Locally finite admissible simple Lie algebras

TL;DR

This paper introduces admissible Lie algebras, a new class of locally finite simple Lie algebras, and demonstrates their structural properties including the existence of maximal toral subalgebras and irreducible root systems.

Contribution

The paper defines admissible Lie algebras and establishes their key structural features, expanding the understanding of locally finite simple Lie algebra classifications.

Findings

Existence of nonzero maximal toral subalgebras in admissible Lie algebras

Root systems are irreducible and locally finite

Structural characterization of admissible Lie algebras

Abstract

We introduce a class of Lie algebras called admissible Lie algebras. We show that a locally finite admissible simple Lie algebra contains a nonzero maximal toral subalgebra and the corresponding root system is an irreducible locally finite root system.

Equations

$$(\ast )\mathcal{L}=(\mathcal{G}\otimes \mathcal{A})\oplus (\mathcal{S}\otimes \mathcal{B})\oplus (\mathcal{V}\otimes \mathcal{C})\oplus \mathcal{D}$$

$$\mathcal{M}={\oplus }_{\alpha \in {\mathcal{H}}^{\star }}{\mathcal{M}}_{\alpha }where{\mathcal{M}}_{\alpha }:=\{x\in \mathcal{M}\mid h\cdot x=\alpha (h)x;\forall h\in \mathcal{H}\};\alpha \in {\mathcal{H}}^{\star }.$$

$$\mathcal{H}\subseteq \sum _{\alpha \in R}[{\mathcal{L}}_{\alpha },{\mathcal{L}}_{-\alpha }]$$

$$\beta (k_{\beta })=2\; and{k}_{\beta }=[u,v]for\; someu\in {\mathcal{L}}_{\beta },v\in {\mathcal{L}}_{-\beta }.$$

$$\mathcal{T}={span}_{\mathbb{F}}\{k_{\beta }\mid \beta \in \Phi \}and{\mathcal{T}}^{\star }={span}_{\mathbb{F}}\{\beta \mid \beta \in \Phi \}.$$

$$\alpha ,{\beta }_1,\dots ,{\beta }_n\in \Phi$$

$$R_{{}_I}=-R_{{}_I}and{\mathcal{L}}_{\alpha }\subseteq I;\alpha \in R_{{}_I}.$$

$$\begin{array}[]{l}\;[\mathcal{L}_{\alpha},\mathcal{L}_{\beta}]\subseteq\mathcal{L}_{\alpha+\beta}\cap I\hbox{ if $\alpha\in R_{{}_{I}}$ or $\beta\in R_{{}_{I}}$}\\ \\ \;[\mathcal{L}_{\alpha+\beta},\mathcal{L}_{-\alpha}]\subseteq\mathcal{L}_{\beta}\cap I\hbox{ if $\alpha\in R_{{}_{I}}$ or $\alpha+\beta\in R_{{}_{I}}$}.\end{array}$$

$$[\mathcal{L}_{\alpha},\mathcal{L}_{\beta}]\subseteq\left\{\begin{array}[]{ll}I&\hbox{if $\alpha,\beta\in R_{{}_{I}},$}\\ J&\hbox{if $\alpha,\beta\in\Delta,$}\\ \{0\}&\hbox{if $\alpha\in R_{{}_{I}},\;\beta\in\Delta.$}\end{array}\right.$$

$$[\sum _{\alpha \in R_{{}_I}}[{\mathcal{L}}_{\alpha },{\mathcal{L}}_{-\alpha }]+\sum _{\alpha \in R_{{}_I}}{\mathcal{L}}_{\alpha },\sum _{\beta \in \Delta }[{\mathcal{L}}_{\beta },{\mathcal{L}}_{-\beta }]]=\{0\}.$$

$$[\mathcal{L},J]=[\sum _{\alpha \in R}{\mathcal{L}}_{\alpha }+\sum _{\alpha \in R}[{\mathcal{L}}_{\alpha },{\mathcal{L}}_{-\alpha }],J]$$

$$\mathcal{L}=I+J.$$

$$[x,\sum _{\alpha \in R_{{}_I}}{\mathcal{L}}_{\alpha }\oplus \sum _{\alpha \in R_{{}_I}}[{\mathcal{L}}_{\alpha },{\mathcal{L}}_{-\alpha }]]=\{0\}.$$

$$[x,{\mathcal{L}}_{\alpha }]=\{0\};\alpha \in \Delta .$$

$$\begin{array}[]{c}\mathcal{L}_{cc}=(\mathcal{L}_{cc})_{0}\oplus\sum_{\alpha\in R}(\mathcal{L}_{cc})_{\alpha}\hbox{ with }\vspace{3mm}\\ (\mathcal{L}_{cc})_{0}=\mathcal{L}_{0,0}/Z(\mathcal{L}_{c}),(\mathcal{L}_{cc})_{\alpha}=\mathcal{L}_{\alpha},$ $\alpha\in R.\end{array}$$

$$({\mathcal{L}}_{{}_{\Delta }}{)}_{\pi (\alpha )}=\{x\in {\mathcal{L}}_{{}_{\Delta }}\mid [h,x]=\alpha (h)x;\forall h\in {\mathcal{H}}_{{}_{\Delta }}\}=\underset{\{\beta \in \Delta \cup \{0\},\pi (\beta )=\pi (\alpha )\}}{⨁}({\mathcal{L}}_{{}_{\Delta }}{)}_{\beta }.$$

$$[{\mathcal{L}}^i,{\mathcal{L}}^j]\subseteq {\delta }_{i,j}{\mathcal{L}}^i.$$

$$\begin{array}[]{c}[\pi_{i}(h),x]=[t_{i},x]=[\sum_{j\in I}t_{j},x]=[h,x]=\alpha(h)x.\end{array}$$

$$h,h^{\prime }\in \mathcal{H}$$

$$f_{t,\alpha }:=\alpha (h)$$

$${\alpha }^i(h+t)=\alpha (h)+f_{t,\alpha };h\in \mathcal{H},t\in T_i^{\prime }.$$

$$\beta (k_{\beta })=2\; and{k}_{\beta }=[u,v]for\; someu\in {\mathcal{L}}_{\beta }^{\prime },v\in {\mathcal{L}}_{-\beta }^{\prime }.$$

$$h=[x,y]=[x,\sum _{\gamma \in A_{\alpha }}{y}_{-\gamma }]=\sum _{\gamma \in A_{\alpha }}[x,y_{-\gamma }]$$

$$k_{\beta }\in \mathcal{H};for\beta \in \Phi \; with\beta {\mid }_{{}_{\mathcal{H}}}\ne 0.$$

$$\alpha (h_{\alpha })=2and[x_{\alpha },x_{-\alpha }]=h_{\alpha }for\; some{x}_{\pm \alpha }\in {\mathcal{L}}_{\pm \alpha }.$$

$$\alpha \in {\Delta }_{{}_M}$$

$$\gamma_{i}:=\left\{\begin{array}[]{ll}\alpha_{i},&\hbox{if }1\leq i\leq n\\ \beta_{i-n},&\hbox{if }n+1\leq i\leq n+m,\\ \end{array}\right.$$

$$\sum _{i=1}^n{\check{\alpha }}_j({\alpha }_i)x_i=k_j;1\le j\le n$$

$$\sum _{i=1}^n{r}_i{\check{\alpha }}_j({\alpha }_i)=\sum _{i=1}^n{r}_i{\alpha }_i(h_{{}_{{\alpha }_j}})=\eta (h_{{}_{{\alpha }_j}}).$$

$$\sum _{i=1}^n{\check{\alpha }}_j({\alpha }_i)x_i=\eta (h_{{}_{{\alpha }_j}});1\le j\le n.$$

Full text

2010 Mathematics Subject Classification(s):17B05, 17B20, 17B60, 17B70, 17B67 *

0. introduction

In 1976, G.B. Seligman [Se] showed that finite dimensional simple Lie algebras containing a nonzero maximal toral subalgebra have a decomposition as

[TABLE]

in which ${\mathcal{G}}$ is a finite dimensional split simple Lie algebra and ${\mathcal{V}},$ $\mathcal{S}$ are specific irreducible ${\mathcal{G}}$-modules. The Lie algebra structure on $\mathcal{L}$ induces an algebra structure on the vector space $\mathfrak{b}:=\mathcal{A}\oplus\mathcal{B}\oplus{\mathcal{C}}$ and $\mathcal{D}$ is a Lie subalgebra of $\mathcal{L}$ isomorphic to a specific subalgebra of inner derivations of $\mathfrak{b}.$ Following G.B. Seligman, S. Berman and R. Moody [BM] introduced the notion of a Lie algebra graded by an irreducible reduced finite root system $R$ which is a Lie algebra containing a finite dimensional split simple Lie subalgebra ${\mathcal{G}}$ of type $R$ with a Cartan subalgebra ${\mathcal{H}}$ such that $\mathcal{L}$ has a weight space decomposition with respect to ${\mathcal{H}}$ and as an algebra, $\mathcal{L}$ is generated by weight spaces corresponding to ”nonzero” weights. Next Allison, Benkart and Gao generalized [BM]’s definition to non reduced case by letting ${\mathcal{G}}$ be a finite dimensional split simple Lie algebra of type $B,C$ or $D$ if $R$ is of type $BC.$ In [BM], [BZ] and [ABG], the authors stated recognition theorems for root graded Lie algebras to classify such Lie algebras up to centrally isogeny, indeed any root graded Lie algebra $\mathcal{L}$ has a decomposition as $(*)$ with a prescribed algebra structure on the so called coordinate algebra $\mathfrak{b}:=\mathcal{A}\oplus\mathcal{B}\oplus{\mathcal{C}}.$ In 1999, N. Stumme studied locally finite Lie algebras containing a splitting Cartan subalgebra and called such algebras, locally finite split Lie algebras. She showed that, a reduced locally finite root system, the root system of a locally finite split semisimple Lie algebra, is the direct union of finite root subsystems of semisimple types. She also proved that a locally finite split simple Lie algebra is the direct union of finite dimensional simple subalgebras. Locally finite root systems appear in the theory of Kac-Moody Lie algebras as well, more precisely, countable irreducible reduced locally finite root systems are the root systems of infinite rank affine Lie algebras [K, §7.11]. In 2003, O. Loos and E. Neher [LN] introduced locally finite root systems axiomatically and gave a complete description of these root systems. Next Neher [N] generalized the definition of a root graded Lie algebra to Lie algebras graded by a locally finite root system. In Neher’s sense, if $R$ is a locally finite root system, $S$ is a subsystem of $R$ and $\Lambda$ is an abelian group, an $(R,S,\Lambda)$-graded Lie algebra is a compatible $\hbox{span}_{\mathbb{Z}}(R)$-graded and $\Lambda$-graded Lie algebra whose support with respect to $\hbox{span}_{\mathbb{Z}}(R)$-grading is contained in $R,$ for every $0\neq\alpha\in S,$ the homogeneous space $\mathcal{L}_{\alpha}^{0}$ contains a so called invertible element and $\mathcal{L}_{0}=\sum_{0\neq\alpha\in R}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}].$ In this work, we first study the direct union $\cup_{n\in\mathbb{N}}{\mathcal{G}}_{n}$ of finite dimensional simple Lie algebras ${\mathcal{G}}_{n},$ $n\in\mathbb{N},$ containing a nonzero maximal toral subalgebra, these Lie algebras are $(R,R_{sdiv},0)$-graded Lie algebras (see Definition 1.1) in Neher’s sense for a locally finite root system $R.$ Here we study a class of Lie algebras, called admissible Lie algebras. An admissible Lie algebra $\mathcal{L}$ has a nonzero toral subalgebra ${\mathcal{H}}$ contained in the subalgebra of $\mathcal{L}$ generated by the weight spaces such that for any nonzero weight vector $x,$ there is a weight vector $y$ such that $[x,y]\in{\mathcal{H}}$ and $(x,[x,y],y)$ is an $\mathfrak{sl}_{2}$-triple. We show that if the Lie algebra $\mathcal{L}$ is locally finite simple Lie algebra whose weight spaces are finite dimensional, then ${\mathcal{H}}$ is a maximal toral subalgebra, the root system of $\mathcal{L}$ with respect to ${\mathcal{H}}$ is a locally finite root system and $\mathcal{L}$ is the direct union of finite dimensional simple subalgebras.

1. admissible Lie algebras

Throughout this work $\mathbb{N}$ denotes the set of all nonnegative integers and $\mathbb{F}$ is a field of characteristic zero. Unless otherwise mentioned, all vector spaces are considered over $\mathbb{F}.$ In the present paper, we denote the dual space of a vector space $V$ by $V^{\star}$ and by $GL(V),$ we mean the group of automorphisms of $V.$ For a matrix $A,$ $tr(a)$ denotes the trace of $A.$ Also for a Lie algebra $\mathcal{L},$ we mean by $Z(\mathcal{L}),$ the center of $\mathcal{L}$ and if $\mathcal{L}$ is finite dimensional, we denote the Killing form of $\mathcal{L}$ by $\kappa.$ We also make a convention that for elements $x_{1},\ldots,x_{m}$ of a Lie algebra, by an expression of the form $[x_{1},\ldots,x_{m}],$ we always mean $[x_{1},\ldots[x_{m-1},x_{m}]\ldots].$ to be zero.

Definition 1.1**.**

Let ${\mathcal{V}}$ be a nontrivial vector space and $R$ be a subset of ${\mathcal{V}},$ $R$ is said to be a locally finite root system in ${\mathcal{V}}$ if

(i) $0\not\in R,$ $R$ is locally finite and spans ${\mathcal{V}},$

(ii) for every $\alpha\in R,$ there exists $\check{\alpha}\in{\mathcal{V}}^{\star}$ such that $\check{\alpha}(\alpha)=2$ and $s_{\alpha}(\beta)\in R$ for $\alpha,\beta\in R$ where $s_{\alpha}:{\mathcal{V}}\longrightarrow{\mathcal{V}}$ maps $v\in{\mathcal{V}}$ to $v-\check{\alpha}(v)\alpha.$

(iii) $\check{\alpha}(\beta)\in{\mathbb{Z}},$ for $\alpha,\beta\in R.$

Set $R_{sdiv}:=R\setminus\{\alpha\in R\mid 2\alpha\in R\}$ and call it the semi-divisible subset of $R,$ the root system $R$ is called reduced if $R=R_{sdiv}.$

Suppose that $R$ is a locally finite root system. A nonempty subset $S$ of $R$ is said to be a subsystem of $R$ if $s_{\alpha}(\beta)\in S$ for $\alpha,\beta\in S.$ Following [LN, §2.6], we say two roots $\alpha,\beta$ are connected if there exist finitely many roots $\alpha_{1}=\alpha,\alpha_{2},\ldots,\alpha_{n}=\beta$ such that $\check{\alpha}_{i+1}(\alpha_{i})\neq 0,$ $1\leq i\leq n-1.$ Connectedness is an equivalence relation on $R.$ The root system $R$ is the disjoint union of its connected components. A nonempty subset $X$ of $R$ is called irreducible, if each two elements $x,y\in X$ are connected and it is called closed if for $x,y\in X$ with $x+y\in R,$ we would have $x+y\in X.$ It is easy to see that each connected component of the locally finite root system $R$ is a closed subsystem of $R.$ Also using [LN, Cor. 3.15], $R$ is a direct limit of its finite subsystems, and if $R$ is irreducible, it is a direct limit of its irreducible finite subsystems.

Definition 1.2**.**

Let ${\mathcal{H}}$ be a Lie algebra. We say an ${\mathcal{H}}-$module $\mathcal{M}$ has a weight space decomposition with respect to ${\mathcal{H}},$ if

[TABLE]

The set $R:=\{\alpha\in{\mathcal{H}}^{\star}\setminus\{0\}\mid\mathcal{M}_{\alpha}\neq\{0\}\}$ is called the set of weights of $\mathcal{M}$ with respect to ${\mathcal{H}}$ and $\mathcal{M}_{\alpha},$ $\alpha\in R,$ is called a weight space, also any element of $\mathcal{M}_{\alpha}$ is called a weight vector of weight $\alpha.$ If a Lie algebra $\mathcal{L}$ has a weight space decomposition with respect to a subalgebra of $\mathcal{L}$ via the adjoint representation, the set of weights is called the *root system * and weight spaces are called root spaces.

An easy verification proves the following lemma.

Lemma 1.3**.**

Let ${\mathcal{H}}$ be a Lie algebra and $\mathcal{M}$ be an ${\mathcal{H}}$-module admitting a weight space decomposition $\mathcal{M}=\oplus_{\alpha\in{\mathcal{H}}^{\star}}\mathcal{M}_{\alpha}$ with respect to ${\mathcal{H}}.$ Let $T$ be a subalgebra of ${\mathcal{H}}$ and take $\pi:{\mathcal{H}}^{\star}\longrightarrow T^{\star}$ to be defined by $\pi(\alpha)=\alpha\mid_{{}_{T}},$ the restriction of $\alpha$ to $T.$ For $\beta\in T^{\star},$ define $\mathcal{M}^{\prime}_{\beta}:=\{v\in\mathcal{M}\mid t\cdot v=\beta(t)v;\;\forall t\in T\},$ then $\mathcal{M}=\oplus_{\beta\in T^{\star}}\mathcal{M}^{\prime}_{\beta}$ and for $\beta\in T^{\star},$ $\mathcal{M}^{\prime}_{\beta}=\displaystyle{\bigoplus_{\{\alpha\in{\mathcal{H}}^{\star},\;\pi(\alpha)=\beta\}}\mathcal{M}_{\alpha}.}$

Definition 1.4**.**

Let $\mathcal{L}$ be a Lie algebra. A nontrivial subalgebra ${\mathcal{H}}$ of $\mathcal{L}$ is called a split toral subalgebra if $\mathcal{L}$ as an ${\mathcal{H}}-$module, via the adjoint representation, has a weight space decomposition with respect to ${\mathcal{H}}.$ One can see that a split toral subalgebra of a Lie algebra is abelian. Throughout the present paper by a toral subalgebra, we always mean a split toral subalgebra.**

Now suppose that $\mathcal{L}$ is a Lie algebra containing a nonzero toral subalgebra ${\mathcal{H}}$ with corresponding root system $R.$ The Lie algebra $(\mathcal{L},{\mathcal{H}})$ (or $\mathcal{L}$ for simplicity) is called *admissible * if $\mathcal{L}$ satisfies the following property:

[TABLE]

An element $h\in{\mathcal{H}}$ is called a splitting element corresponding to $\alpha\in R$ if there are $x\in\mathcal{L}_{\alpha}$ and $y\in\mathcal{L}_{-\alpha}$ such that $(x,h:=[x,y],y)$ is an $\mathfrak{sl}_{2}$-triple. A subset $\Delta$ of $R$ is called connected with respect to a fix set $\{h_{\alpha}\mid\alpha\in R\}$ of splitting elements if for any $\beta,\gamma\in\Delta,$ there is a finite sequence $\alpha_{1},\ldots,\alpha_{n}$ of elements of $\Delta$ such that $\alpha_{1}=\beta,$ $\alpha_{n}=\gamma$ and $\alpha_{i+1}(h_{\alpha_{i}})\neq 0,$ $1\leq i\leq n-1.$ A root $\alpha\in R$ is called integrable if there are $e_{\alpha}\in\mathcal{L}_{\alpha},$ $f_{\alpha}\in\mathcal{L}_{-\alpha}$ such that $h_{\alpha}:=[e_{\alpha},f_{\alpha}]\in{\mathcal{H}},$ $(e_{\alpha},h_{\alpha},f_{\alpha})$ is an $\mathfrak{sl}_{2}$-triple and that $ad_{e_{\alpha}}$ and $ad_{f_{\alpha}}$ act locally nilpotently on $\mathcal{L}.$ We denote by $R_{int},$ the set of integrable roots of $\mathcal{L}$ and note that if $\mathcal{L}$ is a locally finite admissible Lie algebra, then $R=R_{int}.$ A subset $\Delta$ of $R$ is called *symmetric * if $\Delta=-\Delta$ and it is called *closed * if $(\Delta+\Delta)\cap R\subseteq\Delta.$

Example 1.6**.**

Let $\mathcal{L}$ be a finite dimensional semisimple Lie algebra containing a maximal toral subalgebra ${\mathcal{T}}.$ Take $\Phi$ to be the root system of $\mathcal{L}$ with respect to ${\mathcal{T}}.$ Using [Se, Lem. I.3 and Lem. I.5], we get that $(\mathcal{L},{\mathcal{T}})$ is an admissible Lie algebra. Moreover Corollary to Lemma I.4 of [Se] shows that for $\beta\in\Phi,$ there is a unique $k_{\beta}\in{\mathcal{T}}$ with

[TABLE]

Using [Se, Lem. I.5 and Lem. I.6], we get that

[TABLE]

Following the proof of Lemma I.5 of [Se], we get that if $\beta\in\Phi,$ then $\kappa(k_{\beta},k_{\beta})\neq 0$ and for $t\in{\mathcal{T}},$ $\beta(t)=2\kappa(t,k_{\beta})/\kappa(k_{\beta},k_{\beta}),$ now (1.7) implies that the linear transformation form ${\mathcal{T}}$ to ${\mathcal{T}}^{\star}$ mapping $t\mapsto\kappa(t,\cdot),$ $t\in{\mathcal{T}},$ is onto and so is one to one, this in particular implies that the Killing form restricted to ${\mathcal{T}}$ is non-degenerate and that for $\beta\in\Phi,$ $2k_{\beta}/\kappa(k_{\beta},k_{\beta})$ is the unique element of ${\mathcal{T}}$ representing $\beta$ through the Killing form. Therefore one concludes that

[TABLE]

From now on we assume $(\mathcal{L},{\mathcal{H}})$ is an admissible Lie algebra with nonempty root system $R.$

Using $\mathfrak{sl}_{2}$-module theory, one can easily prove the following proposition.

Proposition 1.9**.**

Let $\alpha\in R.$ Take $e\in\mathcal{L}_{\alpha}$ and $f\in\mathcal{L}_{-\alpha}$ to be such that $(e,h:=[e,f],f)$ is an $\mathfrak{sl}_{2}$-triple and set $\mathfrak{g}:=\hbox{span}_{\mathbb{F}}\{e,h,f\}.$ Suppose $ad_{e}$ and $ad_{f}$ act locally nilpotently on $\mathcal{L},$ then the followings are satisfied:

(i) For $\beta\in R,$ $\beta(h)\in{\mathbb{Z}}$ and $\beta-\beta(h)\alpha\in R.$

(ii) If $\beta\in R$ is such that $\alpha+\beta\in R,$ we have $[e,\mathcal{L}_{\beta}]\neq\{0\},$ in particular, $[\mathcal{L}_{\alpha},\mathcal{L}_{\beta}]\neq\{0\}.$

(iii) For $\beta\in R,$ $\{k\in{\mathbb{Z}}\mid\beta+k\alpha\in R\cup\{0\}\}$ is an interval.

(iv) For $\beta\in R,$ if $\beta(h)>0,$ then $\beta-\alpha\in R\cup\{0\}$ and if $\beta(h)<0,$ then $\beta+\alpha\in R\cup\{0\}.$

(v) If $k\in\mathbb{F}$ and $k\alpha\in R,$ then $k\in{\mathbb{Z}}/2.$

(vi) $\{k\alpha\mid k\in\mathbb{F}\}\cap R_{int}\subseteq\{\pm\alpha,\pm(1/2)\alpha,\pm 2\alpha\}.$

Now use Proposition 1.9$(i),(iv)$ together with the same argument as in [LN, §10.2] to prove the following Corollary:

Corollary 1.10**.**

If $R=R_{int,}$ $R$ has the partial sum property, in the sense that if $n\in\mathbb{N}\setminus\{0\}$ and $\beta,\alpha_{1},\ldots,\alpha_{n}\in R\cup\{0\}$ are such that $\alpha_{1}+\cdots+\alpha_{n}=\beta,$ there is a permutation $\pi$ of $\{1,\ldots,n\}$ such that $\alpha_{\pi(1)}+\cdots+\alpha_{\pi(i)}\in R\cup\{0\}$ for all $i\in\{1,\ldots,n\}.$

Proposition 1.11**.**

Suppose that $R=R_{int}.$

(a) For an ideal $I$ of $\mathcal{L},$ set $R_{{}_{I}}:=\{\alpha\in R\mid I\cap\mathcal{L}_{\alpha}\neq\{0\}\},$ then $R_{{}_{I}}$ is a symmetric closed subset of $R$ and $I=(I\cap\mathcal{L}_{0})\oplus\sum_{\alpha\in R}(I\cap\mathcal{L}_{\alpha})=(I\cap\mathcal{L}_{0})\oplus\sum_{\alpha\in R_{{}_{I}}}\mathcal{L}_{\alpha}.$ Moreover if $I$ is simple as a Lie algebra, $I\cap\mathcal{L}_{0}=\sum_{\alpha\in R_{{}_{I}}}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}].$

(b) If $\mathcal{L}$ is semisimple (in the sense that it is a direct sum of simple ideals), then $\mathcal{L}_{0}=\sum_{\alpha\in R}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}].$

(c) Let $I$ be an ideal of $\mathcal{L}.$ Define $\Delta:=R\setminus R_{{}_{I}}$ and set $J:=\sum_{\alpha\in\Delta}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}]\oplus\sum_{\alpha\in\Delta}\mathcal{L}_{\alpha}.$ If $\mathcal{L}_{0}=\sum_{\alpha\in R}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}],$ then $J$ is an ideal of $\mathcal{L},$ $\mathcal{L}=I+J$ and $I\cap J\subseteq Z(\mathcal{L}).$

Proof. $(a)$ Since $I$ is an ideal of $\mathcal{L},$ we have $I=(I\cap\mathcal{L}_{0})\oplus\sum_{\alpha\in R}(I\cap\mathcal{L}_{\alpha})$ by [MP, Pro. 2.1.1]. Now if $\alpha\in R_{{}_{I}}$ and $0\neq x\in I\cap\mathcal{L}_{\alpha},$ then by (1.5), there exists $y\in\mathcal{L}_{-\alpha}$ such that $(x,h:=[x,y],y)$ is an $\mathfrak{sl}_{2}$-triple. Now as $x\in I,$ one gets that $h,y\in I$ and for all $z\in\mathcal{L}_{\alpha},$ $2z=\alpha(h)z=[h,z]\in I.$ So we have

[TABLE]

This in particular implies that $I=(I\cap\mathcal{L}_{0})\oplus\sum_{\alpha\in R_{{}_{I}}}\mathcal{L}_{\alpha}.$ Now this together with Proposition 1.9$(ii)$ implies that $R_{{}_{I}}$ is closed. For the last assertion, one can easily check that $I_{c}:=\sum_{\alpha\in R_{{}_{I}}}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}]\oplus\sum_{\alpha\in R_{{}_{I}}}\mathcal{L}_{\alpha}$ is an ideal of $I$ and so we are done as $I$ is simple.

($b$) Since $\mathcal{L}$ is semisimple, there are an index set $A$ and simple ideals $\mathcal{L}^{i},$ $i\in A,$ of $\mathcal{L}$ such that $\mathcal{L}=\oplus_{i\in A}\mathcal{L}^{i}.$ For $i\in A,$ use the notation as in the previous part and set $R_{i}:=R_{{}_{\mathcal{L}^{i}}},$ since $\mathcal{L}^{i}$ is simple, part $(a)$ implies that $\mathcal{L}^{i}=\sum_{\alpha\in R_{i}}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}]\oplus\sum_{\alpha\in R_{i}}\mathcal{L}_{\alpha}.$ Now we are done as $\mathcal{L}=\oplus_{i\in A}\mathcal{L}^{i}.$

($c$) Suppose $\alpha,\beta\in R,$ then thanks to (1.12), we have

[TABLE]

So using Proposition 1.9$(ii),$ we have

[TABLE]

This in particular implies that

[TABLE]

Now (1.14),(1.15) imply that

[TABLE]

which means that $J$ is an ideal of $\mathcal{L}.$ Also as $\sum_{\alpha\in R_{{}_{I}}}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}]\subseteq I,$ one gets that

[TABLE]

Now suppose that $x\in I\cap J,$ then $x\in I\cap\sum_{\alpha\in\Delta}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}],$ so by (1.15),

[TABLE]

On the other hand since $x\in I\cap\mathcal{L}_{0},$ we have

[TABLE]

This in particular implies that $[x,\sum_{\alpha\in\Delta}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}]+\sum_{\alpha\in\Delta}\mathcal{L}_{\alpha}]=\{0\}.$ Now this together with (1.17) and (1.18) implies that $x\in Z(\mathcal{L}).$ $\Box$

Corollary 1.19**.**

Suppose that $R=R_{int}.$ Set $\mathcal{L}_{c}:=\mathcal{L}_{0,0}\oplus\sum_{\alpha\in R}\mathcal{L}_{\alpha}$ where $\mathcal{L}_{0,0}:=\sum_{\alpha\in R}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}].$ Then $\mathcal{L}_{cc}:=\mathcal{L}_{c}/Z(\mathcal{L}_{c})$ is a semisimple Lie algebra. In particular, if $\mathcal{L}$ is finite dimensional, $\mathcal{L}_{c}$ is semisimple.

**Proof. **It follows from (1.5) that the canonical projection map $\mathcal{L}_{c}\longrightarrow\mathcal{L}_{cc}$ restricted to $\sum_{\alpha\in R}\mathcal{L}_{\alpha}$ is injective, so we identify $\sum_{\alpha\in R}\mathcal{L}_{\alpha}$ as a subspace of $\mathcal{L}_{cc},$ also it is easy to see that $(\mathcal{L}_{cc},\frac{{\mathcal{H}}+Z(\mathcal{L}_{c})}{Z(\mathcal{L}_{c})})$ is an admissible Lie algebra whose root system can be identified with $R.$ More precisely, we have the following weight space decomposition for $\mathcal{L}_{cc}$ with respect to $({\mathcal{H}}+Z(\mathcal{L}_{c}))/{Z(\mathcal{L}_{c})}:$

[TABLE]

Moreover we have $R=R_{int},$ $(\mathcal{L}_{cc})_{0}=\sum_{\alpha\in R}[(\mathcal{L}_{cc})_{\alpha},(\mathcal{L}_{cc})_{-\alpha}]$ and $Z(\mathcal{L}_{cc})=\{0\}.$ Using Proposition 1.11($c$), we get that for any ideal $I$ of $\mathcal{L}_{cc},$ there is an ideal $J$ of $\mathcal{L}_{cc}$ such that $\mathcal{L}=I\oplus J,$ this results in that $\mathcal{L}_{cc}$ is semisimple (see [J, §3.5]). Now suppose $\mathcal{L}$ is finite dimensional and take $\mathfrak{r}$ to be the solvable radical of $\mathcal{L}_{c},$ then $Z(\mathcal{L}_{c})\subseteq\mathfrak{r}.$ Also as $\mathfrak{r}$ is solvable, we get that $\mathfrak{r}/Z(\mathcal{L}_{c})$ is a solvable ideal of the semisimple Lie algebra $\mathcal{L}_{cc}.$ So $\mathfrak{r}/Z(\mathcal{L}_{c})=\{0\},$ i.e., $\mathfrak{r}=Z(\mathcal{L}_{c}).$ This means that $\mathcal{L}_{c}$ is a reductive Lie algebra, now as $\mathcal{L}_{c}$ is perfect, one concludes that $\mathcal{L}_{c}=[\mathcal{L}_{c},\mathcal{L}_{c}]$ is semisimple. This completes the proof. $\Box$

Lemma 1.21**.**

For a subset $\Delta$ of $R,$ set $\mathcal{L}_{{}_{\Delta}}:=\sum_{\alpha\in\Delta}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}]\oplus\sum_{\alpha\in\Delta}\mathcal{L}_{\alpha}$ and ${\mathcal{H}}_{{}_{\Delta}}:={\mathcal{H}}\cap\sum_{\alpha\in\Delta}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}].$ Suppose that $\Delta\subseteq R_{int}$ is symmetric and closed and take $\pi_{{}_{\Delta}}:{\mathcal{H}}^{\star}\longrightarrow{\mathcal{H}}_{{}_{\Delta}}^{\star}$ to be defined by $\alpha\mapsto\alpha_{\mid_{{\mathcal{H}}_{{}_{\Delta}}}},$ $\alpha\in{\mathcal{H}}^{\star}.$ Then $\pi_{{}_{\Delta}}$ restricted to $\Delta\cup\{0\}$ is injective. Also identifying $\Delta$ with $\pi_{{}_{\Delta}}(\Delta),$ we have that $(\mathcal{L}_{{}_{\Delta}},{\mathcal{H}}_{{}_{\Delta}})$ is an admissible Lie algebra with root system $\Delta.$ If moreover $\Delta$ is connected with respect to a fixed set of splitting elements, then $\mathcal{L}_{{}_{\Delta}}$ is a simple Lie subalgebra of $\mathcal{L}.$

**Proof. **Set $\pi:=\pi_{{}_{\Delta}}.$ Suppose that $\alpha,\beta\in\Delta\cup\{0\}$ and $\pi(\alpha)=\pi(\beta).$ We first suppose that $\gamma:=\alpha-\beta\in R,$ then since $\Delta$ is symmetric and closed, we have $\gamma\in\Delta,$ so (1.5) guarantees the existence of $t\in{\mathcal{H}}\cap[\mathcal{L}_{\gamma},\mathcal{L}_{-\gamma}]\subseteq{\mathcal{H}}\cap\mathcal{L}_{{}_{\Delta}}={\mathcal{H}}_{{}_{\Delta}}$ with $\gamma(t)=2,$ so $(\alpha-\beta)(t)=2$ which contradicts the fact that $\alpha\mid_{{\mathcal{H}}_{{}_{\Delta}}}=\beta\mid_{{\mathcal{H}}_{{}_{\Delta}}}.$ Next suppose that $\alpha-\beta\not\in R\cup\{0\},$ then $\alpha\neq 0$ and $\beta\neq 0.$ Since $\alpha$ is integrable, there are $e\in\mathcal{L}_{\alpha},$ $f\in\mathcal{L}_{-\alpha}$ such that $(e,h:=[e,f],f)$ is an $\mathfrak{sl}_{2}$-triple and $ad_{e},$ $ad_{f}$ act locally nilpotently on $\mathcal{L}.$ Now since $\alpha-\beta\not\in R\cup\{0\},$ Proposition 1.9$(iv)$ implies that $\beta(h)\leq 0.$ This contradicts the fact that $\pi(\alpha)=\pi(\beta)$ as $h\in{\mathcal{H}}\cap[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}]\subseteq{\mathcal{H}}_{{}_{\Delta}},$ $\alpha(h)=2$ and $\beta(h)\leq 0.$ These all together imply that $\pi\mid_{{}_{\Delta\cup\{0\}}}$ is injective. Next we note that since $\Delta$ is closed, $\mathcal{L}_{{}_{\Delta}}$ is a subalgebra of $\mathcal{L}.$ One also sees that $\mathcal{L}_{{}_{\Delta}}$ is an ${\mathcal{H}}$-submodule of ${\mathcal{H}}$-module $\mathcal{L}$ admitting the weight space decomposition $\mathcal{L}_{{}_{\Delta}}=\sum_{\alpha\in\Delta\cup\{0\}}(\mathcal{L}_{{}_{\Delta}})_{\alpha}$ with respect to ${\mathcal{H}}$ where $(\mathcal{L}_{{}_{\Delta}})_{\alpha}=\mathcal{L}_{\alpha}$ for $\alpha\in\Delta$ and $(\mathcal{L}_{{}_{\Delta}})_{0}=\sum_{\alpha\in\Delta}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}].$ From Lemma 1.3, we know that $\mathcal{L}_{{}_{\Delta}}$ admits a weight space decomposition $\mathcal{L}_{{}_{\Delta}}=\oplus_{\pi(\alpha)\in\pi(\Delta\cup\{0\})}(\mathcal{L}_{{}_{\Delta}})_{\pi(\alpha)}$ with respect to ${\mathcal{H}}_{{}_{\Delta}}$ where for $\alpha\in\Delta\cup\{0\},$

[TABLE]

Now this together with the injectivity of $\pi_{\mid_{{}_{\Delta\cup\{0\}}}}$ and the fact that $(\mathcal{L},{\mathcal{H}})$ is an admissible Lie algebra implies that $(\mathcal{L}_{{}_{\Delta}},{\mathcal{H}}_{{}_{\Delta}})$ is an admissible Lie algebra with root system $\Delta.$ The last sentence follows from Proposition 1.11. $\Box$

Proposition 1.22**.**

Suppose that $\mathcal{L}$ is a semisimple Lie algebra and ${\mathcal{H}}$ is a nonzero maximal toral subalgebra of $\mathcal{L}$ such that $(\mathcal{L},{\mathcal{H}})$ is an admissible Lie algebra with root system $R=R_{int}.$ Take $I$ to be an index set such that $\mathcal{L}=\oplus_{i\in I}\mathcal{L}^{i}$ where for $i\in I,$ $\mathcal{L}^{i}$ is an ideal of $\mathcal{L}$ which is simple as a Lie algebra. Set $R_{i}:=\{\alpha\in R\mid\mathcal{L}_{\alpha}\subseteq\mathcal{L}^{i}\}$ and ${\mathcal{H}}_{i}:=\mathcal{L}^{i}\cap{\mathcal{H}},$ $i\in I,$ then

(a) ${\mathcal{H}}=\oplus_{i\in I}{\mathcal{H}}_{i},$

(b) for $i\in I,$ ${\mathcal{H}}_{i}$ is a maximal toral subalgebra of $\mathcal{L}^{i}$ and $(\mathcal{L}^{i},{\mathcal{H}}_{i})$ is a admissible Lie algebra with root system $R_{i}.$

**Proof. **($a$) We first mention that for $i,j\in I,$ we have

[TABLE]

Using Proposition 1.11($a$), we get that for $i\in I,$ $\mathcal{L}^{i}=\sum_{\alpha\in R_{i}}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}]\oplus\sum_{\alpha\in R_{i}}\mathcal{L}_{\alpha},$ and $\mathcal{L}_{0}=\sum_{i\in I}\sum_{\alpha\in R_{i}}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}].$ For $i\in I,$ take $\pi_{i}:\mathcal{L}\longrightarrow\mathcal{L}^{i}$ to be the natural projection map and set $T_{i}$ to be the image of ${\mathcal{H}}$ under $\pi_{i}.$ Take $T^{\prime}_{i}$ to be a subspace of $T_{i}$ such that $T_{i}=({\mathcal{H}}\cap T_{i})\oplus T^{\prime}_{i}.$ Let $h\in{\mathcal{H}},$ then there is a unique expression $h=\sum_{i\in I}t_{i}$ with a finitely many nonzero terms $t_{i}\in\mathcal{L}^{i},$ $i\in I.$ Now fix $i\in I,$ using (1.23), we have for $x\in\mathcal{L}_{\alpha}$ with $\alpha\in R_{i}$ that

[TABLE]

This in particular implies that

[TABLE]

which allows us to define $f_{t,\alpha}$ for $t\in T^{\prime}_{i}$ and $\alpha\in R_{i}$ as follows:

[TABLE]

also define $f_{t,\alpha}:=0$ if $t\in T^{\prime}_{i}$ and $\alpha\in(R\cup\{0\})\setminus R_{i}.$ Next suppose that $\alpha\in R\cup\{0\}$ and extend $\alpha$ to the functional $\alpha^{i}\in({\mathcal{H}}\oplus T^{\prime}_{i})^{\star}$ defined as follows:

[TABLE]

Now one can see that ${\mathcal{H}}\oplus T^{\prime}_{i}$ is a toral subalgebra of $\mathcal{L}$ with corresponding root system $\{\alpha^{i}\mid\alpha\in R\}.$ But ${\mathcal{H}}\oplus T^{\prime}_{i}$ contains the maximal toral subalgebra ${\mathcal{H}},$ therefore $T^{\prime}_{i}=\{0\}$ and so $T_{i}=T_{i}\cap{\mathcal{H}}$ which implies that $T_{i}\subseteq{\mathcal{H}}.$ This gives that $T_{i}\subseteq{\mathcal{H}}\cap\mathcal{L}^{i}={\mathcal{H}}_{i},$ on the other hand, if $h\in{\mathcal{H}}_{i},$ then $h=\pi_{i}(h)\in T_{i},$ so we get that $T_{i}={\mathcal{H}}_{i}.$ Now we have ${\mathcal{H}}=\oplus_{i\in I}T_{i},$ and so ${\mathcal{H}}=\oplus_{i\in I}{\mathcal{H}}_{i}.$

$(b)$ Suppose that $i\in I.$ Using Preposition 1.11$(a)$ and Lemma 1.21, we get that $(\mathcal{L}^{i},{\mathcal{H}}_{i})$ is an admissible Lie algebra with root system $R_{i}.$ Next we note that any toral subalgebra $T_{i}$ of $\mathcal{L}_{i}$ larger than ${\mathcal{H}}_{i}$ would automatically be toral in $\mathcal{L}$ and centralizes ${\mathcal{H}}_{j}$ for $j\in I\setminus\{i\}.$ Now ${\mathcal{H}}+T_{i}$ is a toral subalgebra of $\mathcal{L}$ larger than ${\mathcal{H}}$ which is a contradiction. This completes the proof. $\Box$

Lemma 1.27**.**

If $(\mathcal{L},{\mathcal{H}})$ is a finite dimensional semi-simple admissible Lie algebra, then ${\mathcal{H}}$ is a maximal toral subalgebra.

**Proof. **Let ${\mathcal{T}}$ be a maximal toral subalgebra of $\mathcal{L}$ containing ${\mathcal{H}}.$ We first suppose that $\mathcal{L}$ is simple. Since ${\mathcal{T}}$ is a toral subalgebra of $\mathcal{L},$ $\mathcal{L}=\sum_{\beta\in{\mathcal{T}}^{\star}}\mathcal{L}^{\prime}_{\beta}$ where for $\beta\in{\mathcal{T}}^{\star},$ $\mathcal{L}^{\prime}_{\beta}:=\{x\in\mathcal{L}\mid[t,x]=\beta(t)x;\;\forall t\in{\mathcal{T}}\}.$ Take $\Phi$ to be the root system of $\mathcal{L}$ with respect to ${\mathcal{T}}.$ Using Example 1.6, one knows that for $\beta\in\Phi,$ there is a unique element $k_{\beta}\in{\mathcal{T}}$ satisfying

[TABLE]

Using Lemma 1.3, we get that for $\alpha\in R\cup\{0\},$ $\mathcal{L}_{\alpha}=\bigoplus_{\beta\in A_{\alpha}}\mathcal{L}^{\prime}_{\beta}$ where $A_{\alpha}:=\{\beta\in{\mathcal{T}}^{\star},\;\beta\mid_{{}_{\mathcal{H}}}=\alpha\}.$ Now let $\alpha\in R$ and $\beta\in\Phi$ be such that $\beta\in A_{\alpha}.$ Suppose that $0\neq x\in\mathcal{L}^{\prime}_{\beta}\subseteq\mathcal{L}_{\alpha},$ then (1.5) guarantees the existence of $y\in\mathcal{L}_{-\alpha}$ such that $h:=[x,y]\in{\mathcal{H}}$ and $(x,h,y)$ is an $\mathfrak{sl}_{2}$-triple. Since $y\in\mathcal{L}_{-\alpha}=\bigoplus_{\gamma\in A_{\alpha}}\mathcal{L}^{\prime}_{-\gamma},$ we have that $y=\sum_{\gamma\in A_{\alpha}}y_{-\gamma}$ with $y_{-\gamma}\in\mathcal{L}^{\prime}_{-\gamma}.$ Now

[TABLE]

and so $h=[x,y_{-\beta}]$ as $h\in{\mathcal{H}}\subseteq{\mathcal{T}}\subseteq\mathcal{L}^{\prime}_{0}.$ Also as $h\in{\mathcal{H}},$ $\beta(h)=\alpha(h)=2.$ Now considering (1.28), we get from the uniqueness of $k_{\beta}$ that $k_{\beta}=h,$ therefore we have proved

[TABLE]

Now set $A:=\{\beta\in\Phi\mid\beta\mid_{{}_{\mathcal{H}}}\neq 0\}$ and $I:=\sum_{\beta\in A}\mathcal{L}^{\prime}_{\beta}\oplus\sum_{\beta\in A}[\mathcal{L}^{\prime}_{\beta},\mathcal{L}^{\prime}_{-\beta}].$ We claim that $I$ is an ideal of $\mathcal{L}.$ For this, it is enough to show that $(\Phi+A)\cap\Phi\subseteq A.$ Suppose that $\gamma\in\Phi$ and $\beta\in A$ are such that $\gamma+\beta\in\Phi.$ If $\gamma\mid_{{}_{\mathcal{H}}}=0,$ then $\gamma+\beta\mid_{{}_{\mathcal{H}}}\neq 0$ and so $\gamma+\beta\in A$ and if $\gamma\mid_{{}_{\mathcal{H}}}\neq 0,$ then (1.8) together with (1.29) implies that $k_{\gamma+\beta}\in{\mathcal{H}},$ now as $(\gamma+\beta)(k_{\gamma+\beta})=2\neq 0,$ we get that $\gamma+\beta\mid_{{}_{\mathcal{H}}}\neq 0,$ i.e., $\gamma+\beta\in A.$ This shows that $I$ is an ideal of $\mathcal{L},$ but $\mathcal{L}$ is simple, so $I=\mathcal{L}$ which in particular implies that $A=\Phi.$ Now (1.7) together with (1.29) implies that ${\mathcal{T}}\subseteq{\mathcal{H}}$ and so ${\mathcal{H}}={\mathcal{T}}$ is a maximal toral subalgebra. Next suppose that $\mathcal{L}=\oplus_{i=1}^{n}\mathcal{L}_{i}$ where for $1\leq i\leq n,$ $\mathcal{L}_{i}$ is a simple ideal of $\mathcal{L}.$ So by Proposition 1.11($a$) and Lemma 1.21, $(\mathcal{L}_{i},{\mathcal{H}}\cap\mathcal{L}_{i})$ is a finite dimensional admissible simple Lie algebra and so using the first part of the proof, we get that ${\mathcal{H}}\cap\mathcal{L}_{i}$ is a maximal toral subalgebra of $\mathcal{L}_{i}.$ Now if ${\mathcal{T}}$ is a maximal toral subalgebra of $\mathcal{L}$ containing ${\mathcal{H}},$ Example 1.6 gets that $(\mathcal{L},{\mathcal{T}})$ is a semi-simple admissible Lie algebra. So Proposition 1.22 implies that ${\mathcal{T}}_{i}:=\mathcal{L}_{i}\cap{\mathcal{T}}$ is a maximal toral subalgebra of $\mathcal{L}_{i}$ and that ${\mathcal{T}}={\mathcal{T}}_{1}\oplus{\mathcal{T}}_{2}\oplus\cdots\oplus{\mathcal{T}}_{n}.$ But ${\mathcal{H}}\cap\mathcal{L}_{i}\subseteq{\mathcal{T}}_{i},$ therefore ${\mathcal{H}}\cap\mathcal{L}_{i}={\mathcal{T}}_{i}.$ Now we have ${\mathcal{T}}={\mathcal{T}}_{1}\oplus{\mathcal{T}}_{2}\oplus\cdots\oplus{\mathcal{T}}_{n}\subseteq{\mathcal{H}}$ and so we are done. $\Box$

Proposition 1.30**.**

(a) Suppose that $\Delta$ is a symmetric closed finite subset of $R_{int}$ such that $\mathcal{L}_{\alpha}$ is finite dimensional for any $\alpha\in\Delta,$ then for $\alpha\in\Delta,$ there is a unique $h_{\alpha}\in{\mathcal{H}}$ such that

[TABLE]

Moreover if $\alpha,\beta_{1},\ldots,\beta_{m}\in\Delta$ are such that $\alpha\in\hbox{span}_{\mathbb{F}}\{\beta_{1},\ldots,\beta_{m}\},$ then $h_{\alpha}\in\hbox{span}_{\mathbb{F}}\{h_{\beta_{i}}\mid 1\leq i\leq m\}.$

(b) If $R=R_{int}$ and all the weight spaces are finite dimensional, then there is a unique set $\{h_{\alpha}\mid\alpha\in R\}$ of splitting elements which we refer to as the splitting subset of $\mathcal{L}.$

Proof. $(a)$ Set $\mathcal{L}_{{}_{\Delta}}:=\sum_{\beta\in\Delta}[\mathcal{L}_{\beta},\mathcal{L}_{-\beta}]\oplus\sum_{\beta\in\Delta}\mathcal{L}_{\beta}$ and ${\mathcal{H}}_{{}_{\Delta}}={\mathcal{H}}\cap\mathcal{L}_{{}_{\Delta}}.$ Using Propositions 1.21 and Corollary 1.19, we get that $(\mathcal{L}_{{}_{\Delta}},{\mathcal{H}}_{{}_{\Delta}})$ is a finite dimensional admissible semi-simple Lie subalgebra with root system $\Delta.$ Using Lemma 1.27, one gets that ${\mathcal{H}}_{{}_{\Delta}}$ is a maximal toral subalgebra of $\mathcal{L}_{{}_{\Delta}}.$ Now the result follows from Example 1.6.

($b$) Let $\alpha\in R$ and take $\Delta:=R\cap\mathbb{F}\alpha.$ Using Proposition 1.9$(vi)$ together with part ($a$), we are done. $\Box$

Proposition 1.31**.**

Suppose that $(\mathcal{L},{\mathcal{H}})$ is a locally finite admissible Lie algebra with corresponding root system $R$ such that all weight spaces are finite dimensional. Take $\{h_{\alpha}\mid\alpha\in R\}$ to be the splitting subset of $\mathcal{L}$ and ${\mathcal{V}}:=\hbox{span}_{\mathbb{F}}R.$ For $\alpha\in R,$ define $\check{\alpha}:{\mathcal{V}}\longrightarrow\mathbb{F}$ by $v\mapsto v(h_{\alpha}),$ $v\in{\mathcal{V}}$ and $s_{\alpha}:{\mathcal{V}}\longrightarrow{\mathcal{V}}$ by $v\mapsto v-\check{\alpha}(v)\alpha,$ $v\in{\mathcal{V}}.$ For $M\subseteq R,$ set $M_{\pm}:=M\cup-M$ and take ${\mathcal{G}}_{{}_{M}}$ to be the subalgebra of $\mathcal{L}$ generated by $\cup_{\alpha\in M_{\pm}}\mathcal{L}_{\alpha},$ finally set $\Delta_{{}_{M}}:=\{\alpha\in R\mid{\mathcal{G}}_{{}_{M}}\cap\mathcal{L}_{\alpha}\neq\{0\}\},$ then we have the followings:

(a) For $M\subseteq R,$ $\Delta_{{}_{M}}$ is a symmetric closed subset of $R,$ in particular $\mathcal{L}_{{}_{\Delta_{{}_{M}}}}=\sum_{\alpha\in\Delta_{{}_{M}}}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}]\oplus\sum_{\alpha\in\Delta_{{}_{M}}}\mathcal{L}_{\alpha}$ is a Lie subalgebra of $\mathcal{L}.$

(b) If $M$ is a subset of $R$ and $\alpha,\beta\in\Delta_{{}_{M}},$ then $s_{\alpha}(\beta)\in\Delta_{{}_{M}}.$ Also if $M$ is a finite subset of $R,$ ${\mathcal{G}}_{{}_{M}}$ is a finite dimensional Lie subalgebra of $\mathcal{L}$ and $\Delta_{{}_{M}}$ is a finite root system in its span where the reflection based on $\alpha\in\Delta_{{}_{M}}$ is $s_{\alpha}$ restricted to $\hbox{span}_{\mathbb{F}}\Delta_{{}_{M}}.$

(c) If $\alpha,\beta\in R,$ then $\beta(h_{\alpha})\in{\mathbb{Z}}\cap[-4,4].$

(d) $R$ is a locally finite root system in its span where the reflection based on $\alpha\in R$ is $s_{\alpha}$ and if $\mathcal{L}$ is semi-simple, the necessary and sufficient condition for $R$ to be irreducible is that $\mathcal{L}$ is simple. Also for $M\subseteq R,$ $\Delta_{{}_{M}}$ is a closed subsystem of $R$ which is irreducible if $M$ is irreducible.

Proof. $(a)$ We first recall that as $\mathcal{L}$ is locally finite, for any weight vector $x,$ $ad_{x}$ acts locally nilpotently on $\mathcal{L}.$ Now we show that $\Delta_{{}_{M}}$ is symmetric. One knows that each element of ${\mathcal{G}}_{{}_{M}}$ is a sum of elements of the form $[x_{n},\ldots,x_{1}]$ where $n\in\mathbb{N}\setminus\{0\}$ and for $1\leq i\leq n,$ $x_{i}\in\mathcal{L}_{\alpha_{i}},$ for some $\alpha_{i}\in M_{\pm}.$ Now if $\alpha\in\Delta_{{}_{M}},$ ${\mathcal{G}}_{{}_{M}}\cap\mathcal{L}_{\alpha}\neq\{0\},$ so there are $\alpha_{1},\ldots,\alpha_{n}\in M_{\pm}$ such that $\alpha=\sum_{i=1}^{n}\alpha_{i}$ and there are $x_{i}\in\mathcal{L}_{\alpha_{i}},$ $1\leq i\leq n,$ such that $0\neq x:=[x_{n},\ldots,x_{1}]\in{\mathcal{G}}_{{}_{M}}\cap\mathcal{L}_{\alpha}.$ We use induction on $n$ to show that $\Delta_{{}_{M}}$ is symmetric. If $n=1,$ then $\alpha\in M_{\pm}$ and so $-\alpha\in M_{\pm}\subseteq\Delta_{{}_{M}}.$ Now let $\alpha\in\Delta_{{}_{M}}$ be such that there are $n\in\mathbb{N}^{\geq 2},$ $\alpha_{1},\ldots,\alpha_{n}\in M_{\pm}$ and $x_{i}\in\mathcal{L}_{\alpha_{i}},$ $1\leq i\leq n,$ such that $\alpha=\sum_{i=1}^{n}\alpha_{i}$ and $0\neq x:=[x_{n},\ldots,x_{1}]\in{\mathcal{G}}_{{}_{M}}\cap\mathcal{L}_{\alpha}.$ Therefore $0\neq[x_{n-1},\ldots,x_{1}]\in{\mathcal{G}}_{{}_{M}}\cap\mathcal{L}_{\beta}$ where $\beta:=\sum_{i=1}^{n-1}\alpha_{i}.$ Now we have $\beta,\alpha_{n},\alpha=\beta+\alpha_{n}\in\Delta_{{}_{M}}\subseteq R.$ Using the induction hypothesis, we get that $-\beta\in\Delta_{{}_{M}}$ and so there is a nonzero $y\in\mathcal{L}_{-\beta}\cap{\mathcal{G}}_{{}_{M}}.$ Now contemplating Proposition 1.9$(ii),$ we have that $\{0\}\neq[y,\mathcal{L}_{-\alpha_{n}}]\subseteq({\mathcal{G}}_{{}_{M}}\cap\mathcal{L}_{-\alpha_{n}-\beta})$ which shows that $-\alpha\in\Delta_{{}_{M}}.$ Next we show that $\Delta_{{}_{M}}$ is closed. We first show that

[TABLE]

Take $\alpha\in\Delta_{{}_{M}}$ and $\beta\in M_{\pm}$ to be such that $\alpha+\beta\in R$ and x to be a nonzero element of ${\mathcal{G}}_{{}_{M}}\cap\mathcal{L}_{\alpha}.$ Using Lemma 1.9$(ii$), we get that $0\neq[x,\mathcal{L}_{\beta}]\subseteq[{\mathcal{G}}_{{}_{M}},{\mathcal{G}}_{{}_{M}}]\subseteq{\mathcal{G}}_{{}_{M}}$ and so we are done as $[x,\mathcal{L}_{\beta}]\subseteq\mathcal{L}_{\alpha+\beta}.$ Now suppose $\alpha,\beta\in\Delta_{{}_{M}}$ and $\alpha+\beta\in R.$ Since $\alpha,\beta\in\Delta_{{}_{M}},$ there is $\{\alpha_{i}\mid 1\leq i\leq n\}\cup\{\beta_{j}\mid 1\leq j\leq m\}\subseteq M_{\pm}$ such that $\alpha=\sum_{i=1}^{n}\alpha_{i}$ and $\beta=\sum_{j=1}^{m}\beta_{j}.$ For $1\leq i\leq m+n,$ set

[TABLE]

then $\gamma_{i}\in M_{\pm},$ $1\leq i\leq m+n$ and since $\alpha+\beta\in R,$ $\sum_{i=1}^{n+m}\gamma_{i}\in R.$ Now by Corollary 1.10, there exists a permutation $\pi$ on $\{1,\ldots,m+n\}$ such that all partial sums $\sum_{i=1}^{t}\gamma_{\pi(i)}\in R\cup\{0\},$ $1\leq t\leq m+n.$ Now using (1.32) together with an induction process, we get that $\alpha+\beta=\sum_{i=1}^{n+m}\gamma_{i}\in\Delta_{{}_{M}}.$ This means that $\Delta_{{}_{M}}$ is a closed. This completes the proof.

$(b)$ Since $\alpha,\beta\in\Delta_{{}_{M}}\subseteq R,$ by Proposition 1.9$(i),$ $\check{\alpha}(\beta)\in{\mathbb{Z}}$ and $s_{\alpha}(\beta)\in R.$ Now take $e_{\alpha}\in\mathcal{L}_{\alpha}$ and $f_{\alpha}\in\mathcal{L}_{-\alpha}$ to be such that $h_{\alpha}=[e_{\alpha},f_{\alpha}]$ and $(e_{\alpha},h_{\alpha},f_{\alpha})$ is an $\mathfrak{sl}_{2}-$triple, setting $\theta_{\alpha}:=exp(ad_{e_{\alpha}})exp(ad_{-f_{\alpha}})exp(ad_{e_{\alpha}}),$ we get that $\{0\}\neq\theta_{\alpha}(\mathcal{L}_{\beta})\subseteq(\mathcal{L}_{s_{\alpha}(\beta)}\cap\mathcal{L}_{{}_{\Delta_{{}_{M}}}})$ and so $s_{\alpha}(\beta)\in\Delta_{{}_{M}}.$ Now if $M$ is a finite set, since $\mathcal{L}$ is locally finite, ${\mathcal{G}}_{{}_{M}}$ is a finite dimensional subalgebra of $\mathcal{L}$ and so $\Delta_{{}_{M}}$ is finite. This completes the proof.

$(c)$ Set $M:=\{\alpha,\beta\},$ by part $($b$),$ $\Delta_{{}_{M}}$ is a finite root system in $\hbox{span}_{\mathbb{F}}\Delta_{{}_{M}}$ where the reflection based on $\gamma\in\Delta_{{}_{M}}$ is defined by $v\mapsto v-\check{\gamma}(v)\gamma,$ $v\in\hbox{span}_{\mathbb{F}}\Delta_{{}_{M}}.$ Using the finite root system theory, we get that $\{\check{\gamma}(\eta)\mid\gamma,\eta\in\Delta_{{}_{M}}\}\subseteq{\mathbb{Z}}\cap[-4,4],$ in particular $\beta(h_{\alpha})=\check{\alpha}(\beta)\in{\mathbb{Z}}\cap[-4,4].$

$(d)$ Using Proposition 1.9$(i),$ it is enough to prove that $R$ is locally finite. Suppose that $M$ is a finite subset of $R,$ then by part ($a$), $\Delta:=\Delta_{{}_{M}}$ is a finite root system in $\hbox{span}_{\mathbb{F}}\Delta$ where the reflection based on $\gamma\in\Delta_{{}_{M}}$ is defined by $v\mapsto v-\check{\gamma}(v)\gamma$ for $v\in\hbox{span}_{\mathbb{F}}\Delta.$ Take $\Pi:=\{\alpha_{1},\ldots,\alpha_{n}\}$ to be a base of $\Delta.$ One knows that the Cartan matrix $(\check{\alpha}_{i}(\alpha_{j}))$ is an invertible matrix. So for any choice of $\{k_{1},\ldots,k_{n}\}\subseteq\mathbb{F},$ the following system of equations

[TABLE]

has a unique solution. Next suppose that $(r_{1},\ldots,r_{n})\in\mathbb{F}^{n}$ is such that $\eta:=\sum_{i=1}^{n}r_{i}\alpha_{i}\in R,$ then for any $1\leq j\leq n,$

[TABLE]

This means that $(r_{1},\ldots,r_{n})$ is a solution for the following system of equations

[TABLE]

But we know that by part ($c$), $\eta(h_{{}_{\alpha_{j}}})\in[-4,4]\cap{\mathbb{Z}},$ so there are a finitely many choice for $\eta(h_{{}_{\alpha_{j}}}),$ $1\leq j\leq n.$ This together with the fact that (1.33) has a unique solution implies that there are a finitely many choice for $(r_{1},\ldots,r_{n})\in\mathbb{F}^{n}$ such that $\sum_{i=1}^{n}r_{i}\alpha_{i}$ is a root. So $(\hbox{span}_{\mathbb{F}}M)\cap R=(\hbox{span}_{\mathbb{F}}\Delta)\cap R=(\hbox{span}_{\mathbb{F}}\{\alpha_{1},\ldots,\alpha_{n}\})\cap R$ is a finite set. Now if ${\mathcal{W}}$ is a finite dimensional subspace of ${\mathcal{V}}=\hbox{span}_{\mathbb{F}}R,$ then there is a finite set $M$ of $R$ such that ${\mathcal{W}}\subseteq\hbox{span}_{\mathbb{F}}M,$ so ${\mathcal{W}}\cap R\subseteq(\hbox{span}_{\mathbb{F}}M)\cap R$ and so ${\mathcal{W}}\cap R$ is a finite set. This shows that $R$ is a locally finite root system in ${\mathcal{V}}.$ It follows easily from Proposition 1.11($a$) that if $\mathcal{L}$ is semi-simple, $\mathcal{L}$ is simple if and only if $R$ is irreducible. Now let $M$ be a subset of $R,$ parts $(a)$ and $(b)$ shows that $\Delta_{{}_{M}}$ is a subsystem of $R.$ Next suppose that $M$ is irreducible but $\Delta_{{}_{M}}$ is not irreducible, then there is $\alpha\in\Delta_{{}_{M}}$ such that $\alpha$ is connected to none of the elements of $M,$ in particular,

[TABLE]

But we know there are $\beta_{1},\ldots,\beta_{m}\in M$ such that $\alpha=\sum_{i=1}^{m}\pm\beta_{i}.$ Take $M^{\prime}:=\{\alpha,\beta_{1},\ldots,\beta_{m}\},$ we know from parts $(a)$ and $(b)$ that $\Delta_{{}_{M^{\prime}}}$ is a finite closed subsystem of $R,$ so Proposition 1.30 implies that $h_{\alpha}=\sum_{i=1}^{m}r_{i}h_{\beta_{i}}$ for some $r_{i}\in\mathbb{F},$ $1\leq i\leq m.$ This together with (1.34) implies that $\alpha(h_{\alpha})=0$ which is a contradiction. Therefore $\Delta_{{}_{M}}$ is irreducible. $\Box$

Proposition 1.35**.**

Suppose that $\mathcal{L}$ is simple and locally finite such that all the weight spaces are finite dimensional, then ${\mathcal{H}}$ is a maximal toral subalgebra of $\mathcal{L}.$

**Proof. **Take $\{h_{\alpha}\mid\alpha\in R\}$ to be the splitting subset of $\mathcal{L}.$ By Proposition 1.31$(d)$, $R$ is an irreducible locally finite root system in its span where the reflection based on $\alpha\in R$ is defined by $v\mapsto v-v(h_{\alpha})\alpha,$ $v\in\hbox{span}_{\mathbb{F}}R.$ Now let $T$ be a toral subalgebra of $\mathcal{L}$ containing ${\mathcal{H}}.$ Take $\Phi$ to be the root system of $\mathcal{L}$ with respect to $T$ and for $\beta\in\Phi\cup\{0\},$ denote by $\mathcal{L}^{\prime}_{\beta},$ the ”weight space” of $\mathcal{L}$ corresponding to $\beta.$ For $\alpha\in R\cup\{0\},$ set $A_{\alpha}:=\{\beta\in\Phi\cup\{0\}\mid\beta\mid_{{}_{\mathcal{H}}}=\alpha\},$ then by Lemma 1.3, we have $\mathcal{L}_{\alpha}=\oplus_{\beta\in A_{\alpha}}\mathcal{L}^{\prime}_{\beta}.$ Now let $t\in T,$ since ${\mathcal{H}}\subseteq T$ and $T$ is abelian, $t\in\mathcal{L}_{0},$ but $\mathcal{L}$ is simple, so Proposition 1.11($b$) implies that $t\in\sum_{\alpha\in R}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}].$ Therefore there is a finite subset $M$ of $R$ such that $t\in\sum_{\alpha\in M}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}].$ Now take $\Delta$ to be a finite irreducible closed subsystem of $R$ containing $M$ (see [LN, Cor. 3.16]). Using Lemma 1.21, we get that

[TABLE]

is an finite dimensional simple admissible Lie algebra. Now take $T_{{}_{\Delta}}:=T\cap\sum_{\alpha\in\Delta}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}].$ It is read from (1.36) that $\mathcal{L}_{{}_{\Delta}}$ has a weight space decomposition with respect to $T_{{}_{\Delta}},$ in other words $T_{{}_{\Delta}}$ is a toral subalgebra of $\mathcal{L}_{{}_{\Delta}}$ containing ${\mathcal{H}}_{{}_{\Delta}}={\mathcal{H}}\cap\mathcal{L}_{{}_{\Delta}}$ which results in ${\mathcal{H}}_{{}_{\Delta}}=T_{{}_{\Delta}}$ contemplating Lemmas 1.21 and 1.27. Now we have

[TABLE]

This completes the proof. $\Box$

Now one can use Propositions 1.35, 1.31($d$) and Lemma 1.21 to get the following theorem:

Theorem 1.37**.**

Suppose that $(\mathcal{L},{\mathcal{H}})$ is a locally finite simple admissible Lie algebra whose weight spaces are finite dimensional. Take $R$ to be the root system of $\mathcal{L}$ with respect to ${\mathcal{H}}.$ Then $R$ is an irreducible locally finite root system. Next take $\{R_{i}\mid i\in I\}$ to be the class of finite irreducible closed subsystems of $R.$ For $i\in I,$ set $\mathcal{L}_{i}:=\sum_{\alpha\in R_{i}}[\mathcal{L}_{\alpha},\mathcal{L}_{-\alpha}]\oplus\sum_{\alpha\in R_{i}}\mathcal{L}_{\alpha}$ and ${\mathcal{H}}_{i}:={\mathcal{H}}\cap\mathcal{L}_{i}.$ Then ${\mathcal{H}}$ is a maximal toral subalgebra of $\mathcal{L}$ and for $i\in I,$ ${\mathcal{H}}_{i}$ is a maximal toral subalgebra of $\mathcal{L}_{i},$ also $(\mathcal{L}_{i},{\mathcal{H}}_{i})$ is a finite dimensional simple admissible Lie algebra with root system $R_{i}.$ Moreover $\{R_{i}\mid i\in I\}$ and $\{\mathcal{L}_{i}\mid i\in I\}$ are directed systems with respect to inclusion, $R$ is a direct limit of $\{R_{i}\mid i\in I\}$ and $\mathcal{L}$ is a direct limit of $\{\mathcal{L}_{i}\mid i\in I\}.$