“Historically,” note Strade and Farnsteiner in [1] , “Lie algebras emerged from the study of Lie groups.” In Section 1.1 of [1] , they give a simple example of the close connection between Lie algebras and Lie groups. In prime characteristic, David Winter [2] has defined maps which mimic the zero-characteristic exponential maps. See also Lemma 1.2 of [3] . In this paper, we focus on the following “Winter maps”: if
is an element of a characteristic-
Lie algebra
such that
we set
![](https://www.scirp.org/html/htmlimages\1-5300714x\fadcd681-a951-4d05-921e-4c42170e3c64.png)
where
is the identity transformation of
. Such ad-nilpotent elements of degree less than
do exist in some graded Lie algebras, as can be seen from Lemma 2.3 and Proposition 2.7 of Chapter 4 of [1] , as well as from Lemma 1 of [4] ; of course, it is well known that non-zero-root vectors of simple classical-type Lie algebras are ad-nilpotent of degree less than or equal to four.
We will show here that for
such that
the inverse of
as a linear transformation of
is
, so that such transformations generate a group
of linear transformations of
. We will also show that
where, for
a linear transformation of
, and
as above, we define
(1)
Thus, like
and
,
is, in a sense, the functional inverse of
.
Lemma 1 If
and
are elements of
such that
and
then
![](https://www.scirp.org/html/htmlimages\1-5300714x\2b59de02-c7f7-45b2-9ce2-c5e44fde2ca8.png)
Proof. We group terms with respect to total degree in
and
![](https://www.scirp.org/html/htmlimages\1-5300714x\8b54293b-00e1-4492-bb4e-e20c7618f5aa.png)
Lemma 2 Let
, and suppose that
is an element of
such that
then
![](https://www.scirp.org/html/htmlimages\1-5300714x\c13f04e2-097f-48c7-bfef-a1c9fc7637c5.png)
Proof. We have by Lemma 1 that
equals
![](https://www.scirp.org/html/htmlimages\1-5300714x\b9846310-e499-4ae8-800b-000515b05c8e.png)
which we can write in terms of binomial coefficients as
![](https://www.scirp.org/html/htmlimages\1-5300714x\3669de3b-5369-4de0-b83f-80ec483b217f.png)
By the Binomial Theorem, the above expression is equal to
![](https://www.scirp.org/html/htmlimages\1-5300714x\d98a514f-b21f-42c6-99c6-5401eb269674.png)
which we can rewrite as
![](https://www.scirp.org/html/htmlimages\1-5300714x\5fee6ad8-0564-4bb1-a4f1-5ca8381b04b6.png)
and recognize as
. ![](https://www.scirp.org/html/htmlimages\1-5300714x\77fe23f2-28c7-451b-8d2c-d9ba87780710.png)
Lemma 3 For any integer
and any integer
,
, we have
![](https://www.scirp.org/html/htmlimages\1-5300714x\add0fc6e-bbe2-4c2b-a291-b43d33721ba1.png)
Proof. We proceed by induction on
and
. When
, we must have
, and we have
For any
, when
, we have
![](https://www.scirp.org/html/htmlimages\1-5300714x\3027387c-71e9-4513-8c56-4a20250e04e5.png)
Now, for any
and any positive integer
less than
, suppose that
for all positive
less than
Then we have
![](https://www.scirp.org/html/htmlimages\1-5300714x\f8b29921-4df5-47b1-bc1b-02995d0562fa.png)
by induction, and the fact that
(the “
case”). ![](https://www.scirp.org/html/htmlimages\1-5300714x\f49741b6-9a82-42b4-84bf-4fc8f52eb7e3.png)
Lemma 4 Let
be an element of
such that
. Define
(2)
Then for any positive integer
less than
,
(3)
Proof. We proceed by induction on
. Since when
, (3) is just (2), the initial step of the induction proof is established. Suppose (3) is true for
. Then
equals
![](https://www.scirp.org/html/htmlimages\1-5300714x\a2dd75d3-e2a4-489b-ad99-9d85cf364f00.png)
We group terms with respect to total degree (
, in this case) in
and get that
.
Rewriting the above expression using another binomial coefficient, we get that
equals
![](https://www.scirp.org/html/htmlimages\1-5300714x\9a42f3f9-65aa-4b34-8d72-cd84907c04cd.png)
We change the order of summation to get
![](https://www.scirp.org/html/htmlimages\1-5300714x\3f873b5a-5b8d-441f-9df4-889d225582dc.png)
We replace the index of summation
by
to get
.
Adding and subtracting terms, we get
![](https://www.scirp.org/html/htmlimages\1-5300714x\08373846-a87a-4c88-91fe-dab8f2d0a14f.png)
Setting
, we see, as in the proof of Lemma 3, that when r ≥ 1,
![](https://www.scirp.org/html/htmlimages\1-5300714x\b77d0339-ba9b-4924-876f-7b66fb4494fb.png)
by that same Lemma 3. Thus,
![](https://www.scirp.org/html/htmlimages\1-5300714x\b09189b9-a035-4410-93fe-464f56639594.png)
so from the Binomial Theorem, we get that
equals
.
We now distribute to get that
equals
![](https://www.scirp.org/html/htmlimages\1-5300714x\15003b24-5256-4626-b8cc-13057a5d1b92.png)
We replace the latter index of summation
by
to get that
equals
![](https://www.scirp.org/html/htmlimages\1-5300714x\f2f33c19-56f3-47fe-9fbf-1ac803612a9e.png)
We change the order of summation and factor to get that
equals
![](https://www.scirp.org/html/htmlimages\1-5300714x\5c224a99-e6b2-48ea-9579-85be6ffe2480.png)
By binomial arithmetic
equals
![](https://www.scirp.org/html/htmlimages\1-5300714x\7fed2df7-dffe-4ef0-927f-ba73ac84a9f3.png)
The above displayed formula is just (3) for
; i.e.,
equals
.
Thus, the induction step is complete. ![](https://www.scirp.org/html/htmlimages\1-5300714x\71e07537-8a65-4579-b840-69533271a429.png)
Theorem The linear transformation
of
has
as its inverse, whereas the map
of
to the group of non-singular linear transformations of
has
as its inverse, in the sense that
(a).
, and
(b).
.
Proof. (a) If, in Lemma 2, we let
and
, we see that (a) is true.
(b) Since
equals the
of Lemma 4, we have that
equals
![](https://www.scirp.org/html/htmlimages\1-5300714x\89060657-9e48-4c44-a593-790c57999fd0.png)
which, by Lemma 4 equals
![](https://www.scirp.org/html/htmlimages\1-5300714x\b4d2634b-740e-46c8-ae4b-ed6b8c712953.png)
We replace the index
by
to get that
![](https://www.scirp.org/html/htmlimages\1-5300714x\aad1e154-6c7c-4394-8c9a-7f7c3d65aa16.png)
We change the order of summation to get that
![](https://www.scirp.org/html/htmlimages\1-5300714x\c9c5d4c2-9244-4402-a23d-1a18144e50b7.png)
We replace the index
by
to get that
![](https://www.scirp.org/html/htmlimages\1-5300714x\dc667581-44f3-4dfa-aa82-cf8d67f2b254.png)
We cancel an
and a
and combine the
factors to get that
![](https://www.scirp.org/html/htmlimages\1-5300714x\e1b6a93d-a8b1-4eb9-ba4b-7d1b6cfe68a4.png)
We replace the index
by
and we replace the index
by
, and we get that
![](https://www.scirp.org/html/htmlimages\1-5300714x\1ea72208-1b48-4402-a92a-b372e057d3d4.png)
We change the order of summation to get that
![](https://www.scirp.org/html/htmlimages\1-5300714x\db6f1cdc-d4e8-42ca-97ce-5520872d8fa0.png)
We now appeal to a little more binomial arithmetic to observe that since
and
, it follows by induction that
![](https://www.scirp.org/html/htmlimages\1-5300714x\04b2b031-1b1f-4461-afc5-fcfab46efa19.png)
from which we obtain that
![](https://www.scirp.org/html/htmlimages\1-5300714x\d3d505c1-3883-49f4-91eb-dd1630b17bf5.png)
We replace the index
by
to get that
![](https://www.scirp.org/html/htmlimages\1-5300714x\938c0243-6feb-40fe-8560-c9e25ed39f85.png)
Finally, we use Lemma 3 to see that we are left with
![](https://www.scirp.org/html/htmlimages\1-5300714x\a4fdc332-89ae-4699-aa5c-9b01147ce6db.png)