a contradiction $K\equiv 0$.
Otherwise we may change the role of $K$ and $K^{\prime}$ to obtain the
absurdity alike.

Next for the case (iii),we change it to the form
(iii)${}^{\prime}K+g_{\alpha}^{-1}K^{\prime}\in\mathfrak{M}_{\chi}$.

We thus have an equation

$x_{\epsilon_{1}-\epsilon_{2}}K+x_{\epsilon_{1}-\epsilon_{2}}g_{\epsilon_{1}-\epsilon_{2}}^{-1}K^{\prime}\equiv
0$ modulo $\mathfrak{M}_{\chi}$. According to the above argument, we are also
led to a contradiction $K\in\mathfrak{M}_{\chi}$.

∎

Now we turn to the classical Lie algebra of $D_{l}$\- type. We let $L$ be any
modular Lie algebra of $D_{l}$-type over any algebraically closed field $F$ of
characteristic $p\geq 7$.

We note first that the Lie algebra of $D_{l}$\- type with rank $l$, i.e., the
$D_{l}$-type Lie algera over $\mathbb{C}$ has its root system
$\Phi$=$\\{\pm(\epsilon_{i}\pm\epsilon_{j})|i\neq j\\}$,where
$\epsilon_{i},\epsilon_{j}$ are linearly independent orthonormal unit vectors
in $\mathbb{R}^{l}$ with $l\geq 4$. The base of $\Phi$ is equal to
$\\{\epsilon_{1}-\epsilon_{2},\epsilon_{2}-\epsilon_{3},\cdots,\epsilon_{l-1}-\epsilon_{l},\epsilon_{l-1}+\epsilon_{l}\\}.$

######  Proposition 4.8.

We let $\alpha$ be any root in the root system $\Phi$ of $L.$ If
$\chi(x_{\alpha})$ $\neq 0,$ then $dim_{F}$$\rho_{\chi}$$(u(L))$ = $p^{2m},$
where $[Q(u(L)):Q(\mathfrak{Z})]$=$p^{2m}$=$p^{n-l}$ with $\mathfrak{Z}$ the
center of $u(L)$ and $Q$ denotes the quotient algebra.

So we claim that the simple module corresponding to this representation has
$p^{m}$ as its dimension and the Lee’s basis for this simple module is given
as follows.

Let $\mathfrak{M}_{\chi}$ be the kernel of this simple representation with
character $\chi$. We may put $\alpha=\epsilon_{1}-\epsilon_{2}$ since only
long roots exist and all roots of the same length are conjugate under the Weyl
group of $\Phi$.

We put $B_{i}:=b_{i1}$ $h_{\epsilon_{1}-\epsilon_{2}}$\+ $b_{i2}$
$h_{\epsilon_{2}-\epsilon_{3}}$+$\cdots$+$b_{i,l-1}$
$h_{\epsilon_{l-1}-\epsilon_{l}}$\+ $b_{il}h_{\epsilon_{l-1}+\epsilon_{l}},$
where ($b_{i1}$,$b_{i2}$ $\cdots,$$b_{il}$) $\in F^{l}$ are chosen so that any
$(l+1)-$$B_{i}$’s are linearly independent in $\mathbb{P}^{l}(F),$ the
$\mathfrak{B}$ below becomes an $F-$ linearly independent set in $u(L)$ if
necessary and $x_{\alpha}$$B_{i}$ $\not\equiv$$B_{i}$$x_{\alpha}$ for
$\alpha$=$\epsilon_{1}-$$\epsilon_{2}.$

In $u(L))/\mathfrak{M}_{\chi}$ we have a basis
$\mathfrak{B}$$:=\\{(B_{1}+A_{\epsilon_{1}-\epsilon_{2}})^{i_{1}}\otimes(B_{2}+A_{-(\epsilon_{1}-\epsilon_{2})})^{i_{2}}\otimes\cdots\otimes(B_{2l-2}+A_{-(\epsilon_{l-1}-\epsilon_{l})})^{i_{2l-2}}\otimes(B_{2l-1}+A_{\epsilon_{l-1}+\epsilon_{l}})^{i_{2l-1}}\otimes(B_{2l}+A_{-(\epsilon_{l-1}+\epsilon_{l})})^{i_{2l}}\otimes(\otimes_{j=2l+1}^{2m}(B_{j}+A_{\alpha_{j}})^{i_{j}})|0\leq
i_{j}\leq p-1\\}$,

where we put

$A_{\epsilon_{1}-\epsilon_{2}}=x_{\alpha}=x_{\epsilon_{1}-\epsilon_{2}}$,

$A_{\epsilon_{2}-\epsilon_{1}}=c_{\epsilon_{2}-\epsilon_{1}}+(h_{\epsilon_{1}-\epsilon_{2}}+1)^{2}+4x_{\epsilon_{2}-\epsilon_{1}}x_{\epsilon_{1}-\epsilon_{2}}$

$A_{\epsilon_{2}\pm\epsilon_{3}}=x_{\pm\epsilon_{1}-\epsilon_{2}}(c_{\epsilon_{2}\pm\epsilon_{3}}+x_{\epsilon_{2}\pm\epsilon_{3}}x_{-(\epsilon_{2}\pm\epsilon_{3})}\pm
x_{\epsilon_{1}\pm\epsilon_{3}}x_{-(\epsilon_{1}\pm\epsilon_{3})})$,

$A_{-(\epsilon_{1}\pm\epsilon_{3})}=(x_{-\epsilon_{1}-\epsilon_{2}}^{2}$ or
$x_{-\epsilon_{1}-\epsilon_{2}}^{3})(c_{-(\epsilon_{1}\pm\epsilon_{3})}+x_{\epsilon_{2}\pm\epsilon_{3}}x_{-(\epsilon_{2}\pm\epsilon_{3})}\pm
x_{\epsilon_{1}\pm\epsilon_{3}}x_{-(\epsilon_{1}\pm\epsilon_{3})})$,

$A_{\epsilon_{2}\pm\epsilon_{k}}=x_{\epsilon_{3}\pm\epsilon_{k}}(c_{\epsilon_{2}\pm\epsilon_{k}}+x_{\epsilon_{2}\pm\epsilon_{k}}x_{-(\epsilon_{2}\pm\epsilon_{k})}\pm
x_{\epsilon_{1}\pm\epsilon_{k}}x_{-(\epsilon_{1}\pm\epsilon_{k})})$

if $k\neq 1$,

$A_{-(\epsilon_{1}\pm\epsilon_{k})}=x_{-(\epsilon_{2}\pm\epsilon_{k})}(c_{-(\epsilon_{1}\pm\epsilon_{k})}+x_{\epsilon_{2}\pm\epsilon_{k}}x_{-(\epsilon_{2}\pm\epsilon_{k})}\pm
x_{\epsilon_{1}\pm\epsilon_{k}}x_{-(\epsilon_{1}\pm\epsilon_{k})})$,

with the signs chosen so that they may commute with $x_{\alpha}$ and with
$c_{\beta}\in F$ chosen so that $A_{\epsilon_{2}-\epsilon_{1}}$ and
parentheses are invertible.
For any other root $\beta$, we put $A_{\beta}=x_{\beta}^{2}$ or
$x_{\beta}^{3}$ or $x_{\beta}^{4}$ if possible.

Otherwise we make use of the parentheses( ) again used for designating
$A_{-\beta}$. So in this case we put $A_{\beta}=x_{\gamma}^{2}$ or
$x_{\gamma}^{3}$ attached to these ( ) so that $x_{\alpha}$ may commute with
$A_{\beta}$.

###### Proof.

Also evident from [KY-4]. We may actually prove this proposition as follows.

Let $\mathfrak{M}_{\chi}$ be the kernel of this simple representation with
character $\chi$. We may put $\alpha=\epsilon_{1}-\epsilon_{2}$ since only
long roots exist and all roots of the same length are conjugate under the Weyl
group of $\Phi$.

The Lee’s basis for this simple module is given as follows.

As suggested in the present proposition, we put $B_{i}:=b_{i1}$
$h_{\epsilon_{1}-\epsilon_{2}}$\+ $b_{i2}$
$h_{\epsilon_{2}-\epsilon_{3}}$+$\cdots$+$b_{i,l-1}$
$h_{\epsilon_{l-1}-\epsilon_{l}}$\+ $b_{il}h_{\epsilon_{l-1}+\epsilon_{l}},$
where ($b_{i1}$,$b_{i2}$ $\cdots,$$b_{il}$) $\in F^{l}$ are chosen so that any
$(l+1)-$$B_{i}$’s are linearly independent in $\mathbb{P}^{l}(F),$ the
$\mathfrak{B}$ below becomes an $F-$ linearly independent set in $u(L)$ if
necessary and $x_{\alpha}$$B_{i}$ $\not\equiv$$B_{i}$$x_{\alpha}$ for
$\alpha$=$\epsilon_{1}-$$\epsilon_{2}.$

In $u(L))/\mathfrak{M}_{\chi}$ we have a basis
$\mathfrak{B}$$:=\\{(B_{1}+A_{\epsilon_{1}-\epsilon_{2}})^{i_{1}}\otimes(B_{2}+A_{-(\epsilon_{1}-\epsilon_{2})})^{i_{2}}\otimes\cdots\otimes(B_{2l-2}+A_{-(\epsilon_{l-1}-\epsilon_{l})})^{i_{2l-2}}\otimes(B_{2l-1}+A_{\epsilon_{l-1}+\epsilon_{l}})^{i_{2l-1}}\otimes(B_{2l}+A_{-(\epsilon_{l-1}+\epsilon_{l})})^{i_{2l}}\otimes(\otimes_{j=2l+1}^{2m}(B_{j}+A_{\alpha_{j}})^{i_{j}})|0\leq
i_{j}\leq p-1\\}$,

where we put

$A_{\epsilon_{1}-\epsilon_{2}}=x_{\alpha}=x_{\epsilon_{1}-\epsilon_{2}}$,

$A_{\epsilon_{2}-\epsilon_{1}}=c_{\epsilon_{2}-\epsilon_{1}}+(h_{\epsilon_{1}-\epsilon_{2}}+1)^{2}+4x_{\epsilon_{2}-\epsilon_{1}}x_{\epsilon_{1}-\epsilon_{2}}$

$A_{\epsilon_{2}\pm\epsilon_{3}}=x_{\pm\epsilon_{1}-\epsilon_{2}}(c_{\epsilon_{2}\pm\epsilon_{3}}+x_{\epsilon_{2}\pm\epsilon_{3}}x_{-(\epsilon_{2}\pm\epsilon_{3})}\pm
x_{\epsilon_{1}\pm\epsilon_{3}}x_{-(\epsilon_{1}\pm\epsilon_{3})})$,

$A_{-(\epsilon_{1}\pm\epsilon_{3})}=(x_{-\epsilon_{1}-\epsilon_{2}}^{2}$ or
$x_{-\epsilon_{1}-\epsilon_{2}}^{3})(c_{-(\epsilon_{1}\pm\epsilon_{3})}+x_{\epsilon_{2}\pm\epsilon_{3}}x_{-(\epsilon_{2}\pm\epsilon_{3})}\pm
x_{\epsilon_{1}\pm\epsilon_{3}}x_{-(\epsilon_{1}\pm\epsilon_{3})})$,

$A_{\epsilon_{2}\pm\epsilon_{k}}=x_{\epsilon_{3}\pm\epsilon_{k}}(c_{\epsilon_{2}\pm\epsilon_{k}}+x_{\epsilon_{2}\pm\epsilon_{k}}x_{-(\epsilon_{2}\pm\epsilon_{k})}\pm
x_{\epsilon_{1}\pm\epsilon_{k}}x_{-(\epsilon_{1}\pm\epsilon_{k})})$

if $k\neq 1$,

$A_{-(\epsilon_{1}\pm\epsilon_{k})}=x_{-(\epsilon_{2}\pm\epsilon_{k})}(c_{-(\epsilon_{1}\pm\epsilon_{k})}+x_{\epsilon_{2}\pm\epsilon_{k}}x_{-(\epsilon_{2}\pm\epsilon_{k})}\pm
x_{\epsilon_{1}\pm\epsilon_{k}}x_{-(\epsilon_{1}\pm\epsilon_{k})})$,

with the signs chosen so that they may commute with $x_{\alpha}$ and with
$c_{\beta}\in F$ chosen so that $A_{\epsilon_{2}-\epsilon_{1}}$ and
parentheses are invertible.
For any other root $\beta$, we put $A_{\beta}=x_{\beta}^{2}$ or
$x_{\beta}^{3}$ or $x_{\beta}^{4}$ if possible.

Otherwise we make use of the parentheses( ) again used for designating
$A_{-\beta}$. So in this case we put $A_{\beta}=x_{\gamma}^{2}$ or
$x_{\gamma}^{3}$ attached to these ( ) so that $x_{\alpha}$ may commute with
$A_{\beta}$.

It is not difficult to see that the rest of this proof is similar to that of
proposition 4.3. We would like to actually complete the proof as follows.

We may see without difficulty that $\mathfrak{B}$ is a linearly independent
set in $u(L)$ by virtue of P-B-W theorem.

We shall prove that a nontrivial linearly dependent equation leads to
absurdity. We assume first that we have a dependence equation which is of
least degree with respect to $h_{\alpha_{j}}\in H$ and the number of whose
highest degree terms is also least.

In case it is conjugated by $x_{\alpha}$, then there arises a nontrivial
dependence equation of lower degree than the given one,which contradicts to
our assumption.

Otherwise it reduces to the following form, which we have only to prove:

$(\ast)x_{\epsilon_{1}-\epsilon_{2}}$$K$ \+ $K^{\prime}$
$\in$$\mathfrak{M}_{\chi}$

, where $K$ and $K^{\prime}$ commute with $x_{\alpha}$ modulo
$\mathfrak{M}_{\chi}$.

We thus deal with and assume
$x_{\epsilon_{2}-\epsilon_{1}}$$x_{\epsilon_{1}-\epsilon_{2}}$$K$ \+
$x_{\epsilon_{2}-\epsilon_{1}}$$K^{\prime\prime}$ $\in$ $\mathfrak{M}_{\chi}$.

From
$w_{\epsilon_{1}-\epsilon_{2}}:=(h_{\epsilon_{1}-\epsilon_{2}}+1)^{2}+4x_{\epsilon_{2}-\epsilon_{1}}x_{\epsilon_{1}-\epsilon_{2}}\in$
the center of $u(\mathfrak{sl}_{2}(F)),$ we get
$4^{-1}\\{w_{\epsilon_{1}-\epsilon_{2}}-(h+1)^{2}\\}K+x_{\epsilon_{2}-\epsilon_{1}}K^{\prime\prime}\equiv
0$ modulo $\mathfrak{M}_{\chi}$.

If $x_{\epsilon_{2}-\epsilon_{1}}^{p}\equiv c$ which is a constant,then

$(\ast\ast)4^{-1}x_{\epsilon_{2}-\epsilon_{1}}^{p-1}\\{w_{\epsilon_{1}-\epsilon_{2}}-(h_{\epsilon_{1}-\epsilon_{2}}+1)^{2}\\}K+cK^{\prime\prime}\equiv
0$

is obtained.
From $(\ast),(\ast\ast)$, we have

$4^{-1}x_{\epsilon_{2}-\epsilon_{1}}^{p-1}\\{w_{\epsilon_{1}-\epsilon_{2}}-(h_{\epsilon_{1}-\epsilon_{2}}+1)^{2}\\}K-cx_{\epsilon_{1}-\epsilon_{2}}K\newline
\equiv 0$ modulo $\mathfrak{M}_{\chi}$.

Multiplying $x_{\epsilon_{1}-\epsilon_{2}}^{p-1}$ to this equation,we obtain

$(\ast\ast\ast)4^{-1}x_{\epsilon_{1}-\epsilon_{2}}^{p-1}x_{\epsilon_{2}-\epsilon_{1}}^{p-1}\\{w_{\epsilon_{1}-\epsilon_{2}}-(h_{\epsilon_{1}-\epsilon_{2}}+1)^{2}\\}K-cx_{\epsilon_{1}-\epsilon_{2}}^{p}K\equiv
0.$

By making use of $w_{\epsilon_{1}-\epsilon_{2}}$, we may deduce from
$(\ast\ast\ast)$ an equation of the form
( a polynomial of degree $\geq 1$ with respect to
$h_{\epsilon_{1}-\epsilon_{2}})K-cx_{\epsilon_{1}-\epsilon_{2}}^{p}K\equiv 0.$

Finally if we use conjugation and subtraction consecutively,then we are led to
a contradiction $K\in\mathfrak{M}_{\chi}.$
It may be necessary for us to change the role of $K$ and $K^{\prime}$ to
obtain the absurdity alike.

∎

######  Proposition 4.9.

We let $\chi$ be a character of any simple $L$-module with
$\chi(h_{\alpha})\neq$ 0 for some $\alpha\in\Phi$,where $h_{\alpha}$ is an
element in the Chevalley basis of $L$ such that
$Fx_{\alpha}+Fh_{\alpha}+Fx_{-\alpha}=\mathfrak{sl}_{2}(F)$ with
$[x_{\alpha},x_{-\alpha}]=h_{\alpha}\in H$(a CSA of $L$).
We then claim that $dim_{F}$$\rho_{\chi}$$(u(L))$ = $p^{2m},$ where
$[Q(u(L)):Q(\mathfrak{Z})]$=$p^{2m}$=$p^{n-l}$ with $\mathfrak{Z}$ the center
of $u(L)$ and $Q$ denotes the quotient algebra.

Namely, we claim that the simple module corresponding to this representation
has $p^{m}$ as its dimension.

###### Proof.

We may let $\alpha=\epsilon_{1}-\epsilon_{2}$ without loss of generality since
all roots are long and are conjugate under the Weyl group of $\Phi$.

If $\chi(x_{\alpha})\neq 0$ or $\chi(x_{-\alpha})\neq 0$, then our claim is
clear from the preceding proposition4.8. So we may let
$x_{\epsilon_{1}}^{p}\equiv x_{-\epsilon_{1}}^{p}\equiv 0$ modulo
$\mathfrak{M}_{\chi}$.

As we have done before, we let $B_{i}$ be defined as in proposition 4.3.

In $u(L)/\mathfrak{M}_{\chi}$, we have a Lee’s basis as
$\mathfrak{B}$$:=\\{(B_{1}+A_{\epsilon_{1}-\epsilon_{2}})^{i_{1}}\otimes(B_{2}+A_{-(\epsilon_{1}-\epsilon_{2})})^{i_{2}}\otimes\cdots\otimes(B_{2l-2}+A_{-(\epsilon_{l-1}-\epsilon_{l})})^{i_{2l-2}}\otimes(B_{2l-1}+A_{\epsilon_{l-1}+\epsilon_{l}})^{i_{2l-1}}\otimes(B_{2l}+A_{-(\epsilon_{l-1}+\epsilon_{l}})^{i_{2l}}\otimes(\otimes_{j=2l+1}^{2m}(B_{j}+A_{\alpha_{j}})^{i_{j}})|0\leq
i_{j}\leq p-1\\}$,

where we put

$A_{\epsilon_{1}-\epsilon_{2}}=g_{\alpha}=g_{\epsilon_{1}-\epsilon_{2}}$,

$A_{\epsilon_{2}-\epsilon_{1}}=c_{\epsilon_{2}-\epsilon_{1}}+(h_{\epsilon_{1}-\epsilon_{2}}+1)^{2}+4x_{\epsilon_{2}-\epsilon_{1}}x_{\epsilon_{1}-\epsilon_{2}}$,

$A_{\epsilon_{2}\pm\epsilon_{3}}=g_{\alpha}^{2or3}c_{\epsilon_{2}\pm\epsilon_{3}}+x_{\epsilon_{2}\pm\epsilon_{3}}x_{-(\epsilon_{2}\pm\epsilon_{3})}\pm
x_{\epsilon_{1}\pm\epsilon_{3}}x_{-(\epsilon_{1}\pm\epsilon_{3})})$,

$A_{-(\epsilon_{1}\pm\epsilon_{3})}=g_{\alpha}^{4or5}(c_{-(\epsilon_{1}\pm\epsilon_{3})}+x_{\epsilon_{1}\pm\epsilon_{3}}x_{-(\epsilon_{1}\pm\epsilon_{3})}\pm
x_{\epsilon_{2}\pm\epsilon_{3}}x_{-(\epsilon_{2}\pm\epsilon_{3})})$,

$A_{\epsilon_{2}\pm\epsilon_{k}}=x_{\epsilon_{3}\pm\epsilon_{k}}(c_{\epsilon_{2}\pm\epsilon_{k}}+x_{\epsilon_{2}\pm\epsilon_{k}}x_{-(\epsilon_{2}\pm\epsilon_{k})}\pm
x_{\epsilon_{1}\pm\epsilon_{k}}x_{-(\epsilon_{1}\pm\epsilon_{k})})$,

$A_{-(\epsilon_{1}\pm\epsilon_{k})}=x_{-(\epsilon_{3}\pm\epsilon_{k})}^{2}(c_{-(\epsilon_{1}\pm\epsilon_{k})}+x_{\epsilon_{2}\pm\epsilon_{k}}x_{-(\epsilon_{2}\pm\epsilon_{k})}\pm
x_{\epsilon_{1}\pm\epsilon_{k}}x_{-(\epsilon_{1}\pm\epsilon_{k})})$,

with the signs chosen so that they may commute with $x_{\alpha}$ and with
$c_{\beta}\in F$ chosen so that $A_{\epsilon_{2}-\epsilon_{1}}$ and
parentheses are invertible.

For any other root $\beta$, we put $A_{\beta}=x_{\beta}^{3}$ or
$x_{\beta}^{4}$ if possible.

Otherwise we make use of the parentheses( ) again used for designating
$A_{-\beta}$. So in this case we put $A_{\beta}=x_{\gamma}^{3}$ or
$x_{\gamma}^{4}$ attached to these ( ) so that $x_{\alpha}$ may commute with
$A_{\beta}$.

We may see without difficulty that $\mathfrak{B}$ is a linearly independent
set in $u(L)$ by virtue of P-B-W theorem.

We shall prove that a nontrivial linearly dependent equation leads to
absurdity. We assume first that we have a dependence equation which is of
least degree with respect to $h_{\alpha_{j}}\in H$ and the number of whose
highest degree terms is also least.

In case it is conjugated by $g_{\alpha}$, then there arises a nontrivial
dependence equation of lower degree than the given one,which contradicts to
our assumption.

Otherwise it reduces to one of the following forms, so that we have only to
prove that

(i)$x_{\beta}K+K^{\prime}\in\mathfrak{M}_{\chi}$,

(ii) $g_{\alpha}K+K^{\prime}\in\mathfrak{M}_{\chi}$

lead to a contradiction, where both $K$ and $K^{\prime}$ commute with
$x_{\pm\alpha}$ modulo $\mathfrak{M}_{\chi}$. In particular $K$ commute with
$g_{\alpha}$.

For the case (i), we may change it to the form
$x_{\alpha}K+K^{\prime\prime}\in\mathfrak{M}_{\chi}$ for some
$K^{\prime\prime}$ commuting with $x_{\alpha}$ modulo $\mathfrak{M}_{\chi}$.

So we have $x_{\alpha}^{p}K+x_{\alpha}^{p-1}K^{\prime\prime}\equiv 0$, and
hence $x_{\alpha}^{p-1}K^{\prime\prime}\equiv 0$.

Subtracting from this
$x_{-\alpha}x_{\alpha}K+x_{-\alpha}K^{\prime\prime}\equiv 0$, we get

$-x_{-\alpha}x_{\alpha}K+g_{\alpha}K^{\prime\prime}\equiv 0$. Recall here that
$g_{\alpha}$ is invertible and $w_{\alpha}$ belongs to the center of
$u(\mathfrak{sl}_{2}(F))$.

So we get
$4^{-1}\\{(h_{\alpha}+1)^{2}-w_{\alpha}\\}K+g_{\alpha}K^{\prime\prime}\equiv
0$, and hence

$(\ast)g_{\alpha}^{p-1}4^{-1}\\{(h_{\alpha}+1)^{2}-w_{\alpha}\\}K+cK^{\prime\prime}\equiv
0$

is obtained and from the start equation we have

$(\ast\ast)cx_{\alpha}K+cK^{\prime\prime}\equiv 0$, where
$g_{\alpha}^{p}-c\equiv 0$.

Subtracting $(\ast\ast)$ from $(\ast)$, we have
$4^{-1}g_{\alpha}^{p-1}\\{(h_{\alpha}+1)^{2}-w_{\alpha}\\}K-cx_{\alpha}K\equiv
0$.

Multiplying this equation by $g_{\alpha}^{1-p}$ to the right, we obtain
$4^{-1}g_{\alpha}^{p-1}\\{(h_{\alpha}+1)^{2}-w_{\alpha}\\}g_{\alpha}^{1-p}K-cx_{\alpha}g_{\alpha}^{1-p}K\equiv
0$

We thus have
$4^{-1}\\{(h_{\alpha}+1-2)^{2}-w_{\alpha}\\}K-x_{\alpha}g_{\alpha}K\equiv 0$.

So it follows that
$4^{-1}\\{(h_{\alpha}-1)^{2}-w_{\alpha}\\}K+x_{\alpha}x_{-\alpha}K\equiv 0$.

Next multiplying $x_{-\alpha}^{p-1}$ to the right of this last equation, we
obtain $\\{(h_{\alpha}-1)^{2}-w_{\alpha}\\}Kx_{-\alpha}^{p-1}\equiv 0$. Now
multiply $x_{\alpha}$ in turn consecutively to the left of this equation until
it becomes of the form

( a nonzero polynomial of degree $\geq 1$ with respect to $h_{\alpha})K$
$\in\mathfrak{M}_{\chi}$.

By making use of conjugation and subtraction consecutively, we are led to a
contradiction.$K\in\mathfrak{M}_{\chi}$.

Finally for the case (ii),we consider
$K+g_{\alpha}^{-1}K^{\prime}\in\mathfrak{M}_{\chi}$. So we have
$x_{\alpha}K+x_{\alpha}g_{\alpha}^{-1}K^{\prime}\equiv 0$ modulo
$\mathfrak{M}_{\chi}$.

By analogy with the argument as in the case (i), we obtain a contraiction
$K\in\mathfrak{M}_{\chi}$.

∎

[Remark] We have seen up to now that classical modular Lie algebras of
$D_{l},C_{l},B_{l},A_{l}$-types have no subregular point under mild
conditions.

So we cannot help believing that all classical Lie algebras and exceptional
type Lie algebras behave like this, and thus we conjectured such a similar
result in [KWa].
In the next paragraphs and followups, we shall preferably deal with the
exceptional type $E_{8}$.

As is well known the $F_{4}$\- type Lie algebra over $F$ has its dimension
$n=52$ with $l=4$ and the $G_{2}$\- type Lie algebra over $F$ has its
dimension $n=14$ with $l=2$. So comparing these with the types $E_{6},E_{7}$,
and $E_{8}$, we may find out their Lee’s bases without difficulty due to the
fact that for any $x_{\beta}$ in the Chevalley basis of $L$ over $F$, elements
$x_{\beta},x_{\beta}^{2},\cdots,x_{\beta}^{6}$ all have distinct weights with
respect to the $ad(H)$\- module action.

Furthermore the root systems of $E_{7}$ and $E_{6}$ are subroot systems of the
root system of $E_{8}$, so we have only to deal exclusively with $E_{8}$\-
type to figure out the Lee’s bases for the exceptional type Lie algebras. We
might refer to the reference [7] for the Lee’s bases of other exceptional
types.

We note that the Lie algebra of $E_{8}$\- type with rank $8$, i.e., the
$E_{8}$-type Lie algera over $\mathbb{C}$ has its root system
$\Phi$=$\\{\pm(\epsilon_{i}\pm\epsilon_{j})|1\leq i\neq j\leq
8;2^{-1}\Sigma_{i=1}^{8}(-1)^{k(i)}\epsilon_{i}$,where $k(i)$= 0,1 and add up
to an even integer }. Here $\epsilon_{i},\epsilon_{j}$ are linearly
independent orthonormal unit vectors in $\mathbb{R}^{8}$ . The base of $\Phi$
is equal to
$\\{\epsilon_{1}+\epsilon_{2},\epsilon_{2}-\epsilon_{1},\epsilon_{3}-\epsilon_{2},\epsilon_{4}-\epsilon_{3},\epsilon_{5}-\epsilon_{4},\epsilon_{6}-\epsilon_{5},\epsilon_{7}-\epsilon_{6},2^{-1}(\epsilon_{1}+\epsilon_{8}-(\epsilon_{2}+\cdots+\epsilon_{7})\\}.$

######  Proposition 4.10.

Let $\alpha$ be any root in the root system $\Phi$ of $L=E_{8}.$ If
$\chi(x_{\alpha})$ $\neq 0,$ then $dim_{F}$$\rho_{\chi}$$(u(L))$ = $p^{2m},$
where $[Q(u(L)):Q(\mathfrak{Z})]$=$p^{2m}$=$p^{n-l}=p^{248-8}$ with
$\mathfrak{Z}$ the center of $u(L)$ and $Q$ denotes the quotient algebra.

So we claim that the simple module corresponding to this representation has
$p^{m}=p^{120}$ as its dimension.

###### Proof.

We give the Lee’s basis for this simple module as follows.

Let $\alpha=\epsilon_{1}+\epsilon_{2}$ without loss of generality due to Weyl
group and let us put $B_{i}$=$b_{i1}$$h_{\epsilon_{1}+\epsilon_{2}}$ \+
$b_{i2}h_{\epsilon_{2}-\epsilon_{1}}+b_{i3}h_{\epsilon_{3}-\epsilon_{2}}+b_{i4}h_{\epsilon_{4}-\epsilon_{3}}+b_{i5}h_{\epsilon_{5}-\epsilon_{4}}+b_{i6}h_{\epsilon_{6}-\epsilon_{5}}+b_{i7}h_{\epsilon_{7}-\epsilon_{6}}+b_{i8}h_{2^{-1}(\epsilon_{1}+\epsilon_{8}-(\epsilon_{2}+\cdots+\epsilon_{7}))}$
for $i=1,2,\cdots,2m$ ,where ($b_{i1},\cdots,b_{il})\in F^{l}$ are chosen so
that arbitrary ($l+1)-B_{i}$’s are linearly independent in
$\mathbb{P}^{l}(F)$,the $\mathfrak{B}$ below becomes an $F$-linearly
independent set in $u(L)$ if necessary and $x_{\alpha}B_{i}$ $\not\equiv
B_{i}x_{\alpha}$ with $\alpha=\epsilon_{1}+\epsilon_{2}$.

Let $\mathfrak{M}_{\chi}$ be the kernel of the irreducible representation
$\rho_{\chi}:L\rightarrow\mathfrak{gl}(V)$ of the restricted Lie algebra
($L,[p])$ associated with any given irreducible $L$-module $V$ with a
character $\chi$.

In $u(L)/\mathfrak{M}_{\chi}$ we give a Lee’s basis as
$\mathfrak{B}$:=$\\{(B_{1}+A_{\epsilon_{1}+\epsilon_{2}})^{i_{1}}\otimes(B_{2}+A_{-(\epsilon_{1}+\epsilon_{2})})^{i_{2}}\otimes\cdots\otimes(B_{16}+A_{-2^{-1}((\epsilon_{1}+\epsilon_{8})-(\epsilon_{2}+\cdots+\epsilon_{7}))})^{i_{16}}\otimes(\otimes_{j=17}^{240}(B_{j}+A_{\alpha_{j}})^{i_{j}}\\}$
for 0 $\leq i_{j}\leq p-1$,

where we put
$A_{\alpha}=A_{\epsilon_{1}+\epsilon_{2}}=x_{\epsilon_{1}+\epsilon_{2}}$,
$A_{-\epsilon_{1}-\epsilon_{2}}=c_{-\epsilon_{1}-\epsilon_{2}}+(h_{\epsilon_{1}+\epsilon_{2}}+1)^{2}+4x_{-\epsilon_{1}-\epsilon_{2}}x_{\epsilon_{1}+\epsilon_{2}},\newline
A_{-(\epsilon_{1}\pm\epsilon_{k})}=x_{\epsilon_{3}-(\pm\epsilon_{k})}(c_{-\epsilon_{1}\pm\epsilon_{k}}+x_{\epsilon_{1}\pm\epsilon_{k}}x_{-(\epsilon_{1}\pm\epsilon_{k})}\pm
x_{-(\epsilon_{2}\mp\epsilon_{k})}x_{\epsilon_{2}+(\mp\epsilon_{k})})$,
$A_{-(\epsilon_{1}\pm\epsilon_{3})}=x_{\epsilon_{2}-(\pm\epsilon_{3})}(c_{-(\epsilon_{1}\pm\epsilon_{3}}+x_{\epsilon_{1}\mp\epsilon_{3}}x_{-(\epsilon_{1}\pm\epsilon_{3})}\pm
x_{-(\epsilon_{2}\mp\epsilon_{3}}x_{\epsilon_{2}+(\mp\epsilon_{3})},$

$A_{-(\epsilon_{2}\pm\epsilon_{k})}=x_{-\epsilon_{3}-(\pm\epsilon_{k})}(c_{-(\epsilon_{2}\pm\epsilon_{k})}+x_{\epsilon_{2}\pm\epsilon_{k}}x_{-(\epsilon_{2}\pm\epsilon_{k})}\pm
x_{-(\epsilon_{1}\mp\epsilon_{k})}x_{\epsilon_{1}\mp\epsilon_{k})}$,
$A_{-(\epsilon_{2}\pm\epsilon_{3})}=x_{\epsilon_{2}-(\pm\epsilon_{3})}^{2}(c_{-(\epsilon_{2}\pm\epsilon_{3})}+x_{\epsilon_{2}\pm\epsilon_{3}}x_{-(\epsilon_{2}\pm\epsilon_{3})\pm
x_{-(\epsilon_{1}\mp\epsilon_{3})}x_{\epsilon_{1}\mp\epsilon_{3}}\epsilon_{j})}\pm\newline
x_{\epsilon_{j}-\epsilon_{1}}x_{\epsilon_{1}-\epsilon_{j}})$,
$A_{\frac{1}{2}\Sigma_{i=1}^{8}(-1)^{k(i)}\epsilon_{i}}=x_{\frac{1}{2}\Sigma_{i=1}^{8}(-1)^{k(i)}\epsilon_{i}}$;
for
$\beta_{k(i)}=-\frac{1}{2}\epsilon_{1}-\frac{1}{2}\epsilon_{2}+\frac{1}{2}\Sigma_{i=3}^{8}(-1)^{k(i)}\epsilon_{i}$
and for
$\gamma_{k(i)}=\frac{1}{2}\epsilon_{1}+\frac{1}{2}\epsilon_{2}+\frac{1}{2}\Sigma_{i=3}^{8}(-1)^{k(i)}\epsilon_{i}$,
we set
$A_{\beta_{k(i)}}=x_{-\frac{1}{2}\Sigma_{i=3}^{8}(-1)^{k(i)}}\epsilon_{i}$$(C_{\beta_{k(i)}}+x_{\beta_{k(i)}}x_{-\beta_{k(i)}}\pm
x_{\gamma_{k(i)}}x_{-\gamma_{k(i)}}),A_{\gamma_{k(i)}}=x_{\frac{1}{2}\Sigma_{i=3}^{8}(-1)^{k(i)}\epsilon_{i}}(c_{\gamma_{k(i)}}+x_{\beta_{k(i)}}x_{-\beta_{k(i)}}\pm
x_{\gamma_{k(i)}}x_{-\gamma_{k(i)}}).$

with the sign chosen so that they commute with $x_{\alpha}$ and with
$c_{\beta}\in F$ chosen so that $A_{-\epsilon_{1}}$ and parentheses( ) are
invertible.
For any other root $\beta$, we put $A_{\beta}={x_{\beta}}^{2}$ or
$x_{\beta}^{3}$ if possible.

Otherwise we make use of the parentheses( )again used for designating
$A_{-\beta}$. So in this case we put $A_{\beta}={x_{\gamma}}^{2}$ or
${x_{\gamma}}^{3}$ attached to these ( ) so that $x_{\alpha}$ may commute with
$A_{\beta}$.

For the rest of the complete proof , refer to the consolidated proof in
proposition 4.12 below.

∎

######  Proposition 4.11.

Let $\chi$ be a character of any simple $L$-module with $\chi(h_{\alpha})\neq$
0 for some $\alpha\in\Phi$,where $h_{\alpha}$ is an element in the Chevalley
basis of $L=E_{8}$ such that
$Fx_{\alpha}+Fh_{\alpha}+Fx_{-\alpha}=\mathfrak{sl}_{2}(F)$ with
$[x_{\alpha},x_{-\alpha}]=h_{\alpha}\in H$.

We then claim that any simple $L$-module with character $\chi$ is of dimension
$p^{m}=p^{n-l\over 2}$,where $n=dimL=2m+l$ for a CSA H with $dimH=l$.

###### Proof.

Let $\alpha$ be any root in the root system $\Phi$ of $L.$ If
$\chi(x_{\alpha})$ $\neq 0,$ or $\chi(x_{-\alpha})\neq 0$, then
$dim_{F}$$\rho_{\chi}$$(u(L))$ = $p^{2m}$ due to the consolidating proof of
proposition 4.12 below, where
$[Q(u(L)):Q(\mathfrak{Z})]$=$p^{2m}$=$p^{n-l}=p^{248-8}$ with $\mathfrak{Z}$
the center of $u(L)$ and $Q$ denotes the quotient algebra.

We give the Lee’s basis for this simple module as follows.

Let $\alpha=\epsilon_{1}+\epsilon_{2}$ without loss of generality due to Weyl
group and let us put $B_{i}$=$b_{i1}$$h_{\epsilon_{1}+\epsilon_{2}}$ \+
$b_{i2}h_{\epsilon_{2}-\epsilon_{1}}+b_{i3}h_{\epsilon_{3}-\epsilon_{2}}+b_{i4}h_{\epsilon_{4}-\epsilon_{3}}+b_{i5}h_{\epsilon_{5}-\epsilon_{4}}+b_{i6}h_{\epsilon_{6}-\epsilon_{5}}+b_{i7}h_{\epsilon_{7}-\epsilon_{6}}+b_{i8}h_{2^{-1}(\epsilon_{1}+\epsilon_{8}-(\epsilon_{2}+\cdots+\epsilon_{7}))}$
for $i=1,2,\cdots,2m$ ,where ($b_{i1},\cdots,b_{il})\in F^{l}$ are chosen so
that arbitrary ($l+1)-B_{i}$’s are linearly independent in
$\mathbb{P}^{l}(F)$,the $\mathfrak{B}$ below becomes an $F$-linearly
independent set in $u(L)$ if necessary and $x_{\alpha}B_{i}$ $\not\equiv
B_{i}x_{\alpha}$ with $\alpha=\epsilon_{1}+\epsilon_{2}$.

Let $\mathfrak{M}_{\chi}$ be the kernel of the irreducible representation
$\rho_{\chi}:L\rightarrow\mathfrak{gl}(V)$ of the restricted Lie algebra
($L,[p])$ associated with any given irreducible $L$-module $V$ with a
character $\chi$.

In $u(L)/\mathfrak{M}_{\chi}$ we give a Lee’s basis as
$\mathfrak{B}$:=$\\{(B_{1}+A_{\epsilon_{1}+\epsilon_{2}})^{i_{1}}\otimes(B_{2}+A_{-(\epsilon_{1}+\epsilon_{2})})^{i_{2}}\otimes\cdots\otimes(B_{16}+A_{-2^{-1}((\epsilon_{1}+\epsilon_{8})-(\epsilon_{2}+\cdots+\epsilon_{7}))})^{i_{16}}\otimes(\otimes_{j=17}^{240}(B_{j}+A_{\alpha_{j}})^{i_{j}}\\}$
for 0 $\leq i_{j}\leq p-1$,

where we put

$A_{\alpha}=A_{\epsilon_{1}+\epsilon_{2}}=x_{\epsilon_{1}+\epsilon_{2}^{p-1}}-x_{-(\epsilon_{1}+\epsilon_{2})}=g_{\epsilon_{1}+\epsilon_{2}}$,

$A_{-\epsilon_{1}-\epsilon_{2}}=c_{-\epsilon_{1}-\epsilon_{2}}+(h_{\epsilon_{1}+\epsilon_{2}}+1)^{2}+4x_{-\epsilon_{1}-\epsilon_{2}}x_{\epsilon_{1}+\epsilon_{2}},$

$A_{-(\epsilon_{1}\pm\epsilon_{k})}=x_{\epsilon_{3}-(\pm\epsilon_{k})}(c_{-\epsilon_{1}\pm\epsilon_{k}}+x_{\epsilon_{1}\pm\epsilon_{k}}x_{-(\epsilon_{1}\pm\epsilon_{k})}\pm
x_{-(\epsilon_{2}\mp\epsilon_{k})}x_{\epsilon_{2}+(\mp\epsilon_{k})})$,

$A_{-(\epsilon_{1}\pm\epsilon_{3})}=x_{\epsilon_{4}-(\pm\epsilon_{3})}(c_{-(\epsilon_{1}\pm\epsilon_{3})}+x_{\epsilon_{1}\mp\epsilon_{3}}x_{-(\epsilon_{1}\pm\epsilon_{3})}\pm
x_{-(\epsilon_{2}\mp\epsilon_{3})}x_{\epsilon_{2}+(\mp\epsilon_{3})},$

$A_{-(\epsilon_{2}\pm\epsilon_{k})}=x_{-\epsilon_{3}-(\pm\epsilon_{k})}(c_{-(\epsilon_{2}\pm\epsilon_{k})}+x_{\epsilon_{2}\pm\epsilon_{k}}x_{-(\epsilon_{2}\pm\epsilon_{k})}\pm
x_{-(\epsilon_{1}\mp\epsilon_{k})}x_{\epsilon_{1}\mp\epsilon_{k})}$,

$A_{-(\epsilon_{2}\pm\epsilon_{3})}=x_{\epsilon_{5}-(\pm\epsilon_{3})}(c_{-(\epsilon_{2}\pm\epsilon_{3})}+x_{\epsilon_{2}\pm\epsilon_{3}}x_{-(\epsilon_{2}\pm\epsilon_{3})}\pm
x_{-(\epsilon_{1}\mp\epsilon_{3})}x_{\epsilon_{1}\mp\epsilon_{3}})$,

$A_{\frac{1}{2}\Sigma_{i=1}^{8}(-1)^{k(i)}\epsilon_{i}}=x_{\frac{1}{2}\Sigma_{i=1}^{8}(-1)^{k(i)}\epsilon_{i}}$;
for
$\beta_{k(i)}=-\frac{1}{2}\epsilon_{1}-\frac{1}{2}\epsilon_{2}+\frac{1}{2}\Sigma_{i=3}^{8}(-1)^{k(i)}\epsilon_{i}$
and for
$\gamma_{k(i)}=\frac{1}{2}\epsilon_{1}+\frac{1}{2}\epsilon_{2}+\frac{1}{2}\Sigma_{i=3}^{8}(-1)^{k(i)}\epsilon_{i}$,

we set

$A_{\beta_{k(i)}}=x_{-\frac{1}{2}\Sigma_{i=3}^{8}(-1)^{k(i)}}\epsilon_{i}$$(C_{\beta_{k(i)}}+x_{\beta_{k(i)}}x_{-\beta_{k(i)}}\pm
x_{\gamma_{k(i)}}x_{-\gamma_{k(i)}}),$

$A_{\gamma_{k(i)}}=x_{\frac{1}{2}\Sigma_{i=3}^{8}(-1)^{k(i)}\epsilon_{i}}(c_{\gamma_{k(i)}}+x_{\beta_{k(i)}}x_{-\beta_{k(i)}}\pm
x_{\gamma_{k(i)}}x_{-\gamma_{k(i)}})$

respectively with the sign chosen so that they commute with $x_{\alpha}$ and
with $c_{\beta}\in F$ chosen so that $A_{-\epsilon_{1}}$ and parentheses( )
are invertible.

For any other root $\beta$, we put $A_{\beta}={x_{\beta}}^{2}$ or
$x_{\beta}^{3}$ if possible.
Otherwise we make use of the parentheses( )again used for designating
$A_{-\beta}$. So in this case we put $A_{\beta}={x_{\gamma}}^{2}$ or
${x_{\gamma}}^{3}$ attached to these ( ) so that $x_{\alpha}$ may commute with
$A_{\beta}$.
For the rest of the complete proof , refer to the consolidated proof in
proposition 4.12 below.

∎

Let $L$ be any modular Lie algebra of classical type or exceptional type as
listed in section 1 over any algebraically closed field $F$ of characteristic
$p\geq 7$.

For a root $\alpha\in\Phi$ for $L$, we put
$g_{\alpha}:=x_{\alpha}^{p-1}-x_{-\alpha}$ and
$w_{\alpha}:=(h_{\alpha}+1)^{2}+4x_{-\alpha}x_{\alpha}.$

According to [RS], we see that $w_{\alpha}$ is contained in the center of
$u(\mathfrak{sl}_{2}(F))$ and must satisfy an irreducible integral equation of
degree $p$ over the center $\mathfrak{Z}$ of $u(L)$.

######  Proposition 4.12.

Let $\chi$ be a character of any simple $L$-module with $\chi(x_{\alpha})\neq
0,\chi(x_{-\alpha})\neq 0$ or $\chi(h_{\alpha})\neq 0$ for some
$\alpha\in\Phi$, where $x_{\pm\alpha}$ and $h_{\alpha}$ are elements in the
Chevalley basis of $L$ such that
$Fx_{\alpha}+Fh_{\alpha}+Fx_{-\alpha}\cong\mathfrak{sl}_{2}(F)$ .

Then we claim that any simple $L$-module with character $\chi\neq 0$ is of
dimension $p^{m}=p^{n-l\over 2}$,where $n=dimL=2m+l$ for a CSA $H$ with
$dimH=l$.

###### Proof.

For any root $\alpha$, assume first that $\chi(x_{\alpha})\neq 0.$

We may see without difficulty that $\mathfrak{B}$ suggested in the previous
sections is a linearly independent set in $u(L)$ by virtue of P-B-W theorem.

We shall prove that a nontrivial linearly dependent equation leads to
absurdity. We assume first that we have a dependence equation which is of
least degree with respect to $h_{\alpha_{j}}\in H$ and the number of whose
highest degree terms is also least.

In case it is conjugated by $x_{\alpha}$, then there arises a nontrivial
dependence equation of lower degree than the given one,which contradicts to
our assumption.

Otherwise it reduces to the following form, which we have only to prove:

$(\ast)x_{\alpha}K$ \+ $K^{\prime}$ $\in$$\mathfrak{M}_{\chi}$ , where $K$ and
$K^{\prime}$ commute with $x_{\alpha}$ modulo $\mathfrak{M}_{\chi}$.

We thus deal with and assume $x_{-\alpha}$$x_{\alpha}$$K$ \+
$x_{-\alpha}$$K^{\prime\prime}$ $\in$ $\mathfrak{M}_{\chi}$.

From $w_{\alpha}:=(h_{\alpha}+1)^{2}+4x_{-\alpha}x_{\alpha}\in$ the center of
$u(\mathfrak{sl}_{2}(F)),$ we get
$4^{-1}\\{w_{\alpha}-(h_{\alpha}+1)^{2}\\}K+x_{-\alpha}K^{\prime\prime}\equiv
0$ modulo $\mathfrak{M}_{\chi}$.

If $x_{-\alpha}^{p}\equiv c$ which is a constant, then

$(\ast\ast)4^{-1}x_{-\alpha}^{p-1}\\{w_{\alpha}-(h_{\alpha}+1)^{2}\\}K+cK^{\prime\prime}\equiv
0$

is obtained.
From $(\ast),(\ast\ast)$, we have

$4^{-1}x_{-\alpha}^{p-1}\\{w_{\alpha}-(h_{\alpha}+1)^{2}\\}K-cx_{\alpha}K\newline
\equiv 0$ modulo $\mathfrak{M}_{\chi}$.

Multiplying $x_{\alpha}^{p-1}$ to this equation,we obtain

$(\ast\ast\ast)4^{-1}x_{\alpha}^{p-1}x_{-\alpha}^{p-1}\\{w_{\alpha}-(h_{\alpha}+1)^{2}\\}K-cx_{\alpha}^{p}K\equiv
0.$

By making use of $w_{\alpha}$ and the properties of prime ideal , we may
deduce from $(\ast\ast\ast)$ an equation of the form
( a polynomial of degree $\geq 1$ with respect to
$h_{\alpha})K-cx_{\alpha}^{p}K\equiv 0.$

Finally if we use conjugation and subtraction consecutively,then we are led to
a contradiction $K\in\mathfrak{M}_{\chi}.$ It may be necessary for us to
change the role of $K$ and $K^{\prime}$ to obtain the absurdity alike.

Next if $\chi(x_{\alpha})\neq 0$ or $\chi(x_{-\alpha})\neq 0$, then our
assertion is evident from the proof mentioned above.

So for the case $\chi(h_{\alpha})\neq 0$, we may assume $x_{\alpha}^{p}\equiv
x_{-\alpha}^{p}\equiv 0$ modulo $\mathfrak{M}_{\chi}$.

We may see without difficulty that $\mathfrak{B}$ as in the previous
propositions is a linearly independent set in $u(L)$ by virtue of P-B-W
theorem.

We shall prove that a nontrivial linearly dependent equation leads to
absurdity. We assume first that we have a dependence equation which is of
least degree with respect to $h_{\alpha_{j}}\in H$ and the number of whose
highest degree terms is also least.

In case it is conjugated by $g_{\alpha}$, then there arises a nontrivial
dependence equation of lower degree than the given one,which contradicts to
our assumption.

Otherwise it reduces to one of the following forms, so that we have only to
prove that

(i)$x_{\beta}K+K^{\prime}\in\mathfrak{M}_{\chi}$,

(ii) $g_{\alpha}K+K^{\prime}\in\mathfrak{M}_{\chi}$

lead to a contradiction, where both $K$ and $K^{\prime}$ commute with
$x_{\pm\alpha}$ modulo $\mathfrak{M}_{\chi}$. In particular $K$ commute with
$g_{\alpha}$.

For the case (i), we may change it to the form
$x_{\alpha}K+K^{\prime\prime}\in\mathfrak{M}_{\chi}$ for some
$K^{\prime\prime}$ commuting with $x_{\alpha}$ modulo $\mathfrak{M}_{\chi}$.

So we have $x_{\alpha}^{p}K+x_{\alpha}^{p-1}K^{\prime\prime}\equiv 0$, and
hence $x_{\alpha}^{p-1}K^{\prime\prime}\equiv 0$.

Subtracting from this
$x_{-\alpha}x_{\alpha}K+x_{-\alpha}K^{\prime\prime}\equiv 0$, we get

$-x_{-\alpha}x_{\alpha}K+g_{\alpha}K^{\prime\prime}\equiv 0$. Recall here that
$g_{\alpha}$ is invertible and $w_{\alpha}$ belongs to the center of
$u(\mathfrak{sl}_{2}(F))$.

So we get
$4^{-1}\\{(h_{\alpha}+1)^{2}-w_{\alpha}\\}K+g_{\alpha}K^{\prime\prime}\equiv
0$, and hence

$(\ast)g_{\alpha}^{p-1}4^{-1}\\{(h_{\alpha}+1)^{2}-w_{\alpha}\\}K+cK^{\prime\prime}\equiv
0$

is obtained and from the start equation we have

$(\ast\ast)cx_{\alpha}K+cK^{\prime\prime}\equiv 0$, where
$g_{\alpha}^{p}-c\equiv 0$.

Subtracting $(\ast\ast)$ from $(\ast)$, we have
$4^{-1}g_{\alpha}^{p-1}\\{(h_{\alpha}+1)^{2}-w_{\alpha}\\}K-cx_{\alpha}K\equiv
0$.

Multiplying this equation by $g_{\alpha}^{1-p}$ to the right, we obtain
$4^{-1}g_{\alpha}^{p-1}\\{(h_{\alpha}+1)^{2}-w_{\alpha}\\}g_{\alpha}^{1-p}K-cx_{\alpha}g_{\alpha}^{1-p}K\equiv
0$

We thus have
$4^{-1}\\{(h_{\alpha}+1-2)^{2}-w_{\alpha}\\}K-x_{\alpha}g_{\alpha}K\equiv 0$.

So it follows that
$4^{-1}\\{(h_{\alpha}-1)^{2}-w_{\alpha}\\}K+x_{\alpha}x_{-\alpha}K\equiv 0$.

Next multiplying $x_{-\alpha}^{p-1}$ to the right of this last equation, we
obtain $\\{(h_{\alpha}-1)^{2}-w_{\alpha}\\}Kx_{-\alpha}^{p-1}\equiv 0$. Now
multiply $x_{\alpha}$ in turn consecutively to the left of this equation until
it becomes of the form

( a nonzero polynomial of degree $\geq 1$ with respect to $h_{\alpha})K$
$\in\mathfrak{M}_{\chi}$.

By making use of conjugation and subtraction consecutively and by virtue of
properties of prime ideals, we are led to a
contradiction.$K\in\mathfrak{M}_{\chi}$.

Finally for the case (ii),we consider
$K+g_{\alpha}^{-1}K^{\prime}\in\mathfrak{M}_{\chi}$. So we have
$x_{\alpha}K+x_{\alpha}g_{\alpha}^{-1}K^{\prime}\equiv 0$ modulo
$\mathfrak{M}_{\chi}$.

By analogy with the argument as in the case (i), we obtain a contraiction
$K\in\mathfrak{M}_{\chi}$.

∎

We are now concerned with some prototype of exemplary subregular points before
closing this section.

Sometimes we are preferably fond of the $A_{l}$\- type because of the
simplicity of its root system. By making use of this type we would like to
easily figure out in this section an indecomposable Lie algebra having certain
subregular points.

######  Proposition 4.13.

Let $L$ be any Lie algebra of $A_{l}$-type over an algebraically closed field
$F$ of characteristic $p\geq 7$. Then L becomes a Hypo- Lie algebra, which is
also clear for other classical type Lie algebras.

###### Proof.

Refer to previous propositions of this paper and the chapter4 in [KY-4]. ∎

Because we proved that all modular $A_{l}$\- type Lie algebras for
characteristic $p\geq 7$ have no subregular point,i.e., $S(L,p,\chi)=\phi$, we
need to find out in connection with this $A_{l}$\- type Lie algebra some
example of nontrivial indecomposable modular Lie algebra which has no
subregular point.

By making use of the fact that $sl_{2}(F)$ has no subregular point for
$p>2$(refer to [RS],[KY-4]), we can extend this one to $L_{5}$ which is
explained below.

We shall show that a certain indecomposable Lie algebra over $F$ has a
subregular point even for $p\geq 7$, whose fact is notable related to Kim’s
conjecture[KY-6].

######  Proposition 4.14.

Let $L_{5}$ be a Lie algebra generated by

$x_{\epsilon_{j}-\epsilon_{2}},x_{\epsilon_{2}-\epsilon_{j}},h_{\epsilon_{2}-\epsilon_{j}},x_{\epsilon_{1}-\epsilon_{2}},x_{\epsilon_{1}-\epsilon_{j}}(j\neq
1,2)$ over an algebraically closed field $F$ of characteristic $p>2$, where
all these generators are included in classical $A_{l}$-type Lie algebra.
Suppose that $\chi$ is a character of $L_{5}$ s.t.
$\chi(x_{\epsilon_{j}-\epsilon_{2}})\neq 0$ or
$\chi(x_{\epsilon_{2}-\epsilon_{j}})\neq 0$ or
$\chi(h_{\epsilon_{2}-\epsilon_{j}})\neq 0$.

Then we claim that $L_{5}$ is centerless and indecomposable with
$S(L_{5},p,\chi)\neq\phi$.

###### Proof.

Because $L_{5}$ is a proper sub-Lie algebra of an $A_{l}$-type Lie algebra, it
is easy to see that $L_{5}$ is a centerless and indecomposable modular Lie
algebra. Let
$<x_{\epsilon_{1}-\epsilon_{2}},x_{\epsilon_{1}-\epsilon_{j}}>_{F}$ denote the
sub- Lie algebra of $L$ generated by $x_{\epsilon_{1}-\epsilon_{2}}$ and
$x_{\epsilon_{1}-\epsilon_{j}}$ over $F$.

It is just a proper ideal of $L_{5}$, so that we have a projective
homomorphism $\pi$ of Lie algebras

$\pi:L_{5}\rightarrow
L_{5}/<x_{\epsilon_{1}-\epsilon_{2}},x_{\epsilon_{1}-\epsilon_{j}}>_{F}$.

Since we have $[Q(u(L_{5})):Q(\mathfrak{Z}(u(L_{5})))]\geq p^{3}$, it follows
that $[Q(u(L_{5})):Q(\mathfrak{Z}(u(L_{5})))]=p^{4}$, and so the maximal
dimension of irreducible $L_{5}$-modules is $p^{2}$.

Hence evidently there exists a subregular point for this $L_{5}$,i.e.,
$S(L_{5},p,\chi)\neq\phi$ in view of definitions in section 3.

∎

We remark here that the codomain of the projective homomorphism $\pi$ is
isomorphic to $sl_{2}(F)$ and the Lie algebra $L_{5}$ over $F$ in the domain
has dimension 5.

In [KWa] we made a remark conjecturing that some extended Lie algebras of
classical modular Lie algebras must be Hypo-Lie algebras relating to
subregular points.

If we have regular points almost everywhere except for a finite number of
subregular points and $p$-points related to a Lie algebra $L$ included in a
classical modular Lie algebra and if we can find Lee’s bases associated to all
these regular points , then we called $L$ a Hypo- Lie algebra.

So in connection with such concepts, the Lie algebra $L_{5}$ above in this
paper is no way Hypo- Lie algebra.

We thus conclude that $L_{5}$ closely related to $A_{l}$\- type is a
nontrivial centerless indecomposable Lie algebra having infinitely many
subregular points over an algebraically closed field of characteristic $p>2$.

##  5\. proof of $P\neq NP$ and a conjecture

Now in this section we come back to our main story related to a conjecture.

Because the $B_{l}$\- type Lie algebra looks like a general prototype out of
all classical Lie algebras and we used it already in [NWK-1,2], so we would
like to still use it preferably.

For a while we fix an algebraically closed field $F$ of characteristic $p\geq
7$ unless otherwise specified until we remark on our conjecture related to
$NP$completeness.

As we are well aware, there exists the universal Casimir element $s\in u(L)$
for any $B_{l}$\- type classical Lie algebra $L$. Let the rank $l$ of $L$ is
$l\geq 3$,i.e., the dimension of CSA $\geq 3.$

We may express the universal Casimir element $s$ as
$s=\sum_{\alpha\in\Phi^{+}}a_{\alpha}x_{\alpha}x_{-\alpha}+\sum_{i=1}^{l}b_{i}h_{\alpha_{i}}+\sum_{i,j}a_{ij}h_{\alpha_{i}}h_{\alpha_{j}}$,
where $0\neq b_{i}\in F$ , $a_{\alpha},a_{ij}\in F$, $H$ is a Cartan
subalgebra(abbreviated as CSA) with a basis $\\{h_{\alpha_{i}}|1\leq i\leq
l\\}$, and $\\{x_{\alpha},\alpha\in\Phi;h_{\alpha_{i}},1\leq i\leq l\\}$ is
the standard Chevalley basis of $L$ with a root system $\Phi$ including
$\Phi^{+}$as a set of positive roots.

Here we rearrange the Chevalley basis as
$\\{h_{\alpha_{1}},h_{\alpha_{2}},\cdots,h_{\alpha_{l}};x_{\alpha_{1}},x_{-\alpha_{1}},\cdots,x_{\alpha_{l}},x_{-\alpha_{l}},$
$x_{\alpha_{l+1}},x_{-\alpha_{l+1}},\cdots,x_{\alpha_{m}},x_{-\alpha_{m}}\\}$,
where dim $L=n=l+2m$, and $l=rankL=dimH$ over $F$.

Clearly we have $x_{\alpha_{i}}^{[p]}=x_{-\alpha_{i}}^{[p]}=0$ for $1\leq
i\leq m$ and $h_{\alpha_{j}}^{[p]}=h_{\alpha_{j}}$ for $1\leq j\l\leq l$. We
shall denote the center of $u(L)$ simply by $\mathfrak{Z}$ as before.

######  Proposition 5.1.

We have the following in the quotient algebra $Q(u(L))$:

(i)$dim_{Q(\mathfrak{Z})}Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})=p^{l}$,
(ii)$Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})$ becomes a Galois
field over $Q(\mathfrak{Z})$.

###### Proof.

Since $Q(u(L))$ becomes a central simple Artinian algebra over
$Q(\mathfrak{Z})$ which is actually a division algebra,we obtain (i)
immediately from the Noether-Skolem theorem and from the identities such as
$h_{\alpha_{i}}x_{\alpha_{i}}=x_{\alpha_{i}}(h_{\alpha_{i}}+2)$ for
$i=1,2,\cdots,l$ and
$[Q(\mathfrak{Z})(h_{\alpha_{1}},h_{\alpha_{2}},\cdots,h_{\alpha_{l}}):Q(\mathfrak{Z})]=[Q(\mathfrak{Z})(h_{\alpha_{1}},h_{\alpha_{2}},\cdots,h_{\alpha_{l}}):Q(\mathfrak{Z})(h_{\alpha_{1}},h_{\alpha_{2}},\cdots,h_{\alpha_{l-1}})]\cdot$

$[Q(\mathfrak{Z})(h_{\alpha_{1}},h_{\alpha_{2}},\cdots,h_{\alpha_{l-1}}):Q(\mathfrak{Z})(h_{\alpha_{1}},h_{\alpha_{2}},\cdots,h_{\alpha_{l-2}})]\cdots$

$[Q(\mathfrak{Z})(h_{\alpha_{1}}):Q(\mathfrak{Z})]$ for $i=1,2,\cdots,l$.

(ii) is an immediate consequence of (i) considering that each bracket [ ] in
the above factorization gives rise to $p$\- distinct conjugates of
$h_{\alpha_{i}}$ for $j=1,2,\cdots,l$.

∎

######  Proposition 5.2.

We may obtain the irreducible polynomial $Irr(s,\mathcal{O}(L))$ of $s$ over
the $p$-center $\mathcal{O}(L)$ by expanding out the following norm

$(\ast)$
$N_{Q(\mathfrak{Z})}^{Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})}$$\\{s_{i}-(b_{1}h_{\alpha_{1}}+\cdots
b_{l}h_{\alpha_{l}}+\sum\limits_{i,j}a_{ij}h_{\alpha_{i}}h_{\alpha_{j}})\\}=N_{Q(\mathfrak{Z})}^{Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})}(\sum\limits_{\alpha\in\Phi^{+}}a_{\alpha}x_{\alpha}x_{-\alpha})$
. Moreover we have $deg(Irr(s,\mathcal{O}(L))=p^{l}$ and $s$ becomes separable
over $\mathcal{O}(L)$ and so over $Q(\mathcal{O}(L))$.

###### Proof.

Firstthing we should perceive that

the left hand side of $(\ast)=s^{p^{l}}+\hat{a}_{1}s^{p^{l}-1}+$
$\cdots+\hat{a}_{p^{l}-1}s+\hat{a}_{p^{l}}$ for some $\hat{a}_{i}\in
Q(\mathfrak{Z})(h_{\alpha_{1}},h_{\alpha_{2}}.\cdots,h_{\alpha_{l}})$ for
$i=1,2,\dots,p^{l}$.

Here we contend that $\hat{a}_{i}\in Q(O(L))$ in fact. We choose any distinct
$p^{l}$\- elements $s_{i}\in\mathcal{O}(L)$ and take norms such as
$N_{Q(\mathfrak{Z})}^{Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})}$$\\{s_{i}-(b_{1}h_{\alpha_{1}}+\cdots+b_{l}h_{\alpha_{l}}+\sum\limits_{i,j}a_{i,j}h_{\alpha_{i}}h_{\alpha_{j}})\\}=N_{Q(\mathfrak{Z})}^{Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})}\\{s_{i}-s+(\sum\limits_{\alpha\in\Phi^{+}}a_{\alpha}x_{\alpha}x_{-\alpha})\\}$.

We know that the Noether -Skolem theorem allows us to extend every
automorphism of $Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})$ to an
inner automorphism of $Q(u(L)).$

So if we conjugate this by some elements in $Q(u(L))$, then we obtain
$p^{l}$\- distinct automorphisms of
$Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})$ over
$Q(\mathfrak{Z})$.

Next we need to note that
$[Q(\mathcal{O}(L))(h_{\alpha_{1}},\cdots,h_{\alpha_{l}}):Q(\mathcal{O}(L)]=[Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}}):Q(\mathfrak{Z})]=p^{l}$
and that isomorphisms of
$Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})$ over $Q(\mathfrak{Z})$
are the same as those of
$Q(\mathcal{O}(L))(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})$ over
$Q(\mathcal{O}(L))$.

So we obtain

$N_{Q(\mathfrak{Z})}^{Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})}$$\\{s_{i}-(b_{1}h_{\alpha_{1}}+\cdots+b_{l}h_{\alpha_{l}}+\sum\limits_{i,j}a_{i,j}h_{\alpha_{i}}h_{\alpha_{j}})\\}=N_{Q(\mathcal{O}(L))}^{Q(\mathcal{O}(L))(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})}$
$\\{s_{i}-(b_{1}h_{\alpha_{1}}+\cdots+b_{l}h_{\alpha_{l}}+\sum\limits_{i,j}a_{i,j}h_{\alpha_{i}}h_{\alpha_{j}})\\}\in
Q(\mathcal{O}(L))$,

which should boil down to the form
$N_{Q(\mathfrak{Z})}^{Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})}\\{s_{i}-s+(\sum\limits_{\alpha\in\Phi^{+}}a_{\alpha}x_{\alpha}x_{-\alpha})\\}$.

On the other hand, if we put

$N_{Q(\mathcal{O}(L))}^{Q(\mathcal{O}(L))(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})}$
$\\{s_{i}-(b_{1}h_{\alpha_{1}}+\cdots+b_{l}h_{\alpha_{l}}+\sum\limits_{i,j}a_{i,j}h_{\alpha_{i}}h_{\alpha_{j}})\\}=:s_{i}^{p^{l}}+\hat{b}_{1}s_{i}^{p^{l-1}}+\cdots,\hat{b}_{p^{l}-1}s_{i}+\hat{b}_{p^{l}}=k_{i}$
for some $k_{i}\in Q(\mathcal{O}(L))$ and $\hat{b}_{j}\in
Q(\mathcal{O}(L))(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})$ for
$j=1,2.\cdots,p^{l}$,

then we have a system in indeterminates $\hat{b}_{j}$ of $p^{l}$\-
nonhomogeneous linear equations with the determinant of coefficients: $\det
A=\begin{vmatrix}1&s_{1}\cdots s_{1}^{p^{l}-1}\\\ \vdots&\ddots\vdots\\\
1&s_{p^{l}}\cdots s_{p^{l}}^{p^{l}-1}\end{vmatrix}$

$=\Pi_{i<j}(s_{j}-s_{i})\neq 0$, which turns out to be the so called
$Vandermonde$ determinant.

If we make use of the Cramer’s formula, then we may have the solutions
$\hat{b}_{j}\in Q(\mathcal{O}(L))$ of the system above. So we get
$\hat{a}_{j}=\hat{b}_{j}$ for $j=1,2,\cdots,p^{l}$, and hence $\hat{a}_{j}\in
Q(\mathcal{O}(L))$ follows.

By the way , since $s$ is integral over $\mathcal{O}(L)$, we have that

the right hand side of
$(\ast)=N_{Q(\mathfrak{Z})}^{Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})}(\sum_{\alpha\in\Phi^{+}}a_{\alpha}x_{\alpha}x_{-\alpha})$
also belongs to $Q(\mathcal{O}(L))$ which is secured by the following lemma.

∎

######  Lemma 5.3.

If we suppose that $s$ satisfies an algebraic equation of the form
$f(\mathbb{X}):=\mathbb{X}^{p^{l}}+\hat{a}_{1}\mathbb{X}^{p^{l}-1}+\cdots+\hat{a}_{p^{l}-1}\mathbb{X}+\hat{a}_{p^{l}}=0$
with $\hat{a}_{j}\in Q(\mathcal{O}(L))$ for $1\leq j\leq p^{l}-1,$ which is
the same form as in the above argument for the left hand side of norm equation
$(\ast)$,

then we have that $\hat{a}_{p^{l}}$ also belongs to $Q(\mathcal{O}(L))$

###### Proof.

Thanks to proposition5.1, we may have $p^{l}$\- distinct automorphisms
$\sigma_{i}$ of $Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})$ over
$Q(\mathfrak{Z})$ , which are represented by inner automorphisms of $Q(L)$.

Next the integral equation of $s$ over $\mathcal{O}(L)$ is just
$Irr(s,Q(\mathcal{O}(L))$ itself because $\mathcal{O}(L)$ becomes the Noether
normalization of $\mathfrak{Z}$ and hence a unique factorization domain.

Now the map
$s\mapsto\sigma_{i}({b}_{1}h_{\alpha_{1}}+\cdots+b_{l}h_{\alpha_{l}}+\sum\limits_{i,j}a_{ij}h_{\alpha_{i}}h_{\alpha_{j}})+\sigma_{j}(\sum\limits_{\alpha\in\Phi^{+}}a_{\alpha}x_{\alpha}x_{-\alpha})$
for $1\leq i,j\leq p^{l}$

gives rise to an isomorphism of the algebra
$Q(\mathcal{O}(L)(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})(s)$ over
$Q(\mathcal{O}(L))$.

Hence the irreducible polynomial $Irr(s,Q(\mathcal{O}(L)))$ must divide the
polynomial $f(\mathbb{X})$, and thus we have that
$f(\mathbb{X})=Irr(s,Q(\mathcal{O}(L)))\cdot g(\mathbb{X})$ for some unique
$g(\mathbb{X}):=\mathbb{X}^{p^{l}-p^{l^{\prime}}}+c_{1}\cdot\mathbb{X}^{p^{l}-p^{l^{\prime}}-1}+\cdots+c_{p^{l}-p^{l^{\prime}}}$
with $c_{j}\in Q(\mathfrak{Z})$ and $p^{l}-p^{l^{\prime}}\geq 0$.

Seeing that $c_{j}$’s are uniquely determined only by the coefficients of
$Irr(s,Q(\mathcal{O}(L)))$ and those of terms
$\mathbb{X}^{p^{l}},\mathbb{X}^{p^{l}-1},\cdots,\mathbb{X}^{p^{l^{\prime}}+1},\mathbb{X}^{p^{l^{\prime}}}$

in $f(\mathbb{X})$ and that those coefficients belong to $Q(\mathcal{O}(L))$,
we know that our claim is proven.

∎

Now coming back to our main proof, we must still show that the algebraic
equation of $s$ obtained just above is the $Irr(s,Q(\mathcal{O}(L)))$, which
is also the integral equation of $s$ over $\mathcal{O}(L).$ Such a fact is
attributed to the following another lemma.

######  Lemma 5.4.

In this lemma only,

let $E$ be a finite extension field of arbitrary field $F$ with nonzero
characteristic $p$ and let $\alpha,\beta,\gamma$ be elements of $E-F$ with
$[F(\alpha)(\gamma):F]=p^{n}$.

Suppose that
$f(\mathbb{X}):=Irr(\alpha,F)=(\mathbb{X}-\sigma_{1}(\alpha))\cdots(\mathbb{X}-\sigma_{p^{m}}(\alpha))$
for some distinct $\sigma_{i}(\alpha)\in F(\alpha)$ for
$i=1,2,\cdots,p^{m}\leq p^{n}$ with
$\sigma_{i}\in{isomorphisms\;of\;E\,over\,F}\;\newline
and\;that\;\beta=\alpha+\gamma$ is an element such that
$\forall\sigma_{i},\beta$ satisfies $\sigma_{i}(\beta)=\beta.$

Suppose further that $\Pi_{i=1}^{p^{m}}\sigma_{i}(\gamma)\in F$ and
$g(\mathbb{X}):=\Pi_{i=1}^{p^{m}}(\mathbb{X}-\sigma_{i}(\alpha))-\Pi_{i=1}^{p^{m}}\sigma_{i}(\gamma)\in
F[\mathbb{X}].$

Then we have $g(\mathbb{X})=Irr(\beta,F)$ which is separable over $F$.

###### Proof.

It is not difficult to see that $\gamma\notin F(\alpha)$, and hence there
exists at least $p^{m+1}$\- distinct isomorphisms of $F(\alpha)\ni\beta$ over
$F$.

Next we consider a field lattice diagram as follows:

$F(\alpha)(\gamma)\ni\beta$

at least p-dimensional $|$

$F(\alpha)$

$p^{m}$-dimensional $|$

$F$

Given any nontrivial isomorphism $\tau$ of $F(\alpha)(\gamma)$ over
$F(\alpha)$, we must have
$\tau(\beta)=\tau(\alpha+\gamma)=\tau(\sigma_{i}(\alpha)+\sigma_{i}(\gamma))=\sigma_{i}(\alpha)+\overline{\tau}(\gamma)$
for some isomorphism $\bar{\tau}$ of $F(\alpha)(\gamma)$ over $F(\gamma)$ and
$\forall i$ with $1\leq i\leq p^{m}$.

Because $F(\alpha)$ is a Galois extension of $F$, we see that there are at
least $p^{m}$\- distinct conjugates of $\beta$ over $F$. However the degree of
$g(\mathbb{X})$ becomes $degg(\mathbb{X})=p^{m}$ and clearly $g(\beta)=0$, so
that $Irr(\beta,F)$ also has the same degre $p^{m}$. In addition no doubt
$g(\mathbb{X})$ becomes separable over $F$. ∎

Now so as to finish our proof of the main proposition5.2, we put
$F:=Q(\mathcal{O}(L)),\beta:=s,\alpha:=b_{1}h_{\alpha_{1}}+\cdots+b_{l}h_{\alpha_{l}}+\sum\limits_{i,j}a_{ij}h_{\alpha_{i}}h_{\alpha_{j}},\gamma:=\sum\limits_{\alpha\in\Phi^{+}}a_{\alpha}x_{\alpha}x_{-\alpha}$
and make use of automorpisms $\\{\sigma_{i}|i=1,2,\cdots,p^{l}\\}$ of
$Q(\mathfrak{Z})(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})$ over $Q(\mathfrak{Z})$
which are the same automorphisms of
$Q(\mathcal{O}(L))(h_{\alpha_{1}},\cdots,h_{\alpha_{l}})$ over
$Q(\mathcal{O}(L))$. Finally applying the preceding lemma , we are done after
all.

######  Proposition 5.5.

We obtain $\mathfrak{Z}=\mathcal{O}(L)[s]$.

###### Proof.

According to proposition5.2, we must have
$Q(\mathfrak{Z})=Q(\mathcal{O}(L)[s])$. Since $\mathfrak{Z}$ becomes a
finitely generated $\mathcal{O}(L)$\- module , we have that
$\mathcal{O}(L)[s]$ is completely closed in $\mathfrak{Z}$,i.e.,any nontrivial
quotients of $\mathcal{O}(L)[s]$ is not contained in
$\mathfrak{Z}-\mathcal{O}(L)[s]$. We can explain this explicitly as follows.

Assume first that some $\mu\in\mathfrak{Z}-\mathcal{O}(L)[s]$ satisfies an
equation of the form $\mu\cdot f=\hat{f}$ for some distinct polynomials
$f,\hat{f}$ in
$F[x_{\alpha_{1}}^{p},x_{-\alpha_{1}}^{p},h_{\alpha_{1}}^{p}-h_{\alpha_{1}},\cdots,x_{\alpha_{l}}^{p},x_{-\alpha_{l}}^{p},h_{\alpha_{l}}^{p}-h_{\alpha_{l}}$,

$x_{\alpha_{l+1}}^{p},x_{-\alpha_{l+1}}^{p},\cdots,x_{\alpha_{m}}^{p},x_{-\alpha_{m}}^{p},s]$

satisfying that a nontrivial fraction $\mu=\hat{f}/f$ is reduced and that
$\mu,s$ are integral over $\mathcal{O}(L)$.

Noticing that
$x_{\alpha_{1}}^{p},x_{-\alpha_{1}}^{p},h_{\alpha_{1}}^{p}-h_{\alpha_{1}},\cdots,x_{\alpha_{l}}^{p},x_{-\alpha_{l}}^{p},h_{\alpha_{l}}^{p}-h_{\alpha_{l}},$

$x_{\alpha_{l+1}}^{p},x_{-\alpha_{l+1}}^{p},\cdots,x_{\alpha_{m}}^{p},x_{-\alpha_{m}}^{p}$
are all algebraically independent variables and that the fraction $\hat{f}/f$
should always be defined for all these variables , we perceive that the
denominator $f$ must be 1, a contradiction to our assumption. So we obtain our
required result.

∎

######  Proposition 5.6.

Suppoose in particular that $L$ is $B_{2}$\- type Lie algebra over an
algebraically closed field of characteristic $p\geq 7$;

then we have that the quotient algebra $Q(u(L))$ becomes a crossed product in
the Brauer group $B(Q(\mathfrak{Z}))$ of
$Q(\mathfrak{Z})(\alpha)(x_{\alpha_{1}}x_{-\alpha_{1}}+\cdots,x_{\alpha_{m}}x_{-\alpha_{m}})$
with $\alpha$ as in the proof of proposition5.4.

###### Proof.

We must notice first that $dimL=5\times rankL$.

So we see immediately that $Q(u(L))$ becomes a crossed product in the Brauer
group $B(Q(\mathfrak{Z}))$ of
$Q(\mathfrak{Z})(\alpha)(x_{\alpha_{1}}x_{-\alpha_{1}}+\cdots,x_{\alpha_{m}}x_{-\alpha_{m}})$
keeping in mind the above propositions. ∎

Now from here on, we fix our field $F$ to be algebraically closed field of
characteristic $p\geq 7$.

Furthermore we assume that this field $F$ is a $p$-adic field with its
valuation ring $A$ whose maximal ideal is denoted by $\mathfrak{p}$.

For the purpose of constructing our serious conjecture, relating to
$NP$-hardness we consider only such a field henceforth.

We let $L$ be any $B_{l}$-type Lie algebra with rank $L$= $l\geq 3$ over an
algebraically closed field $F$ of a $p$-adic field of characteristic $p\geq
7$. Let $A$ be the valuation ring of $F$ with $\mathfrak{p}$ the maximal ideal
of $A$ definitely once and for all.

The Steinberg module $V$ associated with a regular $p$\- point arises from an
irreducible representation $\rho_{\chi}:u(L)\rightarrow End_{F}(V)$ with
$\chi=0$ and dim $V=p^{m}$,where dim $L=n=2m+l$.

We shall use such a particular module for our conjecture as in [NWK-1,2].

######  Proposition 5.7.

Let notations be as in the refernce [NWK-1,2].

Then the counting problem in the reference [NWK-1,2] is an $NP$\- problem.
However it cannot belong to the $P$\- class,i.e., it is not a $P$-problem.

###### Proof.

We made use of a choice fuction which gives rise to a system of nonhomogeneous
linear equations including arbitrary random variables of cardinality $p^{2m}$,
where $[Q(u(L)):Q(\mathfrak{Z}(u(L)))]=p^{2m}$.

Here $L$ is an $n$\- dimensional $B_{l}$\- type classical Lie algebra of rank
$l$ with $n=l+2m$ over an algebraically closed field of a $\mathfrak{p}$\-
adic field of characteristic $p\geq 7$ and $Q$ indicates a quotient algebra of
a noncommutative algebra.

Let $s$ be the universal Casimir element of $u(L)$. Any maximal ideal
$\mathfrak{M}_{\chi}$ of $u(L)$ associated with a character $\chi\in L^{\ast}$
must contain an ideal of the form

$\\{\sum_{i=1}^{n}u(L)(\breve{x}_{i}-\breve{x}_{i}^{[p]}-\xi_{i})\\}+u(L)(s-\xi_{n+1})$,
where $\xi_{j}$’s are numbers in $F$, and $\breve{x}_{i}$ represents the basis
elements of $L$ written as ordered elements like
$x_{\alpha_{1}},x_{\alpha_{-1}},h_{\alpha_{1}},\cdots,x_{\alpha_{i}},x_{-\alpha_{i}},h_{\alpha_{i}},x_{\alpha_{i+1}},x_{-\alpha_{i+1}},\cdots,x_{\alpha_{m}},x_{-\alpha_{m}}$.

Moreover $dim_{F}u(L)/\mathfrak{M}_{\chi}\leq p^{n-l}$
considering $[Q(u(L)):Q(\mathfrak{Z}(u(L)))]=p^{n-l}=p^{2m}$.

For the purpose of proving our claim, we may make use our earlier propositions
in section3. However we would like to take, a regular $p$\- point associated
with a $B_{l}$\- type classical Lie algebra, which was used earlier in
[NWK-1,2].

We see the Steinberg module $V$ over $L$ has the maximal dimension $p^{m}$
with its associated point $(0,\cdots,0,\xi_{n+1})$ and with its associated
character $\chi=0$.

So the Steinberg module is closely connected with the restricted
representation. We may assume $\xi_{i+1}=0$ without loss of generality.

We put $S_{\alpha_{i}}:=Fx_{\alpha_{i}}+Fx_{-\alpha_{i}}+h_{\alpha_{i}}$ for
$1\leq i\leq m,g_{\alpha_{i}}:=x_{\alpha_{i}}^{p-1}-x_{-\alpha_{i}}$ for
$1\leq i\leq l$ and
$w_{\alpha_{i}}:=(h_{\alpha_{i}}+1)^{2}+4x_{-\alpha_{i}}x_{\alpha_{i}}$ for
$1\leq i\leq m$.

There exists $w_{\alpha_{j}}$ for $1\leq j\leq m$ with $2m=n-l$ such that
$w_{\alpha_{j}}$ acts as zero for each irreducible quotient $S_{\alpha_{j}}$\-
module for the composition series of $S_{\alpha_{j}}$\- module $V$.

In this case $g_{\alpha_{j}}$ is invertible in $u(L)/\mathfrak{M}_{\chi}$.We
exibit the basis of $u(L)/m_{\chi}$ as follows.

As a preliminary step we put
$B_{i}:=b_{i_{1}}h_{\epsilon_{1}-\epsilon_{2}}+b_{i_{2}}h_{\epsilon_{2}-\epsilon_{3}}+\cdots+b_{i_{l}}h_{\epsilon_{l}-\epsilon_{l+1}}$for
$l\geq 2$ and for $1\leq i\leq 2m$, where $(b_{i_{1}},\cdots,b_{i_{l}})\in
F^{l}$ is chosen so that any $(l-)B_{j}$’s are linearly independent in
$P^{l}(F)$ and the $\mathfrak{B}$ below becomes an $F$-linearly independent
set in $u(L)$ if necessary for this particular $\alpha_{j}$ and $B_{i}\cdot
x_{\alpha_{j}}\not\equiv x_{\alpha_{j}}\cdot B_{i}$ modulo
$\mathfrak{M}_{\chi}$.

(I) We assume first $\alpha_{j}$ is a long root.

We may assume $\alpha_{j}=\epsilon_{1}-\epsilon_{2}$ without loss of
generality.

Next we put
$\mathfrak{B}:=\\{(B_{1}+A_{\epsilon_{1}-\epsilon_{2}})^{i_{1}}\otimes(B_{2}+A_{-(\epsilon_{1}-\epsilon_{2})})^{i_{2}}\otimes\cdots\otimes(B_{2i-2}+A_{-(\epsilon_{l-1}-\epsilon_{l})})^{i_{2l-2}}\otimes(B_{2l-1}+A_{\epsilon_{l}})^{2l-1}\otimes(B_{2l}+A_{-\epsilon_{l}})^{i_{2l}}\otimes(\otimes_{j=2l+1}^{2m}(B_{j}+A_{\alpha_{j}})^{i_{j}})\\}$
for $0\leq i_{j}\leq p-1$,

where we designate

$A_{\epsilon_{1}-\epsilon_{2}}=g_{\epsilon_{1}-\epsilon_{2}},$

$A_{\epsilon_{2}-\epsilon_{1}}=g_{\epsilon_{1}-\epsilon_{2}}^{2}(C_{\epsilon_{2}-\epsilon_{1}}+w_{\epsilon_{1}-\epsilon_{2}}),$

$A_{\epsilon_{1}\pm\epsilon_{3}}=g_{\epsilon_{1}-\epsilon_{2}}^{3or4}(C_{\epsilon_{1}\pm\epsilon_{3}}+x_{\epsilon_{1}\pm\epsilon_{3}}x_{-(\epsilon_{1}\pm\epsilon_{3})}\pm
x_{\epsilon_{2}\pm\epsilon_{3}}\cdot x_{-(\epsilon_{2}\pm\epsilon_{3})}),$

$A_{-\epsilon_{1}-\epsilon_{3}}=g_{\epsilon_{1}-\epsilon_{2}}^{5}(C_{-\epsilon_{1}-\epsilon_{3}}+x_{\epsilon_{1}+\epsilon_{3}}\cdot
x_{-\epsilon_{1}-\epsilon_{3}}\pm x_{\epsilon_{2}+\epsilon_{3}}\cdot
x_{-\epsilon_{2}-\epsilon_{3}},$

$A_{\epsilon_{3}-\epsilon_{1}}=(C_{\epsilon_{3}-\epsilon_{1}}+x_{\epsilon_{1}-\epsilon_{3}}\cdot
x_{-(\epsilon_{1}-\epsilon_{3})}\pm x_{\epsilon_{2}-\epsilon_{3}}\cdot
x_{-(\epsilon_{2}-\epsilon_{3})}),$

$A_{\epsilon_{2}}=x_{\epsilon_{3}}(C_{\epsilon_{2}}+x_{\epsilon_{1}-\epsilon_{3}}\cdot
x_{-(\epsilon_{1}-\epsilon_{3})}\pm x_{\epsilon_{2}-\epsilon_{3}}\cdot
x_{-(\epsilon_{2}-\epsilon_{3})}),$

$A_{-\epsilon_{2}}=x_{-\epsilon_{3}}^{2}(C_{-\epsilon_{2}}+x_{\epsilon_{1}-\epsilon_{3}}\cdot
x_{-(\epsilon_{1}-\epsilon_{3})}\pm x_{\epsilon_{2}-\epsilon_{3}}\cdot
x_{-(\epsilon_{2}-\epsilon_{3})}),$

$A_{\epsilon_{1}}=x_{\epsilon_{3}}^{2}(C_{\epsilon_{2}-\epsilon_{3})}+x_{\epsilon_{1}-\epsilon_{3}}\cdot
x_{-(\epsilon_{1}-\epsilon_{3})}\pm x_{\epsilon_{2}-\epsilon_{3}}\cdot
x_{-(\epsilon_{2}-\epsilon_{3})}),$

$A_{-\epsilon_{1}}=x_{-\epsilon_{3}}(C_{-\epsilon_{1}}+x_{\epsilon_{1}-\epsilon_{3}}\cdot
x_{-(\epsilon_{1}-\epsilon_{3})}\pm x_{\epsilon_{2}-\epsilon_{3}}\cdot
x_{-(\epsilon_{2}-\epsilon_{3})})$,

$A_{\epsilon_{2}\pm\epsilon_{j}}=x_{-\epsilon_{j}}^{1or2}(C_{\epsilon_{2}\pm\epsilon_{j}}+x_{\epsilon_{2}\pm\epsilon_{j}}\cdot
x_{-(\epsilon_{2}\pm\epsilon_{j})}\pm x_{\epsilon_{1}\pm\epsilon_{j}}\cdot
x_{-(\epsilon_{1}\pm\epsilon_{j})}),$

$A_{-\epsilon_{2}-\epsilon_{j}}=x_{-\epsilon_{j}}^{3}(C_{-\epsilon_{2}-\epsilon_{j}}+x_{\epsilon_{2}+\epsilon_{j}}\cdot
x_{-(\epsilon_{2}+\epsilon_{j})}\pm x_{\epsilon_{1}+\epsilon_{j}}\cdot
x_{-(\epsilon_{1}+\epsilon_{j})}),$

$A_{\epsilon_{j}-\epsilon_{2}}=x_{\epsilon_{j}(}(C_{\epsilon_{j}-\epsilon_{2}}+x_{\epsilon_{2}-\epsilon_{j}}\cdot
x_{-(\epsilon_{2}-\epsilon_{j})}\pm x_{\epsilon_{1}-\epsilon_{j}}\cdot
x_{-(\epsilon_{1}-\epsilon_{j})})$,

$A_{\epsilon_{1}\pm\epsilon_{j}}=x_{\epsilon_{3}-\epsilon_{j}}^{1or2}(C_{\epsilon_{1}\pm\epsilon_{j}}+x_{\epsilon_{2}\pm\epsilon_{j}}\cdot
x_{-(\epsilon_{2}\pm\epsilon_{j})}\pm x_{-(\epsilon_{1}\pm\epsilon_{j})})$,

$A_{-\epsilon_{1}-\epsilon_{j}}=x_{\epsilon_{1}-\epsilon_{j}}^{2or3}(C_{-\epsilon_{1}-\epsilon_{j}}+x_{\epsilon_{1}+\epsilon_{j}}\cdot
x_{-\epsilon_{1}-\epsilon_{j}}\pm x_{\epsilon_{2}+\epsilon_{j}}\cdot
x_{-\epsilon_{2}-\epsilon_{j}})$,

$A_{\epsilon_{j}-\epsilon_{1}}=x_{\epsilon_{j}-\epsilon_{3}}(C_{\epsilon_{j}-\epsilon_{1}}+x_{\epsilon_{1}-\epsilon_{j}}\cdot
x_{-(\epsilon_{1}-\epsilon_{j})}\pm x_{\epsilon_{2}-\epsilon_{j}}\cdot
x_{-(\epsilon_{2}-\epsilon_{j})}),$

and $A_{\alpha}=x_{\alpha}^{2}$ or $x_{\alpha}^{3}$or $x_{\alpha}^{4}$

suitably for other roots $\alpha$ so that they all have distinct and different
weights with respect to $(adH)$\- module $u(L)$,

where $H$ is the cartan subalgebra consisting of all linear combination of
$h_{\alpha}$ for any root $\alpha$. Here $C_{\alpha}\in F$ are chosen so that
parentheses are invertible and signs are chosen so that they commute with
$x_{\alpha_{j}}=x_{\epsilon_{1}-\epsilon_{2}}$

(II) Next we assume $\alpha_{j}$ is a short root. Without loss of generality
we may put $\alpha_{j}=\epsilon_{1}$. Likewise as in (I), we suggest a basis
of $u(L)/\mathfrak{M}_{\chi}$ like the following.

$\mathfrak{B}:=\\{(B_{1}+A_{\epsilon_{1}})^{i_{1}}\otimes(B_{2}+A_{-\epsilon_{1}})^{i_{2}}\otimes(B_{3}+A_{\epsilon_{1}-\epsilon_{2}})^{i_{3}}\otimes(B_{4}+A_{-(\epsilon_{1}-\epsilon_{2})})^{i_{4}}\otimes\cdots\otimes(B_{2l}+A_{-(\epsilon_{l-1}-\epsilon_{l})})^{i_{2l}}\otimes(B_{2l+1}+A_{\epsilon_{l}})^{i_{2l+1}}\otimes(B_{2l+2}+A_{-\epsilon_{l}})^{i_{2l+2}}\otimes(\otimes_{j=2l+3}^{2m}(B_{l}+A_{\alpha_{j}})^{i_{j}})\\}$

for $0\leq i_{j}\leq p-1$,

where $A_{\epsilon_{1}}=g_{\epsilon_{1}}$,

$A_{-\epsilon_{1}}=(C_{-\epsilon_{1}}+w_{\epsilon_{1}}),$

$A_{-\epsilon_{1}\pm\epsilon_{2}}=x_{\epsilon_{3}\pm\epsilon_{2}}(C_{-\epsilon_{1}\pm\epsilon_{2}}+x_{-\epsilon_{1}\pm\epsilon_{2}}\cdot
x_{-(-\epsilon_{1}\pm\epsilon_{2})}\pm x_{\pm\epsilon_{2}}\cdot
x_{-(\pm\epsilon_{2})}\pm x_{\epsilon_{1}\pm\epsilon_{2}}\cdot
x_{-(\epsilon_{1}\pm\epsilon_{2})}),$

$A_{\epsilon_{1}+\epsilon_{2}}=x_{\epsilon_{3}-\epsilon_{2}}^{2}(C_{\epsilon_{1}+\epsilon_{2}}+x_{-\epsilon_{1}-\epsilon_{2}}\cdot
x_{\epsilon_{1}+\epsilon_{2}}\pm x{-\epsilon_{2}}\cdot x_{\epsilon_{2}}\pm
x_{\epsilon_{1}-\epsilon_{2}}\cdot x{\epsilon_{2}-\epsilon_{1}}),$

$A_{-\epsilon_{1}\pm\epsilon_{j}}=x_{-\epsilon_{2}\pm\epsilon_{j}}(C_{-\epsilon_{1}\pm\epsilon_{j}}+x_{-\epsilon_{1}\pm\epsilon_{j}}\cdot
x_{-(-\epsilon_{1}\pm\epsilon_{j})}+x_{\epsilon_{1}\pm\epsilon_{j}}\cdot
x_{-(\epsilon_{1}\pm\epsilon_{j})}\pm x_{\pm\epsilon_{j}}\pm
x_{-(\pm\epsilon_{j})}),$

$A_{\epsilon_{1}+\epsilon_{j}}=x_{-\epsilon_{2}-\epsilon_{j}}^{2}(C_{\epsilon_{1}+\epsilon_{j}}+x_{-\epsilon_{j}-\epsilon_{1}}\cdot
x_{\epsilon_{1}+\epsilon_{j}}\pm x{-\epsilon_{j}}\cdot x_{\epsilon_{j}}\pm
x_{\epsilon_{1}-\epsilon_{j}}\cdot x{-(\epsilon_{1}-\epsilon_{j})},$

$A_{\pm\epsilon_{2}}=x_{\epsilon_{3}\pm\epsilon_{2}}^{2}(C_{\pm\epsilon_{2}}+x_{\epsilon_{2}}\cdot
x_{-\epsilon_{2}}\pm x_{\epsilon_{1}+\epsilon_{2}}\cdot
x_{-(\epsilon_{1}+\epsilon_{2}}\pm x_{\epsilon_{3}-\epsilon_{1}}\cdot
x_{-(\epsilon_{3}-\epsilon_{1})})$,

$A_{\epsilon_{j}}=x_{\epsilon_{2}+\epsilon_{j}}(C_{\epsilon_{j}}+x_{\epsilon_{1}}\cdot
x{-\epsilon_{1}}\pm x_{\epsilon_{1}+\epsilon_{j}}\pm
x_{-(\epsilon_{1}+\epsilon_{j})}\pm x_{\epsilon_{j}-\epsilon_{1}}\cdot
x_{-(\epsilon_{j}-\epsilon_{1})}),$

$A_{-\epsilon_{j}}=x_{\epsilon_{2}-\epsilon_{j}}(C_{-\epsilon_{j}}+x_{-\epsilon_{j}}\cdot
x_{\epsilon_{j}}\pm x_{\epsilon_{1}-\epsilon_{j}}\cdot
x_{-(\epsilon_{1}-\epsilon_{j})}\pm x{-\epsilon_{j}-\epsilon_{1}}\cdot
x_{\epsilon_{1}+\epsilon_{j}},$

and $A_{\alpha}=x_{\alpha}^{2}$ or $x_{\alpha}^{3}$ or $x_{\alpha}^{4}$ for
the remaining roots $\alpha$.

We proved in the references [KY-4],[NWK-1,2] that $\mathfrak{B}$’s really form
a basis of the factor algebra $u(L)/\mathfrak{M}_{\chi}$ under consideration.

However we can exhibit its proof in detail differently from those as below.

We built up the bases $\mathfrak{B}$ in both cases of the factor algebra
$u(L)/\mathfrak{M}_{\chi}$ above

with the sign chosen so that they commute with $x_{\alpha}$ and with
$c_{\alpha}\in F$ chosen so that $A_{\epsilon_{2}-\epsilon_{1}}$ and
parentheses are invertible. For any other root $\beta$ we put $A_{\beta}$=
$x_{\beta}^{2}$ or $x_{\beta}^{3}$ if possible.

Otherwise we may attach to these sorts the parentheses( ) used for designating
$A_{-\beta}$ so that $A_{\gamma}\forall\gamma\in\Phi$ may commute with
$x_{\alpha}$.

We shall prove that $\mathfrak{B}$ is a basis in
$u(L))$/$\mathfrak{M}_{\chi}$.

By virtue of P-B-W theorem, it is not difficult to see that $\mathfrak{B}$ is
evidently a linearly independent set over $F$ in $u(L)$. Furthermore $\forall$
$\beta$ $\in\Phi$, $A_{\beta}\notin\mathfrak{M}_{\chi}$(see detailed proof
below).

We shall prove that a nontrivial linearly dependent equation leads to
absurdity.

We assume first that there is a dependence equation which is of least degree
with respect to $h_{\alpha}\in H$ and the number of whose highest degree terms
is also least.

In case it is conjugated by $g_{\alpha_{j}}$, then there arises a nontrivial
dependence equation of lower degree than the given one, which contradicts our
assumption.

Otherwise it reduces to one of the following forms:

(i)$x_{\pm\epsilon_{j}}K+K^{\prime}\in\mathfrak{M}_{\chi},$

(ii)$x_{\pm\epsilon_{j}\pm\epsilon_{k}}K+K^{\prime}\in\mathfrak{M}_{\chi},$

(iii)$g_{\epsilon_{1}-\epsilon_{2}}K+K^{\prime}\in\mathfrak{M}_{\chi},$

where $K,K^{\prime}$ commute with $x_{\alpha}$ and $x_{-\alpha}$ modulo
$\mathfrak{M}_{\chi}$.
Here we assumed first that $\alpha_{j}$ is a long root , and we may put
$\alpha_{j}=\alpha$ for brevity.

By making use of proofs of propositions in section 4 or in [KY-7], we may
reduce (i) and (ii) to the equation of the form

$x_{\epsilon_{1}-\epsilon_{2}}K+K^{\prime}\in\mathfrak{M}_{\chi},$

where $K$ commute with $x_{\pm(\epsilon_{1}-\epsilon_{2})}$ and $K^{\prime}$
commute with $x_{\epsilon_{1}-\epsilon_{2}}$ modulo $\mathfrak{M}_{\chi}$.

We have
$x_{\epsilon_{1}-\epsilon_{2}}^{p}K+x_{\epsilon_{1}-\epsilon_{2}}^{p-1}K^{\prime}\equiv
0$, so we get $x_{\epsilon_{1}-\epsilon_{2}}^{p-1}K^{\prime}\equiv 0$.

Subtracting
$x_{\epsilon_{2}-\epsilon_{1}}x_{\epsilon_{1}-\epsilon_{2}}K+x_{\epsilon_{2}-\epsilon_{1}}K^{\prime}\equiv
0$ from this equation, we obtain
$-x_{\epsilon_{2}-\epsilon_{1}}x_{\epsilon_{1}-\epsilon_{2}}K+g_{\alpha}K^{\prime}\equiv
0$. We should remember that $g_{\alpha}$ is invertible in
$u(L)/\mathfrak{M}_{\chi}$ by virtue of [RS].

By the way we use $w_{\alpha}:=(h_{\alpha}+1)^{2}+4x_{-\alpha}x_{\alpha}\in$
the center of $u(\mathfrak{sl}_{2}(F))$. Hence we have
$-4^{-1}\\{w_{\alpha}-(h_{\alpha}+1)^{2}\\}K+g_{\alpha}K^{\prime}\equiv 0$. So
we obtain

$4^{-1}g_{\alpha}^{p-1}\\{(h_{\alpha}+1)^{2}-w_{\alpha}\\}+cK^{\prime}\equiv
0\cdots(\ast)$

and from the start equation we get

$cx_{\alpha}K+cK^{\prime}\equiv 0\cdots(\ast\ast)$.

Subtracting $(\ast\ast)$ from $(\ast)$, we get
$4^{-1}g_{\alpha}^{-1}\\{(h_{\alpha}+1)^{2}-w_{\alpha}\\}K-cx_{\alpha}K\equiv
0$. Multiplying this equation by $g_{\alpha}^{1-p}$ to the right, we have

$4^{-1}g_{\alpha}^{p-1}\\{h_{\alpha}+1)^{2}-w_{\alpha}\\}g_{\alpha}^{1-p}K-cx_{\alpha}g_{\alpha}^{1-p}K\equiv
4^{-1}g_{\alpha}^{p-1}\\{(h_{\alpha}+1)^{2}-w_{\alpha}\\}g_{\alpha}^{1-p}K+x_{\alpha}x_{-\alpha}K\equiv
0$.

Conjugation of the brace of this equation $(p-1)$\- times by $g_{\alpha}$
gives rise to
$4^{-1}\\{(h_{\alpha}-1)^{2}-w_{\alpha}\\}K+x_{\alpha}x_{-\alpha}K\equiv 0$.
Next mutiplying $x_{-\alpha}^{p-1}$ to the right of the last equation, we
obtain

$\\{(h_{\alpha}-1)^{2}-w_{\alpha}\\}Kx_{-\alpha}^{p-1}\equiv 0$ modulo
$\mathfrak{M}_{\chi}$.

Now we multiply $x_{\alpha}$ to the left of this equaion consecutively until
it becomes of the form

(a nonzero polynomial of degree $\geq 1$ with respect to $h_{\alpha})K\newline
\equiv 0$ modulo $\mathfrak{M}_{\chi}$.

If we make use of conjugation and subtraction consecutively, then we arrive at
a contradiction $K\equiv 0$.

Next for the case (iii),we change it to the form
(iii)${}^{\prime}K+g_{\alpha}^{-1}K^{\prime}\in\mathfrak{M}_{\chi}$.

We thus have an equation

$x_{\epsilon_{1}-\epsilon_{2}}K+x_{\epsilon_{1}-\epsilon_{2}}g_{\epsilon_{1}-\epsilon_{2}}^{-1}K^{\prime}\equiv
0$ modulo $\mathfrak{M}_{\chi}$. According to the above argument, we are also
led to a contradiction $K\in\mathfrak{M}_{\chi}$.

For a short root $\alpha$ , the proof is very similar to the one just above.

Hence any coset of $u(L)/\mathfrak{M}_{\chi}$ must be of the form :( a linear
combination of all elements of $\mathfrak{B}+\mathfrak{M}_{\chi})$.

Now we let $v$ be the nonarchimedean valuation of $F$ and $w$ the additive
valuation of rank 1 satisfying that $v(x)=p^{-w(x)},\forall x\in F$.

For any given $x=(x_{1},x_{2},\cdots,x^{p^{2m}})\in F^{p^{2m}}$, where
$x_{i}$’s are coefficients of the above linear combination, we put
$v(x):=sup(v(x_{i}))$ and we put $U_{\lambda}:=\\{x\in A:\lambda\geq
0,v(x)\leq p^{-\lambda}\\},$ which is an ideal of $A$.

We let $G_{U_{\lambda}}:=\\{x\in F^{p^{2m}}:x_{i}\in U_{\lambda},1\leq
p^{2m}\\}=\\{x\in F^{p^{2m}}:v(x)\leq p^{-\lambda}\\}$.

Next we consider an infinite tower of $G_{U_{\lambda}}$’s such as
$G_{U_{1}}\supset G_{U_{2}}\supset G_{U_{3}}\supset\cdots$

and pore over the infinite family of sets as follows:
$S_{1}:=G_{U_{1}}-G_{U_{2}},S_{2}:=G_{U_{2}}-G_{U_{3}},\cdots,S_{n}:=G_{U_{n}}-G_{U_{n+1}},\cdots.$

It is obvious that $G_{U_{1}}=\cup_{i=1}^{\infty}S_{i}$. We put here
$S^{\prime}_{i}:=\\{x\in S_{i}:v(x)=p^{-i}\\}$ for $i\in\mathbb{N}$(the
natural number system).

Here at this juncture by the axiom of choice we may get a choice function
$\psi$ of the family $\\{S^{\prime}_{i}:i\geq 1\\}$

satisfying that $\psi(S^{\prime}_{i})\in S^{\prime}_{i}$.

We consider a counting problem asking whether or not a given particular
element $x$ of $u(L)$ is contained in a coset of the form
$\psi(S^{\prime}_{i})+\mathfrak{M}_{\chi}$ and how many coefficients are there
in $e\in\mathfrak{M}_{\chi}$ modulo $\sum_{i}u(L)(x_{i}^{p}-x_{i}^{[p]})$
having their absolute values equal to $v(x)$.

First thing in order to solve this problem we must check whether $x+e$ with
some $e\in\mathfrak{M}_{\chi}$ becomes of the form $\psi(S^{\prime}_{i})$.

In the mean time we may delete any term having a factor $x_{i}^{p}-x^{[p]}$
for some $1\leq i\leq n$ no matter where they may exist in $x$, $e$, or
$\psi(S^{\prime}_{i})$ in terms of P-B-W basis.

Hence we let bars in the expression, say
$\bar{x}+\bar{e}\in\overline{\psi(S^{\prime}_{i})}$ indicate the expression
after such deletion in prederence to the original counting problem.

We must show that it requires at most a polynomial time of the input size to
check whether or not $\bar{x}+\bar{e}\in\overline{\psi(S^{\prime}_{i})}$.

Next we may define $\forall x\in u(L)$, $v(\bar{x})$:= max
$\\{v(c_{j}):\bar{x}=\sum{c_{j}}\bar{b_{j}}\\}$ with $c_{j}\in F$ for P-B-W
basis $b_{j}$ of $u(L)$. We consider first that if
$\bar{x}\in\overline{\psi(S^{\prime}_{i})}+\bar{e}$, then we should get
necessarily $v(\bar{\psi}(S^{\prime}_{i}))\leq v(\bar{x})$.

Let $c_{i}$ be coefficients of $\beta_{i}^{\prime}s$ of $\mathfrak{B}$ for
$1\leq i\leq p^{2m}$. Further let $C_{j_{1}j_{2}\cdots j_{n}}$ be coefficients
of the term $x_{j_{1}j_{2}\cdots j_{n}}$ of a linear combination $\bar{x}$ of
elements of the form $x_{j_{1}j_{2}\cdots
j_{n}}=b_{1}^{j_{1}}b_{2}^{j_{2}}\cdots b_{n}^{j_{n}}$,

where $b_{1},\cdots,b_{n}$ are basis elements of $L$ and $0\leq j_{k}\leq
p-1$.

We can rearrange the element $s$, $\sum\overline{c_{i}\beta_{i}}$,and
$\bar{e}=\sum\overline{e_{j_{1}j_{2}\cdots
j_{n}}}$$\overline{x_{j_{1}J_{2}\cdots j_{n}}}$ with respect to P-B-W basis in
order to get a system of linear equations of the form

$C_{j_{1}j_{2}\cdots j_{n}}=\sum_{i=1}^{p^{2m}}C^{\prime}_{ij_{1}\cdots
j_{n}}\cdot C_{ij_{1}\cdots j_{n}}+\sum e^{\prime}_{j_{1}\cdots j_{n}}\cdot
e_{j_{1}\cdots j_{n}}$

with $C^{\prime}_{ij_{1}\cdots j_{n}},e^{\prime}_{j_{1}\cdots j_{n}}$ integers
in $\\{1,2,\cdots,p-1\\}$.

We should note here that such integers arise due to rearranging
$\overline{\psi(S_{i})}$ and $\bar{e}$ according to P-B-W basis.

Next the substitution of $C_{ij_{1}\cdots j_{n}}$ by $\psi(S^{\prime}_{i})$
gives rise to a system of nonhomogeneous linear equations in indeterminates
$e_{j_{1}\cdots j_{n}}$.

Obviously this system has an algorithm by the Cramer’s formula. We perceive
that the input time is at most $p^{n}$ which is the number of elements of
P-B-W basis.

So as to put forth flow charts for this algorithm, we need only polynomial
time in view of the fact that the following system

$(\ast)\left\\{\begin{array}[]{cc}a_{11}x_{1}+\cdots+a_{1p^{n}}x_{p^{n}}=C_{j_{1}\cdots
j_{n}}-\sum_{i=1}^{p^{2m}}C^{\prime}_{ij_{1}\cdots j_{n}}\psi_{ij_{1}\cdots
j_{n}}$\newline , \par\par$\\\ \cdots\cdots$ $\cdots\cdots$\newline
\par\par$\\\
a_{p^{n}1}x_{1}+\cdots+a_{p^{n}p^{n}}x_{p^{n}}=C_{p-1,\cdots,p-1}\par-\sum_{i=1}^{p^{2m}}C^{\prime}_{i(p-1,\cdots,p-1)}\cdot\psi_{i(p-1,\cdots,p-1)}\end{array}\right.$

is solvable if and only if $v(x_{i})\leq v(x),\forall{i}\leq p^{n}$,

where $a_{ij}$ represents $e^{\prime}_{j_{1}\cdots j_{n}}$, and
$x_{1},\cdots,x_{p^{n}}$ represents $e_{j_{1}\cdots j_{n}}$ respectively with
order $j_{1}\cdots j_{n}\leq j^{\prime}_{1}\cdots j^{\prime}_{n}$ defined if
and only if $j_{1}\leq j^{\prime}_{1},\cdots,j_{k-1}\leq j^{\prime}_{k-1},$

and $j_{k}<j^{\prime}_{k}$ for some $k\leq p^{n}.$

So it requires at most a polynomial time of the input size to check whether or
not $\bar{x}\in\overline{\psi(S^{\prime}_{i})+e}$.

Moreover it is evident that within a big fixed constant time of the input
size, the number of coefficients in $e\in\mathfrak{M}_{\chi}$ modulo $\sum
u(L)(x_{i}^{p}-x_{i}^{[p]})$ suitable for our bid can be easily detected by
vrtue of Cramer’s formula.

Hence the given counting problem is evidently contained in $NP$\- class.

Here we should recollect that an algorithm of a problem in $NP$\- class means
at most polynomial time logical processes regarding input sizes for the
individual check of solutions of the problem,

whereas an algorithm of a problem in $P$\- class means at most polynomial time
logical processes regarding the input sizes for the general solution of the
problem.

We must also note that algorithm shoud be determined only by input data with
other data finite and definitely determined and at most countable.

We may take an example for this matter. Consider the problem asking if a
positive integer is a prime number or not.

It is already known that any specific integer can be checkted by a calculator
in order to know whether it is prime or not within at most polynomial time
regarding the number of digits representing the integer.

So we know that the problem is contained in the $NP$\- class.

However we must still compute to know whether or not the algorithm for the
solution is generally determined within at most polynomial time regarding the
input sizes consisting of information only about any given integer.

It means that we have to determine whether or not the problem is in the $P$\-
class. But it is well known that the probem is contained in the $P$\- class,
which was proved by the Indian Institute of Technology.

Finally, we insist that our builtup problem cannot belong to the $P$\- class.

Suppose now that the counting problem under consideration is contained in the
$P$\- class. If we consider the cardinality of $\hat{n}(x):=\\#\\{j_{1}\cdots
j_{n}$ : $v(e^{\prime}_{j_{1}\cdots j_{n}}\cdot e_{j_{1}\cdot
j_{n}})=v(x)\\}$, then the cardinality must be determined only by the input
data of $x$.

However by the former system $(\ast)$ of nonhomogeneous linear equations, we
see that $\hat{n}(x)$ is determined not only by the data $C_{j_{1}\cdot
j_{n}}$ of $\bar{x}$ but also by the coefficients $\psi_{ij_{1}\cdot j_{n}}$
which are random variables in the field $F$ regardless of $x$ except for
$v(x)$.

Note also that the field $F$ is an uncountable set, so we cannot find out
clearly any element in $F$ in polynomial time of the input size.

So if we identify the coefficients of $\bar{x}$ with $\bar{x}$ with respect to
P-B-W basis for brevity, then the coefficiets time becomes the input time
$\bar{x}$ and the output time $f(\bar{x})$ is obviously not less than any
exponential function time $R^{\bar{x}}$. In other words we must have
$R^{\bar{x}}=\mathcal{O}(f(\bar{x}))$, i.e., $f(\bar{x})=\Omega(R^{\bar{x}})$
in terms of definition 2.1.

In fact it may take an infinite time even to find out a particular choice
function out of all choice functions because the ground field $F$ is
uncountable.

Hence we meet with a contradiction. So our given counting problem is not
contained in the $P$\- class. Since it is known that $P$-class $\subseteq
NP$-class, it turns out that $P$-class is properly contained in $NP$-class.

Hence we conclude that $P\neq NP$ after all.

∎

It may or may not be true that the counting problem is equivalent to the the
subset sum problem.

In other words the subset sum problem may or may not reduce to the counting
problem in polynomial time.

However it looks like we may conjecture the following, the conjecture being
compared to the subset sum problem.

[conjecture]
We conjecture that the counting problem under consideration could be an $NP$\-
complete problem. We would like to find out the heuristic reason for this
conjecture somehow.

We know that the subset sum problem requires at most the time
$\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\cdots+\binom{n}{n}=2^{n}$

for the input time $n$.

We don’t know for now whether or not we can reduce the time required. If it is
possible to say that the counting problem requires generally at least $2^{n}$
for the input time $n$, then we might say that the latter algorithm is
absolutely harder than the former one.

So this could be roughly the heuristic reason for the conjecture above.

##  6\. Questions and answers

Before closing our paper, we naturally raise two more questions attached to
$P\neq NP$.

[Question-1]Does $NP$-class form a distributive lattice with respect to the
usual logical operations V (disjunction) and (conjunction)?

[Question-2]Does $P$-class form a proper distributive sublattice of the
$NP$-class?

We have found earlier that the answers to both questions are affirmative.
Hence the rest part of this section focuses on the proofs of such facts.

Firstthing definitions related to these questions should be recalled. Suppose
that a certain collection of statements or a certain collection of problems is
closed under conjunction operation $\Lambda$ and disjunction operation V.

Generally speaking, these collections are not necessarily Boolean algebras.
However these may be sometimes distributive lattices.

Now let $x$ and $y$ be elements of an ordered set $S_{o}$. If there exists the
supremum of $\\{x,y\\}$, then we call it the join of $a,b$ and we denote it by
$x\cup y$.

Analogously in case the infimum of $\\{x,y\\}$ exists, then we call it the
meet of $x,y$ and denote it by the symbol $x\cap y$.

######  Definition 6.1.

We shall call an ordered set $S_{o}$ a lattice in case every pair of its
elements has a join and a meet. If $S_{o}$ satisfies in addition any of the
following equivalent conditions, we shall call $S_{o}$ a distributive lattice:

(i) $x\cup(y\cap z)=(x\cup y)\cap(x\cup z)$,

(ii)$x\cap(y\cup z)=(x\cap y)\cup(x\cap z)$,

(iii)$(x\cup y)\cap(y\cup z)\cap(z\cup x)=(x\cap y)\cup(y\cap z)\cup(z\cap
x)$.

######  Proposition 6.2.

Any Boolean algebra becomes a distributive lattice.

###### Proof.

We assume that $<B,+,\ast>$ is a Boolean algebra. Next we define
$inf\\{a,b\\}:=a\ast b$ and $sup\\{a,b\\}:=a+b$.

Such definitions are well defined so long as we define a partial order
$a\leq^{\otimes}$ in case that the following equivalent equations are
satisfied:

(i)$a\ast b^{\prime}=0,$

(ii)$a+b=b,$

(iii)$a^{\prime}+b=1,$

(iv)$a\ast b=a$,

where the notation ′ denotes the complement. It is well known that these four
equations are always satisfied in any Boolean algebra.

Furthermore it is not difficult to check that $<B,+,\ast>$ satisfies the
axioms of definition6.1 mentioned above.

∎

Now we pause for a while to think of a problem in decent way in connection
with a mathematical statement.

As we are well aware, a mathematical problem usually has one of the forms such
as ′′ Prove some fact, or prove or disprove something, or compute something
$\cdots$ etc.′′

However the important point is whether or not we can solve such a problem
within a polynomial time relative to input size. Actually in practice we
cannot dispense with time limit as is well known.

So in this regard we change any problem $p\in NP$ into the mathematical
statement ′′ we can solve $p$ within a polynomial time.′′

No doubt we we may still put put this new statement as the same letter $p$
incurring no harm to the structure of the class $NP$.

In this situation we may thus discuss about the structure more explicitly.
Hence in the next proposition we shall show that $NP$-class becomes a
distributive lattice in reality.

######  Proposition 6.3.

Under the operations disjunction $V$ and conjunction $\Lambda$ just described
above, we insist that $NP$\- class has the distributive lattice structure

###### Proof.

By virtue of proposition6.1, and proposition6.2, we are well aware that the
algebra of propositions constitutes a Boolean algebra and so a distributive
lattice.

However it is clear that the $NP$\- class is closed under these two operations
disjunction $V$ and conjunction $\Lambda$ , so that the $NP$\- class becomes a
distributive lattice as a sublattice of the algebra of propositions obviously.
∎

######  Proposition 6.4.

$P$-class also becomes a distributive lattice under the two operations given
above.

###### Proof.

We know that the $P$\- class forms a proper subclass of $NP$\- class by virtue
of the proof of proposition5.7.

Moreover $P$-class also becomes stable with two operations conjunction
$\Lambda$ and disjunction $V$, so that $P$-class becomes a proper sublattice
of the $NP$\- class obviously. ∎

$\mathbf{[Acknowledgement]}$
We give special many thanks to all persons who have helped us establish such a
pretty long story relating to a Clay problem.

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