International Journal of Emerging Technology and Advanced Engineering

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 10, October 2017)

395

Algebraic-Groups and Elucidation of Their Structure

Dr. Ramawatar Prasad

1

, Sri Ram Nandan Yadav

2

,

Dr. H.C. Jha3,

Dr. Abhishek Kumar Singh

4

1,3_{Associate Prof. & Head,} 2_{Research Scholar,} 4_{PhD, M.Sc., (Gold-Medalist)}

1,2_{University Department of Mathematics, K.S. College, L.Sarai, L.N.M.U, Darbhanga, Bihar (INDIA)} 3

University Department of Mathematics, C.M. Science College, L.N.M.U, Darbhanga, Bihar (INDIA)

4

University Department of Mathematics, Magadh University, Bodhgaya -824234, Bihar (INDIA)

Abstract: Since Algebraic Groups are groups defined by Polynomials. As the group of Matrices of Determinant one is an algebraic group. Algebraic Groups play a vital role in most branches of Mathematics and also in Quantum Mechanics and in Lie Groups. Also Arithmetic Groups are groups of Matrices with integer entries and an important source of discrete groups acting on manifolds. Our aim is to provide a modern Exposition of theory of algebraic groups and also the Extension of Duality between Algebraic groups & their categories of representations.

Keywords: Algebraic Groups, Polynomials, Quantum Mechanics, Lie-Group, Discrete, Manifold, Duality, Categories.

I. INTRODUCTION

The classification of Algebraic Groups and their structures are the great achievements in Mathematics. Algebraic Groups are used in most branches of Mathematics and they hence also played an important role in Quantum Mechanics and their branches of physics usually as Lie-Group.

As per the definition of Algebraic Group, the co-ordinate Ring is allowed to have nilpotent elements. But in the years the Tannkien Duality between Algebraic Groups similar to that of Pontryagin Duality in the theory of locally compact abelian groups.

A connected Algebraic Group G is simple if it is not commutative and has no normal algebraic subgroups and also its centre Z is finite and G/Z is simple.

SLn is simple for n > 1 because its Centre



































1

0

0

_n

Z

is finite and the quotient PSLn

= SLn/Z is simple.

An Algebraic Group G is solvable if there is a sequence of Algebraic subgroups.

1

...

...

2 1

0













G

G

G

G

n

G

such that

each

G

_i1 is Normal and

1 

i

i

G

G

is commutative.

Theorem 1.2 : Let

G

_mbe the functor RR (multiplicative group). Each

a



R

has a unique inverse, and so

 



a

b

R

ab



Hom



k

X

Y

XY

R



R

G

m

(

)

~

,

1

~

k alg

[

,

]

/(

1

),

2











_

Therefore

G

_mis an affine algebraic group, called the

multiplicative group.

Proof: Let

k

(

X

)

be the field of fractions of

k

(

X

)

, and let

k

[

X

,

X

1

]

be the subring of polynomials in

X

and

1 

X

. The homomorphism

1 1

,

],

,

[

]

,

[

X

Y



k

X

X



X

X

Y

X



k





defines an isomorphism

k

[

X

,

Y

]

/(

XY



1

)

~



k

[

X

,

X

1

]

, and so

G

_m

(

R

)



~

Hom

_kalg

(

k

[

X

,

X

1

],

R

).

Thus

,

)

(

]

,

[

];

,

[

)

(



1



1









G

m

k

X

X

for

f

k

X

X

and

a

G

m

R

R

f

_R

(

a

)



f

(

a

,

a

1

).

Theorem 1.3 : Let G be the functor such that G(R) = {1}

for all k-algebras R. Then

),

,

(

~

)

(

R

Hom

lg

k

R

G



ka

And so G is an affine algebraic group, called the trivial algebraicgroup.

Proof : More generally, for any finite group G, let

g G

g

k

G

O

(

)





_ (product of copies of k indexed by the elements of G). Then

R



Hom

kalg

(

O

(

G

),

R

)

is an

affine algebraic group

(

G

)

_kover k such that

(

G

)

_k(R)= G

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396

Theorem 1.4 : For any k-vector space V, the functor of k-algebras3

Da(V) : R  Homk-lin (V,R), (additive group)

is represented by the symmetric algebra Sym(V) of V :

),

,

(

~

)

),

(

(

lg

Sym

V

R

Hom

V

R

Hom

_ka



_klin

R a k-algebra,

Therefore Da(V) is an affine group over k (and even an

affine algebraic group when V is finite dimensional).

Proof: Now assume that V is finite dimensional. Then

Hom

_klin

(

V



,

R

)

~



R



_k

V

,

and so

R

~



R



k

V

(additive group) is an affine

algebraic group with coordinate ring Sym(

V

). Moreover,

EndR-lin

(

R



_k

V

)



~

R



_k Endk-lin

),

(

~

)

(

V



R



k

V



k

V



and so the functor

R

~



EndR-lin

(

R



_k

V

)

(additive group)

is an algebraic group Va with coordinate ring Sym

)

(

V



k

V

 . The choice of a basis e1, ..., en for V

defines isomorphisms EndR-lin

(

R



_k

V

)

~



M

_n

(

R

)

and

Sym

₍

_V



_V



₎

k

~



k

[

X

11

,

X

12

,...

...,

X

nn

]

(polynomial algebra in the n2 symbol (Xij)1



i, j



n). For

f



_[

_,

_,...

_...,

_]

12

11

X

X

nn

X

k

and

),

(

)

(

a

M

R

a



_ij



_n

f

_R

(

a

)



f

(

a

₁₁

,

a

₁₂

,

...,

a

_nn

).

Theorem 1.5 : For

n



n

matrices M and N with entries in a k-algebra R,

det(MN) = det(M). det(N)

and

adj(M).M = det(M). I = M. adj(M) by (Cramer's rule),

where I denotes the identity matrix and

adj(M) = ((-1)i + j det Mji)



Mn(R)

With Mij the matrix obtained from M by deleting the ith

row and jth column.

Proof: These formulas can be proved by the same argument as for R a field, or by applying the principle of permanence of identities. So, there is a functor SLn sending a k-algebra

R to the group of

n



n

matrices with entries in R and with determinant 1. Hence

SLn(R)

,

,

)

1

)

(det(

]

...,

,

,

[

~

11 12

lg





















_

R

X

X

X

X

k

Hom

ij n n a k

where det(Xij) is the polynomial, and so SLn is an affine

algebraic group with O(SLn)=

.

)

1

)

(det(

]

...,

,

,

[

_{11 12}



















ij n n

X

X

X

X

k

It is called the special

linear group. For

f



O(SLn) and

a



(

a

ij

)



SLn(R),

).

...,

,

(

)

(

_{11 nn}

R

a

f

a

a

f



Theorem 1.6 : Similar arguments show that the

n



n

matrices with entries in a k-algebra R and with determinant a unit in R form a group GLn(R), and that

R

~



GLn(R) is a functor. Moreover,

GLn(R)

,

,

)

1

)

(det(

]

,

...,

,

,

[

~

11 12

lg





















_

R

Y

X

Y

X

X

X

k

Hom

ij n n a k

and so GLn is an affine algebraic group with coordinate

ring4

,

.

)

1

)

(det(

]

,

...,

,

,

[

11 12



















R

Y

X

Y

X

X

X

k

ij n n

It is called the

general linear group. For

f



O

(

GL

_n

)

and

a



(

a

ij

)



GLn(R),

).

)

det(

,

...,

,

(

)

(

_ij



_{11 nn ij} 1

R

a

f

a

a

a

f

Proof: Again let A be the k-algebra in 2n2 symbols, X11,

X12, ..., Xn n, Y11, ..., Ynn modulo the ideal generated by

the n2 entries of the matrix (Xij) (Yij)-I. Then

Homk-alg(A, R) =



(

A

,

B

)

A

,

B



M

n

(

R

),

AB



I



.

The map (A, B)



A projects this bijectively onto {A



Mn(R)/A is invertible} (because a right inverse of a

square matrix is unique if it exists, and is also a left

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397

II. CONCEPT OF STRUCTURE OF SOLVABLE ALGEBRAIC

GROUPS

Let G be an abstract group and commutator of

G

y

x

,



is

 

1 1 1

)

(

)

(

,

y



x

y

x



y





xy

yx



x

So,

[

x

,

y

]



1

if and only if

xy



yx

,

and G is commutative if and only if every commutator equals 1. The

derived group G1 of G is the subgroup generated by commutators.

Since every automorphism of G maps commutators to commutators, and So G' is a characteristic subgroup of G

(in particular, it is normal). In fact, it is the smallest normal subgroup such that G/G' is commutative.

Let us define a suitable mapping such as :

  

x

y

x

y



G

G

y

x

y

x

₁

,

₁

,...

..,

_n

,

_n

)

₁

,

₁

...

_n

,

_n

:

2n



(



has image the set of elements of G that can be written as a product of at most n commutators, and so DG is the union

of the images of these maps. The map

G

2n2



G

factors through

G

2n



G

.



1 1

 

1 1



1 1 1

1 1

1 1

1

,

,...

..,

,

)

(

,

,...

..,

,

,

1

,

1

)

,

...

,

(

x

y

x

n_

y

n_



x

y

x

n_

y

n_



x

y

x

n_

y

n_ Also a group G is said

to be solvable if the derived series

...

2







DG

D

G

G

terminates with 1. For example, if

n



5

,

then

S

_n is not solvable because its derived series

S

_n



A

_nterminates with

A

_n

.

Now we are going to extend the idea of algebraic

groups.

Commutative groups are trangulizable :

At first we prove by linear algebra, then

Theorem 1.7 : If V be a finite -dimensional vector space over an algebraically closed field k, and let S be a set of commuting endomorphisms of V. There exists a basis of V

for which S is contained in the group of upper triangular matrices, i.e., a basis e1, ..., en such that



e

1

,...

.

e

i







e

1

,...

..

e

i





for all i.

Let S be a set of commutating

n



n

matrices : then there exists an invertible matrix P such that PAP-1 is upper triangular for all

A



S

.

Proof : We prove this by induction on the dimension of V.

If every





S

is a scalar multiple of the identity map, then there is nothing to prove. Otherwise, there exists an

S





and an eigenvalue

a

for



such that the eigenspace

V

_a



V

.

Because every element of S

commutes with



,

V

_ais stable under the action of the elements of S : for





S

and

x



V

_a

.

)

(

)

(

)

(

)

(



x





x



ax

a



x









The induction hypothesis applied to S acting on

V

_a and

a

V

V

/

shows that there exist bases

e

₁

,...

e

_mfor

V

_a

and

e

_m1

,...,

e

_n for

V

/

V

_a such that



e

1

,...

.

e

i







e

1

,...

..

e

i





for all

i



m



e

m1

,...

.

e

mi







e

m1

,...

.

e

mi





for all

m

n

i





Let

e

_mi



e

_mi



V

_a with

e

_mi



V

.

Then e1, ...

en is a basis for V .

Theorem-1.8 : Let V be a finite-dimensional vector space over an algebraically closed field k, and let G be a

commutative smooth algebraic subgroup of

GL

V

.

There

exists a basis of V for which G is contained in

T

_n

.

Proof : We have from above then , there exists a basis for

V such that the map

GL

(

V

)



GL

_n

(

k

)

defined as

)

(

k

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398

)

(

k

G

GL

(

V

)

)

(

k

T

n

GL

n

(

k

)

Now

G



T

_n is an algebraic subgroup of G such that (

G



T

_n) (k) = G(k), and so

G



T

_n= G. So,

.

n

T

G



Theorem-1.9 : Every commutative smooth algebraic group

G over an algebraically closed field is a direct product of two algebraic subgroups

u

s

G

G

G

~

–



such that

G

_s

(

k

)



G

(

k

)

_sand

G

_u

(

k

)



G

(

k

)

_u.

Proof : First note that the subgroups

D

_nand

U

_n of

T

_n

have trivial intersection, because

}

{

)

(

)

(

n n

n

R

U

R

I

D





(inside

T

_n

(

R

)

) for all R.

Choose an embedding

G



T

_n for some n, and let

n

D

G

Gs





and .

G

_u



G





_n Because G is commutative.

G

G

G

_s



_u



is a homomorphism with kernel

G

_s



G

_u. Because

}

1

{



n

n

U

D



as algebraic groups,

G

_s



G

_u



{

1

}

, and so is injective; because

G

_s

(

k

)

G

_u

(

k

)



G

(

k

)

and G is smooth, is subjective ; therefore it is an isomorphism.

Theorem- 1.10: The quotient

G

DG

is commutative

(hence DG is the smallest normal sub-group with this property).

Proof : Let N1 and N2 be normal subgroups of G. Then

N1(R) and N2(R) are normal subgroups of G(R) and

)

(

)

(

)

)(

(

N

₁



N

₂

R



N

₁

R



N

₂

R

. Therefore

)

(

)

(

)

(

)

(

–

~

)

)(

(

)

(

R

N

1

N

2

R

G

R

N

1

R

G

R

N

2

R

G





on varying R and passing to the associated sheaves, we get an isomorphism

2 1

2

1

N

~

G

N

G

N

N

G







Therefore,

G

N

₁



N

₂ is commutative if

G

N

₁

and

G

N

₂ are commutative. Similarly, for any finite

family

(

N

i

)

i



I

of

G



N

i is commutative if each

quotient

G

N

_iis commutative, and it suffices to consider

finite families.

DG, it is analogous to the description of the derived group as the subgroup generated by commutators. As for abstract groups, there exist maps of functors

.

...

2

4 2

G

G

G

G







n



Let In be the kernel of the homomorphism

)

(

)

(

2n

G

O

G

O



of k-algebras defined by

.

2

G

G

n



Then

...

...

2

1



I





I

n



I

and we let

I





I

_n

Theorem -1.11 : The coordinate ring of DG is O(G)/I.

Proof : We have from the figure of set-valued functors

G

G

G

G

G

G

mult

n n

n





















2 4

2

We get a diagram of k-algebras

)

(

)

(

)

(

)

(

)

(

)

(

2

G

O

G

O

G

O

I

G

O

I

G

O

I

G

O

n n n





















(because O(G)/In is the image of O(G) in O(G 4n

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399

I

G

O

I

G

O

G

O

(

)

(

)

/

(

)

/

:







factors through

O

(

G

)



O

(

G

)

I

, and defines a Hopf algebra structure on

O

(

G

)

I

, which corresponds to the smallest algebraic subgroup G' of G such that G'(R)

contains all the commutators for all R. Clearly, this is also the smallest normal subgroup such that G/G' is commutative.

III. APPLICATION

The application of the main paper is in-depth study of modern Mathematics especially in modern Algebra which has the importance in Algebraic groups and Lie-Groups.

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