [e_1,e_1]=e_3, \] \[ L_3:\ [e_1,e_1]=e_3,\ [e_1,e_2]=e_3. \] Their nilpotency data are recorded there as class \(3\) for \(L_1\), and class \(2\) for \(L_2\) and \(L_3\), with \(\dim Z(L_2)=2\) and \(\dim Z(L_3)=1\) [2305.00592]. The automorphism groups admit explicit upper-triangular matrix realizations. For \(L_1\), \[ \Aut(L_1)\cong \left\{ \begin{pmatrix} a&b&c\\ 0&a^2&2ab\\ 0&0&a^3 \end{pmatrix} \right\} \simeq F^2\rtimes F^\times, \] with \(a\in F^\times\), \(b,c\in F\) [2305.00592]. For \(L_2\), \[ \Aut(L_2)\cong \left\{ \begin{pmatrix} a&b&c\\ 0&q&r\\ 0&0&a^2 \end{pmatrix} \right\} \simeq F^3\rtimes(F^\times\times F^\times), \] with \(a,q\in F^\times\) and \(b,c,r\in F\) [2305.00592]. For \(L_3\), the automorphism group is \[ \Aut(L_3)\cong \left\{ \begin{pmatrix} \sigma&t-1&\alpha_3\\ 0&t&\beta_3\\ 0&0&\sigma t \end{pmatrix} \right\}\subset \GL_3(F), \qquad \sigma,t\in F^\times, \] and it decomposes as \[ \Aut(L_3)\cong S_3\rtimes D_3,\qquad S_3=T_3\rtimes J_3, \] hence \[ \Aut(L_3)=(T_3\rtimes J_3)\rtimes D_3, \] with \(T_3\simeq F^2\), \(J_3\simeq F^\times\), \(D_3\simeq F^\times\) [2305.00592]. The same paper records the dimensions \[ \dim\Aut(L_1)=3,\qquad \dim\Aut(L_2)=5,\qquad \dim\Aut(L_3)=4 \] and emphasizes that the size and decomposition of \(\Aut(L)\) reflect the central and derived series of \(L\) [2305.00592]. More specifically, invariance of \([L,L]\) under automorphisms and the induced scalings on successive quotients account for the toral factors in the group structure [2305.00592]. For non-nilpotent examples, the 2024 automorphism study computes two explicit families. For \[ [e_1,e_1]=e_3,\qquad [e_1,e_2]=e_2,\qquad [e_1,e_3]=e_3, \] one obtains \(\Aut(L_1)\cong \GL_2(F)\) [2407.14692]. For \[ [e_1,e_1]=e_3,\qquad [e_1,e_2]=e_2+X e_3,\qquad [e_1,e_3]=e_3,\qquad X\neq 0, \] one obtains an abelian group \[ \Aut(L_2)\cong (F,+)\times(F^\times,\cdot) \] with representatives \[ f_{\lambda,s}:\ \begin{cases} e_1\mapsto e_1+(\lambda-1)e_3,\\ e_2\mapsto \lambda e_2+s e_3,\\ e_3\mapsto \lambda e_3, \end{cases} \qquad \lambda\in F^\times,\ s\in F \] [2407.14692]. These formulas provide concrete instances of how small changes in the bracket can alter the automorphism group from a non-abelian linear group to a direct product. ## 5. Degenerations, irreducible components, and rigidity over \(\mathbb{C}\) The geometric study of the affine variety \(\mathrm{Leib}_3\) of all complex three-dimensional Leibniz algebras identifies four Lie algebras and nine non-Lie Leibniz algebras up to isomorphism [1808.00907]. The non-Lie algebras are denoted \(L_1(\beta),L_2,L_3,L_4(\alpha),L_5(\gamma),L_6,L_7,L_8,L_9\), with explicit multiplication tables [1808.00907]. In each of the one-parameter families, distinct parameters outside finitely many resonant values give non-isomorphic algebras; among the Lie families, the only further identifications are \(g_2(\alpha)\cong g_2(1/\alpha)\) [1808.00907]. Primary degenerations are given by one-parameter changes of basis \(E_i(t)\) with polynomial entries in \(t\), such that the structure constants converge to those of the target algebra as \(t\to 0\) [1808.00907]. The list of primary degenerations includes, among others, \[ g_2(\alpha)\to g_3,\qquad g_3\to g_1,\qquad g_4\to g_3, \] \[ L_1(\beta)\to g_3,\qquad L_1(\beta)\to L_2,\qquad L_2\to L_3,\qquad L_3\to L_1(0), \] \[ L_4(\alpha)\to L_2,\qquad L_4(\alpha)\to L_3,\qquad L_5(\gamma)\to L_3, \] \[ L_6\to L_2,\qquad L_7\to L_6,\qquad L_8\to L_7,\qquad L_9\to L_7, \] and every algebra degenerates to the zero algebra [1808.00907]. The variety \(\mathrm{Leib}_3\) has exactly five irreducible components [1808.00907]. They are the closures of the families or rigid points: \[ \mathcal C_1=\overline{\{g_2(\alpha)\}},\qquad \mathcal C_2=\overline{\{g_4\}},\qquad \mathcal C_3=\overline{\{L_1(\beta)\}}, \] \[ \mathcal C_4=\overline{\{L_5(\gamma)\}},\qquad \mathcal C_5=\overline{\{L_6\}}, \] with respective dimensions \(8,6,7,8,7\) [1808.00907]. The rigid algebras are exactly \(g_4\) and \(L_5(\gamma)\) [1808.00907]. Since a rigid algebra has a Zariski-open \(\GL(V)\)-orbit, this gives an explicit geometric distinction between structurally stable and non-stable multiplication laws in dimension three. This degeneration picture complements the isomorphism classification. The classification enumerates orbit representatives; the degeneration poset describes orbit closures. A plausible implication is that, in low dimension, deformation-theoretic and geometric invariant-theoretic viewpoints are unusually close to explicit computation. ## 6. Polynomial invariants and Nijenhuis operators Over \(\mathbb{C}\), the invariant-theoretic study of the nine non-Lie isomorphism classes gives explicit automorphism groups, rings of polynomial invariants, and low-degree trace invariants [2505.07125]. The classification used there consists of the algebras \(L_1,L_2,L_3,L_4^\alpha,L_5,L_6,L_7^\beta,L_8,L_9\), with exactly three nilpotent algebras: \[ (L_4^\alpha)=3,\qquad (L_5)=3,\qquad (L_8)=4, \] while all others have infinite nilpotency class [2505.07125]. The polynomial invariant rings \(I_m(L)\) are computed explicitly. Representative cases include \[ I_m(L_1)=\mathbb{C}[z_1,\dots,z_m],\qquad I_m(L_2)=\mathbb{C}[z_1,\dots,z_m], \] \[ I_m(L_3)= \mathbb{C}\bigl[z_r,\ x_r z_s-z_r x_s\mid 1\le r