INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group 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 Mathematics 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 1 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 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear groups GL(n;R) and GL(n;C) : ..................................................................... 6 6.2.The special linear group SL(n;R) and SL(n;C) ............................................................................ 6 6.3.Orthogonal O(n) and special orthogonal SO(n) groups: .......................................................... 8 O(n)=set of all nxn orthogonal matrices ..................................................................................... 8 6.4. The Heisenberg group H: ....................................................................................................... 9 7. Sub-group of GL(n;C)which is not matrix 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 3 Lie group and Geometry on the Lie Group SL2(R) 1.Introduction We are all very familiar with the group of real number, group complex number 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 addition to discussing Lie groups and Lie algebras, Borel pays considerable attention to algebraic groups, beginning with the little known work of Ludwig Maurer in the 1890s up through the work of Claude Chevalley, Ellis Kolchin , Borel and others in the post-World War II era. In the beginning, Sophus Lie hoped to develop a Galois Theory for differential equations. Felix Klein's famous \Erlangen Program," which aimed to connect firmly geometry with group theory, 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, Wilhelm Killing had succeeded in classifying (modulo 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 space present upon which the Lie group acted. The notion of an abstract group with the compatible structure of a complex manifold 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. 4 Lie group and Geometry on the Lie Group SL2(R) 2.Definition of general linear group: The general linear group over the real numbers, 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 multiplication. 3.Definition of a general Lie group: A matrix lie group is any subgroup G of GL(n;C) with the following property: If Am is any sequence 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 invertible matrix 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 function 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 time. 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. 5 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 real number 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 determinant 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 matrix multiplication. 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 identity matrix plays the role of identity element 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 6 Lie group and Geometry on the Lie Group SL2(R) lim (det An)= det A (as determinant is a continuous function) 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 Iwasawa decomposition 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 7 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 <x,y>=<Ax,Ay> ,where x,y are in Rn Another way , A is said to be orthogonal if AAt=AtA=I , where At is transpose of A.