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INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR

Lie and on the SL2(R)

PROJECT REPORT – SEMESTER IV

MOUSUMI MALICK

2-YEARS MSc(2011-2012)

Guided by –Prof.DEBAPRIYA BISWAS

Lie group and Geometry on the Lie Group SL2(R)

CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision.

Dr. Debapriya Biswas

Department of Mathematics

Date-

Indian Institute of Technology Khargpur

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Lie group and Geometry on the Lie Group SL2(R)

ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement.

Date- place-IIT Kharagpur Mousumi Malick

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Lie group and Geometry on the Lie Group SL2(R)

CONTENTS

1.Introduction ...... 4

2. of general : ...... 5

3.Definition of a general Lie group:...... 5

4.Definition of : ...... 5

5. Definition of under a group action: ...... 5

6.1.The general linear groups GL(n;R) and GL(n;C) : ...... 6 6.2.The SL(n;R) and SL(n;C) ...... 6 6.3.Orthogonal O(n) and special orthogonal SO(n) groups: ...... 8 O(n)= of all nxn orthogonal matrices ...... 8 6.4. The H: ...... 9

7. Sub-group of GL(n;C)which is not Lie group:...... 10

8. The geometry on SL2(R): ...... 11

8.1 EPH –classification of action (1) on SL2(R): ...... 12 8.2 Some Properties: ...... 18

9. Importance of Lie group and its Gemetry in Mathematics: ...... 27

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Lie group and Geometry on the Lie Group SL2(R)

1.Introduction

We are all very familiar with the group of real , group and group of matrices. In my report I will discuss about matrix lie group and some geometry dealing with the special linear group SL2(R).

In to discussing Lie groups and Lie , Borel pays considerable attention to algebraic groups, beginning with the little known work of Ludwig Maurer in the 1890s up through the work of , , Borel and others in the post-World War II era.

In the beginning, hoped to develop a Galois for differential equations. 's famous \," which aimed to connect firmly geometry with , also influenced Lie's thinking. By 1893, Lie (together with Friedrich Engel) had completed the final, third volume of the massive treatise Theorie der Transformationsgruppen. Also by 1890, had succeeded in classifying ( a few gaps in his arguments) the complex simple Lie algebras. Killing's work was rigorously treated and extended in Elie Cartan's 1894 thesis.

A Lie group was a transformation group, that is, there was some geometric present upon which the Lie group acted. The notion of an abstract group with the compatible structure of a complex came later; mathematicians such as Klein resisted this notion, feeling that it would distance the subject too far from its applications. In addition, Lie's theory was largely local. Early workers were aware of global examples (such as SL(n;C)), but the theory was mostly developed for what we today would call \germs" of Lie groups. A global theory emerged only with the work of Weyl and Cartan in the twentieth century.

For discussing the geometry on SL2(R) first we have to know what is lie group and some example of Lie group.

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Lie group and Geometry on the Lie Group SL2(R)

2.Definition of : The general linear group over the real , denoted by GL(n;R),is the group of all nxn invertible matrices with real entries. The general linear group over the complex numbers, denoted by GL(n;C) ,is the group of all nxn invertible matrices with complex entries. The general linear groups are indeed groups under the operation of matrix .

3.Definition of a general Lie group: A matrix lie group is any G of GL(n;C) with the following property:

If Am is any of matrices in G, and Am Converges to some matrix A then either A in G ,or A is not invertible.

Or equivalent to saying that a matrix lie group is a closed sub group of GL(n;C),and this property is necessary. We can prove it by give an example.

The set of all nxn invertible all of whose entries are real and rational. This is a subgroup of GL(n;C) but not closed. That is one can have a sequence of invertible matrices with rational entries converging to an with some irrational entries. So this sub group is not a matrix lie group.

4.Definition of group action: Let G be a group and S a nonempty set. Then G is said to act on S if there is a from G x S to S, usually denoted (g; s)  gs, such that es = s for all s belongs S, and for all g, h  G and s S, that (gh)s = g(hs).

There can be diﬀerent ways for a group to act on a set. The notation gs for the imageof (g; s) is ambiguous, but won’t cause problems since we will not consider two diﬀerent actions of a group on a set at a . Before we get into properties of group actions, we give

5. Definition of orbit under a group action:

Suppose G is a group which acts on a set S. If s S, let O(s) = {gs : g G}.The set O(s) is called the orbit of s.

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Lie group and Geometry on the Lie Group SL2(R)

6.Some examples of matrix lie group:

6.1.The general linear groups GL(n;R) and GL(n;C) :

The general linear groups over R and C are themselves Matrix lie group. Now GL(n;C) is a subgroup of itself. Again, if Am is a sequence of matrices in GL(n;C) and Am converges to A, then either A is in GL(n;C),or A is not invertible.

Therefore GL(n;C) is matrix lie group. Moreover GL(n;R) is a subgroup of GL(n;C),and if Am is a sequence of converges to some matrix A, then A is also a real matrix. Thus A is not invertible or A is in GL(n;R).Hence GL(n;R) is a matrix lie group.

6.2.The special linear group SL(n;R) and SL(n;C)

The special linear group over real or complex is the group of nxn invertible matrices with one .This is one of the most important matrix lie group and my work is going on in this matrix lie group.

Now we have to show that SL(n;R) and SL(n;C) are matrix lie group. At first prove that SL(n;C) is a group under . i.For any A,B in SL(n;C) with determinant 1. AB also in SL(n;C),as det(AB)=detA.det B =1. Therefore SL(n;C) is closed under matrix multiplication. ii. Matrix multiplication is associative, hence SL(n;C) is associative. iii. nxn plays the role of identity here.

adjA iv. Now det A≠0 ,so A-1 exit. We know that A-1= and det(adj A) = (detA)n-1= 1 det A that implies det A-1=1. Therefore A-1 belongs to SL(n;C) .Hence SL(n;C) is a group under matrix multiplication.

Next we have to show that SL(n;C) is a matrix lie group.SL(n;C)is a subgroup of GL(n;C) and let Am is a sequence in SL(n;C) converges to some matrix A ,we now show that A belongs to SL(n;C) .

That is lim An=A n

Taking determinant on both side we have

det(lim An)= det A n

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Lie group and Geometry on the Lie Group SL2(R)

 lim (det An)= det A (as determinant is a ) n

 det A= 1

Hence SL(n;C) is a matrix lie group. Similarly We have to prove that SL(n;R)is also a matrix lie group.

Here we want to introduce one most important thing is called the of SL(2;R).By this decomposition any element S of SL(2;R) can be written in the form S=ANK where A,N and K are subgroup of SL(2;R)

a b   0  1 v cos  sin  Let S=  =         1       c d   0   0 1  sin cos 

Where, A , N and K

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Lie group and Geometry on the Lie Group SL2(R)

6.3.Orthogonal O(n) and special orthogonal SO(n) groups:

O(n)=set of all nxn orthogonal matrices SO(n)=set of all nxn orthogonal matrices with determinant one.

An nxn real matrix A=(aij)nxn is said to be orthogonal if the column vectors that make up A orthonormal.

Equivalently a matrix A is orthogonal if it preserves the inner product.

That is = ,where x,y are in Rn

Another way , A is said to be orthogonal if AAt=AtA=I , where At is transpose of A.

Now we have to show that O(n) is a matrix lie group. For this first we proof that O(n) is a group under matrix multiplication.

iCloser proper y: let A ,B O(n) then AB also in O(n) . (by the property ) iiAssociate property: matrix multiplication is associate. Therefore associativity property holds. iii Identity: here the nxn is the identity. iv : for any A in O(n) the inverse of A is always exist as determinant of A is not zero. The inverse is also in O(n) as inverse of orthogonal matrix is also orthogonal.

Hence from the above four property it is clear that O(n) is a group. O(n) is a of GL(n;C) and let Am is a sequence in O(n) which converges to some matrix A ,we have to prove that A is in O(n). that is , lim Am=A m Taking transpose on both sides we have,

t t (lim Am) =A m

t t  (Am) =A

t t Now, Am (Am) =A A =I

Therefore A is in O(n).hence O(n) is a matrix lie group. Similarly we can show that SO(n) is a matrix lie group only requirement is that the matrix in SO(n) are of determinant one.

We know that the determinant of a orthogonal matrix is  1 .That implies SO(n) is half of O(n) .

O(2) is the orthogonal matrix lie group of 2 and any element of O(2) can written in the form

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Lie group and Geometry on the Lie Group SL2(R)

cos  sin  cos sin    or    sin cos   sin  cos     

6.4. The Heisenberg group H: The set of all 3x3 real matrices A of the form

1 a b   A=0 1 c where a, b and c are arbitrary real numbers, is the Heisenberg group.   0 0 1 This lie group is one of the most important lie group its lie gives a realization of the Heisenberg commutation relations of quantum .That’s why this group named as Heisenberg group. For showing that it is a matrix lie group first we prove that it is a group under matrix multiplication.

1 a b  1 a b   1 1   2 2  0 1 c 0 1 c i For any A= 1  and B= 2  in H     0 0 1  0 0 1 

1 a  a b  b  a c   1 2 1 2 1 2  AB=0 1 c1  c2  is also belongs to H. Therefore H is closed under usual   0 0 1  matrix multiplication.

ii Associate property holds trivially.

iii  Here 3x3 identity matrix play the role of identity element.

1  a ac  b   ivInverse of any element A = of H is A-1=0 1  c  also belongs to H.   0 0 1 

Hence H is a group, and H is sub group of GL(3;C).Let {Am} is a sequence in H which converges to some matrix A. Now A is also in H as the sequence of real number converges to some real number

Therefore H is a matrix lie group.

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Lie group and Geometry on the Lie Group SL2(R)

7. Sub-group of GL(n;C)which is not matrix Lie group: The following sub group of of GL(2;C) defined by

eit 0  G=   , tR  ita   0 e  where ‘a’ be an irrational real number.Now G is a subgroup of GL(n;C),for any g1, belongs to G it it e 0  e 0  ei(tt) 0  g +g -1   +  =  is in G,as t’+t”R. But it is not a matrix 1 2 =  iat   iat   ia(tt)   0 e   0 e   0 e  lie group . And if we choose t=(2n+1) ,be an odd multiple of then eit=-1,but eita  -1,as a is an . We can make ta arbitrarily close to an odd multiple of .Hence we can find a sequence {Am} in G which is converges to -I .But –I is not in G .So by the definition of lie group G is not a matrix lie group. For this let {am}=1/2n( 7 7 7...... ) for n=1 2 3 4...... eit 0  ’ Am =   and t =(2n+1) then the sequence converges to –I . But –I do not belong to the  iamt   0 e  group G .Hence G is not a matrix lie group.

There are another examples of matrix lie group such as (U(n)) , ((SU(n)), symplectic groups over real and complex ,and Lorentz groups.

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Lie group and Geometry on the Lie Group SL2(R)

8. The geometry on SL2(R):

Now we will discuss what kind of geometry is related to the group SL2(R).The special general linear group SL2(R) of order 2 is the group of 2x2 matrices with determinant 1.

The simplest non liner transformation of the real is the Mobious transformation or the bilinear transformation. This bilinear transformation can be associated with 2x2 matrices .

ax  b We know that w= is the form of the bilinear transformation, where a, b, c, d R and x is cx  d also a real number. But it can be possible that those are complex number.

Now the above transformation can be written in this form:

a b  g: x g.x= where g=   c d    ------(1)

Let g is in SL2(R) then (ad-bc)=1.

a1 b1  a2 b2  Suppose g1 and g2 are in SL2(R) , g1=   and g2=   c1 d1   c2 d 2 

a2 x  b2 (a1a2  a1c2 )x  b1b2  b1d 2 g1g2:x g1g2.x=g1 = c2 x  d 2 (c1c2  c1a2 )x  d1b2  d1d 2

Where a1d1-b1c1=1 and a2d2-b2c2=1.

We see that g1g2 is again a Mobious transformation .In other way we can say that (1) is an left action on the group SL2(R).According to the mathematician F. Klein’s Erlangen program any geometry is dealing with properties under a certain group action. Then for this case some geometry are related to the action (1) on SL2(R).

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Lie group and Geometry on the Lie Group SL2(R)

8.1 EPH –classification of action (1) on SL2(R): Mobious transformation is also useful for complex number. It is easy to see that the action (1) on 2 the group SL2(R) makes sense also a of complex numbers z=x+iy ,where i =-1 .We also see that for z=x+iy if y>0 then g.z has positive imaginary part.

a b  az  b ax  b  iay 1 2 Let g=   and g.z= = = [(ax+b)(cx+d)+acy +iy] where k>0.  c d  cz  d cx  d  icy k That is the action (1) defines a map from upper half plane to itself.

Normally we know that any complex number can be written in the form z=x+iy , where i2=-1 . But if we take three different value of i2 we get different result .

The three cases are:

case-1: for i2=-1 ,this named as elliptic case. case-2: for i2=0 , this named as parabolic case. case-3: for i2=1 , this named as hyperabolic case.

we will abbreviate this separation as EPH- classification .we repeatedly meet such a division in various mathematical objects. For example in geometry, partial and many other mathematical parts also. We can take i2=

The common origin of this fundamental division can seen from the simple picture of a coordinate line split by zero into negative and positive half axes.

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Lie group and Geometry on the Lie Group SL2(R)

The figure of all three cases: case: i2=-1

FIGURE (1)

Elliptic case: Under the action (1) of the K subgroup of the group SL2(R) the orbit are . We can see this from the above picture.

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Lie group and Geometry on the Lie Group SL2(R)

Case 2: when i2=0

Figure (2)

Parabolic case: Under the action (1) of the K subgroup of the group SL2(R) gives the orbit are system of parabolas. we can see this from the figure(2). As the orbit is the system parabolas for this it named as parabolic case.

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Lie group and Geometry on the Lie Group SL2(R)

Case 3: when i2=1

Figure (3)

Hyperbolic case: : Under the action (1) of the K subgroup of the group SL2(R) gives the orbit are system of hyperbolas. we can see this from the figure(2). As the orbit is the system hyperbolas for this it named as hyperbolic case.

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Lie group and Geometry on the Lie Group SL2(R)

We can understand the action (1) in all three cases we use the Iwasawa decomposition of SL2(R), which we are discussing earlier. By Iwasawa decomposition any element S of SL2(R) can written in the form

a b   0  1 v cos  sin  S=   =         1       c d   0   0 1  sin cos 

Where, A , N and K where A, N, K are the subgroup

of SL2(R).

That is we can write SL2(R)=ANK.

Subgroup A and N acts in (1) irrespective of the value of i2 , A makes a dilation by  2 ,that is the point z on the are becomes z and N shifts points to left by v ,that is zz+v . The action of the third matrix that is the subgroup K depends the value of the i2.

,

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Lie group and Geometry on the Lie Group SL2(R)

Figure (2)

In figure (2) we can see that under the action (1) of the K subgroup the orbits as conic sections circles are sections by the plane EE’, parabolas are sections by the plane PP’ and hyperbolas are sections by the plane HH’.

The common name cycle is used to denote circles parabolas and hyperbolas in the respective EPH cases. All three types of orbits are generated by the of the along the cone.

Since the action of both A and N for any value of i2are extremely shape preserving ,we find natural invariants objects of the Mobious map.

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Lie group and Geometry on the Lie Group SL2(R)

8.2 Some Properties: Now we are discussing only for the elliptic case that is for i2=-1 We proof some result which we obtain from this discussion.

We know that Mobious transformation is preserving and we proof it in complex algebra .Again we proof there by another way. That is under the action (1)on SL2(R)any circle map in a circle.

Theorem : Circle are invariant under the action (1) on SL2(R) proof:

We will show that for a given g in SL2(R)and a circle C its gC again a circle. We know that

SL2(R) can be decomposed in SL2(R)= ANK (Iwasawa decomposition) where A, N, K are subgroup of

SL2(R).Suppose gn ,gn’ belongs to N and ga ,ga’ belongs to A, g belongs to SL2(R).

-1 Then g( ga’gn’) is in SL2(R) , since SL2(R) is a group.

-1 And we can write by Iwasawa decomposition g( ga’gn’) = ga gn gk , where gk is in K

therefore g= gagngkga’gn’

figure(3)

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Lie group and Geometry on the Lie Group SL2(R)

For a fixed circle C there is always a unique pair of transformation gn’ from the subgroup N of SL2(R) and ga’ from the subgroup A such that ga’gn’C exactly a K orbit.

Then we have gk(ga’gn’C)= ga’gn’C

Therefore gC=gagngkga’gn’C

or, =gagnga’gn’C

Since we are earlier says that the subgroup A and N obviously preserve the shape of any cycle. Here the cycle is the circle and gC again a circle .

Note: The result is also holds for the parabolic and hyperbolic case.

The standard mathematical method is to declare objects under investigation to be simply points of some bigger space. This space should be equipped with an appropriate structure to hold externally information that was previously inner properties of our objects.

A circle is the set of points on (u,v) R2 defined for i2=-1 by the equation

k(u2-i2v2)-2lu-2nv+m=0 ,where i2=-1

This equation is defined by a point (k, l, n, m) from a P3, since for a scaling factor   0 the point ( k, l, n, m) defines the same equation. We call p3 the circle space and refer to the initial R2 as the point space.

In order to get a with the mobious action (1) we arrange the numbers (k, l, n, m) into the matrix.  l  i sn  m   S   i C     with a new imaginary and an additional  k  l  i sn parameter s usually equal to  1 . The values of the new imaginary unit are -1, 0, 1 independently of the value of the previous imaginary unit. This matrix is the cornerstone of the Fillmore-Springer- Cnops construction in short FSCc .

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Lie group and Geometry on the Lie Group SL2(R)

 Here we see that FSCs uses an new imaginary unit i which is not related the old one. In other words any EPH type geometry cycle space P3 allows one to draw cycles in the point space R2 as circles parabolas or hyperbolas.

figure 4(a)

In figure 4(a) we can see that the same cycles drawn in different EPH styles.

Where ce=(l/k ,n/k) ,cp=(l/k ,0) , ch=(l/k ,-n/k) are the centers of the circle, parabola and hyperbola respectively. They are related each other by the equations

1 ce=c and cp= (c  c ) h 2 e h

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Lie group and Geometry on the Lie Group SL2(R)

figure4 (b)

In this figure two parabolas are drawn and they have the same focal length n/2k. There ce are on the same level. In other words we can say that concentric parabolas are obtained by a vertical shift.

ALGEBRIC AND GEOMETRIC INVARIENTS:

For 2x2 matrices there are only two essentially different invariants the Mobious action (1) , the and the determinant. We can easily show that the determinant is invariants under the Mobious action (1).

 x x   1 2  Let X=  be a 2x2 matrix whose determinant is ( x1x4  x2 x3 ). Suppose gSL2(R) then  x3 x4  determinant of g is 1.

The Mobious action g:X g  X determinant ( g.X)= determinant g.determinant X (by the property determinant)

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Lie group and Geometry on the Lie Group SL2(R)

=determinant X

Hence determinant is invariant under the Mobious action (1).

We denote (k, l, n, m) as a point of the projective space P3. If k  0 we may normalize the quadruple l n m to (1, , , ) . More over in this case determinant of C S is equal to the squre of the cycle’s k k k  radius .

ORTHOGONALITY:

When we discuss the orthogonality of some matrices or some vectors first we define the inner product between them. The inner product between two cycles is calculated by the trace of product of two cycles as matrices.

s  s s  s   that is C C  trC C 

 s  s s  s   Two cycles are called  - orthogonal if C C  trC C  0.  For the case when 1, of the cycle and point space are both elliptic or hyperbolic.In this case an orthogonality is the standard one.

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Lie group and Geometry on the Lie Group SL2(R)

figure 5(a)

Elliptic case: The orthogonality between the big circle and any from the middle or small families is given in the usual Euclidean sense.

figure 5(b)

Parabolic case: The orthogonality between the big circle and any from the middle or small families is given in the usual Euclidean sense.

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Lie group and Geometry on the Lie Group SL2(R)

figure 5(c)

Hyperbolic case: The orthogonality between the big circle and any from the middle or small families is given in the usual Euclidean sense.

In the above three picture middle and small group of cycles orthogonal to big cycle group. Point b s belongs to the big cycle C and the family of middle cycle passing through b is orthogonal to . They all also intersect in the point d which is inverse point of b in .

DISTANCE LENGTH AND PERPENDICULARITY:

For discussing geometry the first words comes on our mind is distance and length. Here the question is can we obtain distances and lengths from the cycles.

The distance between two points is the extremum of of all cycles passing through both points.

The length of a directed interval AB is the radius of the cycle with its center or focus at the point A that passes through B.

We mentioned already that for circles normalized by the condition k=1 the value of determinant of is equal to

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Lie group and Geometry on the Lie Group SL2(R)

s s s s s   det C = C C  trC C  gives the of the circle radius. Thus we may keep it as the definition of the radius of the cycle. But then we need to accept that in parabolic case the radius is the distance between roots of the parabola.

figure 6(a)

In figure 6(a) the square of the parabolic is the square of the distance between roots if they are real (z1 and z2) otherwise the negative square of the distance between the adjoint roots (z3 and z4).

 If    the distance in all EPH cases is the following expression

2 2 2 2 de, p,h (u,v)  (u  iv)(u  iv)  u  i v

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Lie group and Geometry on the Lie Group SL2(R)

figure 6(b)

In this figure distance as the extremum of diameters in elliptic (z1 and z2) and parabolic (z3 and z4) cases.

figure 6(c)

In this figure we can see that perpendicular as the shortest route to a line.

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Lie group and Geometry on the Lie Group SL2(R)

9. Importance of Lie group and its Gemetry in Mathematics:

Lie groups are smooth and, therefore, can be studied using differential , in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups is to replace the global object, the group, with its local or linearized version, which Lie himself called its " group" and which has since become known as its .

Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus corresponds to the choice of the group E(3) of distance-preserving transformations of the R3, corresponds to enlarging the group to the , whereas in one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" of a manifold. On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a , this action provides a measure of rigidity and yields a rich . The presence of continuous symmetries expressed via a on a manifold places strong constraints on its geometry and facilitates on the manifold. Linear actions of Lie groups are especially important, and are studied in .

In the 1940s–1950s, Ellis Kolchin, and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary . This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in . The theory of automorphic forms, an important branch of modern , deals extensively with analogues of Lie groups over adele rings; p-adic Lie groups play an important role, via their connections with Galois representations in number theory.

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Lie group and Geometry on the Lie Group SL2(R)

10.Conclusion:

Lie algebra and Lie group is an important part of mathematics, many research are going on in this section. This is useful in as well as .It has very huge application in partial differential equation, statistic, number theory etc. is also applicable in .

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Lie group and Geometry on the Lie Group SL2(R)

11.References:

1.Brian C. Hall ; Lie groups ,Lie algebras ,and Representations ; Springer International

Edition.

2.Vladimir V. Kisil ,Starting with the group SL2(R) , Notices Amer.Math .Sos , volume 54(2007)

number 11 ,page no.-1458-1465.

3.K.B. Dutta ; Matrix and ; Eastern Economy Edition.

4.Frank Ayres ; Theory and Problems of Matrices ; Schaum’s outline series.

5. ALAN F. BEARDON , Algebra and Geometry,Cambridge University Press, Cambridge, 2005. MR 2153234 (2006a:00001)

6. BULLETIN(New Series)OF THE American Mathematical Society, volume 40(Feb 12 ,2007) number -2, pages 253-257 .

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