Search Results for “radical of an algebraic group”

Source: Radical of an algebraic group

The radical of an algebraic group is the identity component of its maximal normal solvable subgroup.
For example, the radical of the general linear group

GL

n

⁡
(
K
)

{\displaystyle \operatorname {GL} _{n}(K)}

(for a field K) is the subgroup consisting of scalar matrices, i.e. matrices

(

a

i
j

)

{\displaystyle (a_{ij})}

with

a

11

=
⋯
=

a

n
n

{\displaystyle a_{11}=\dots =a_{nn}}

and

a

i
j

=
0

{\displaystyle a_{ij}=0}

for

i
≠
j

{\displaystyle i\neq j}

.
An algebraic group is called semisimple if its radical is trivial, i.e., consists of the identity element only. The group

SL

n

⁡
(
K
)

{\displaystyle \operatorname {SL} _{n}(K)}

is semi-simple, for example.
The subgroup of unipotent elements in the radical is called the unipotent radical, it serves to define reductive groups.