Matrix Lie groups. and their Lie algebras. Mahmood Alaghmandan. A project in fulfillment of the requirement for the Lie algebra course

Matrix Lie groups and their Lie algebras Mahmood Alaghmandan A project in fulfillment of the requirement for the Lie algebra course Department of Mathematics and Statistics University of Saskatchewan March 2012 Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 1 / 36

Special orthogonal group An example of Lie groups A matrix A is orthogonal if its transpose is equal to its inverse: Equivalently A tr = A 1. AA tr = I If for orthogonal matrix A, det(a) = 1, A is call special orthogonal matrix. Orthogonal and special orthogonal groups O(n) = {A M n (R) : A tr = A 1 } and SO(n) = {A M n (R) : A tr = A 1, det(a) = 1}. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 2 / 36

Example. SO(2) is the set of all matrices [ cos θ sin θ sin θ cos θ for different angels θ. ] The rotation with angle θ. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 3 / 36

Example. SO(3) forms the set all of rotations in R 3 which preserves the length and orientation. Every rotation in SO(3) fixes a unique 1-dimensional linear subspace of R 3, due to Euler s rotation theorem. cos θ sin θ 0 sin θ cos θ 0 0 0 1 is the rotation with angel θ with the axis of rotation z-axis. (0.1) Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 4 / 36

Applying complex matrices! σ 1 = [ 0 1 1 0 ] [ 0 i, σ 2 = i 0 ] [ 1 0, σ 3 = 0 1 called Pauli matrices are linearly independent. So for each x = (x i ) 3 i=1 R3, x = x 1 x 2 x 3 = x 1 σ 1 + x 2 σ 2 + x 3 σ 3. ] Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 5 / 36

A new interpretation Note that [ e iθ/2 0 0 e iθ/2 ] [ e iθ/2 0 (x 1 σ 1 + x 2 σ 2 + x 3 σ 3 ) 0 e iθ/2 ] 1 = (x 1 cos θ + x 2 sin θ)σ 1 + ( x 1 sin θ + x 2 cos θ)σ 2 + x 3 σ 3. As clearly we can see this is the rotation that appeared in a matrix in SO(3), (0.1). Special unitary group SU(2) Let SU(2) be the set of all matrices [ ] a + di b ci A = where det(a) = 1 and a, b, c, d R. b ci a di We can see that these matrices form a group, we call them special unitary matrices. Although this is a complex counterpart of SO(2), it has a strong relation with SO(3). Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 6 / 36

Special unitary group A complex n n matrix A is called unitary if its columns are orthogonal. (Linear algebra) over Hilbert spaces H = C n it forms a unitary operator i.e. A = A 1 when A is the adjoint operator of A. (Functional Analysis) Special Unitary group SU(n) = {A M n (C) : A A = AA = I and det(a) = 1.} Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 7 / 36

Outline 1 Matrix Lie groups 2 Matrix exponential 3 Lie algebra of a matrix Lie group 4 Lie algebra of some famous matrix Lie groups 5 Geometric interpretation Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 8 / 36

General linear group GL(n, F) General linear group Set of all invertible matrices on field F also form a group with the regular multiplication of matrices denoted by GL(n, F). Special linear group All invertible matrices A GL(n, F) s.t. det(a) = 1 denoted by SL(n, F). All groups GL(n, R), SL(n, R), SL(n, C), SU(n), and SO(n) are all subgroups of GL(n, C) for an appropriate n. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 9 / 36

Topology on M n (C) Hilbert-Schmidt norm for matrix A = [a i,j ] i,j 1,,n is n A 2 := a i,j 2 i,j=1 1/2. Convergence Let (A m ) m be a sequence in M n (C). A m converges to A if A m A 2 0. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 10 / 36

Matrix Lie group Matrix (linear) Lie group A matrix Lie group G is a subgroup of GL(n, C) such that if (A m ) m is any convergent sequence in G, either A G or A is not invertible. O(n), SO(n), SU(n), SL(n, R), SL(n, C), GL(n, R), and GL(n, C) are all matrix Lie groups. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 11 / 36

Matrix exponential For A M n (C), we define e A := k=0 1 k! Ak where A 0 = I. It is well defined: e A 2 k=0 1 k! A k 2 = e A 2 < ; Main properties of e A e 0 = I, ( e A) = e (A ), ( e A) tr = e (A tr ), and e (α+β)a = e αa e βa. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 13 / 36

More properties Matrix exponential essential properties If AB = BA, then e A+B = e A e B = e B e A. e A is invertible and (e A ) 1 = e A. If A is invertible, then e ABA 1 = Ae B A 1. Even if [A, B] 0, we have ( e A+B = lim e 1 k A e 1 B) k k k (2.1) called Lie product formula. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 14 / 36

Calculation of the exponential of a matrix Calculation of the exponential of nilpotent matrices is easy. For diagonalizable matrices also we have a good trick to simplify the calculations. For an arbitrary matrix we can write as summation of one nilpotent matrix and one diagonalizable matrix which commute. The relation of determinant and trace through matrix exponential Let A M n (C). Then det(e A ) = e trace(a). Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 15 / 36

Derivative of matrix exponential function Moreover A e A is a continuous map: If A m A, then e Am e A. So we can talk about the limit of the following matrices where t 0 e (t 0+ t)a e t 0A t Therefore, d e (t0+ t)a e t 0A dt (eta ) t=t0 = lim t 0 t = Ae t 0A Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 16 / 36

Lie algebra of a matrix Lie group Definition of Lie algebra of G For G be a matrix Lie group. g := {A M n (C) : e ta G for all t R}. Note that: t should be real. For A g, e ta G for all real t. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 18 / 36

Question. Is the Lie algebra of a matrix Lie group really a Lie algebra? Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 19 / 36

YES! It is really a (REAL) Lie algebra Lie bracket over matrices! For A g, ta g for all t R. A + B g if A, B g. Proof. If [A, B] = 0, then e A+B = e A e B. In general we apply Lie product formula, (2.1). ( (1) For each k, e 1 k A e 1 B) k k G. (2) Its limit (e A+B ) must belong to G. Proof. [A, B] g if A, B g. G e ta e B e ta = e eta Be ta = e ta Be ta g Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 20 / 36

[A, B] g if A, B g. Rest of the proof. But d dt (eta Be ta ) t=0 = d dt (eta ) t=0 Be 0A + e 0A d dt (Be ta ) t=0 = AB BA. Why does it show that AB BA g? Because: d dt (eta Be ta e ta Be ta B ) t=0 = lim t 0 t AND g is a complete vector space! ( g A k A then lim k e A k = e A. So e A G. Hence g is complete!) Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 21 / 36

Lie algebra of a matrix Lie group Anticommutativity and Jacobi identity are achievable because of the definition of the Lie bracket over matrices. Summary Let G be a matrix Lie group. Then g is a vector space over R. As a subspace of M n (C), g is a complete topological space. Moreover, g is a Lie algebra with Lie bracket over matrices over field R. In the following we will see that for a variety of matrix Lie groups, Lie algebra will actually form a complex Lie algebra. We call these groups, complex matrix Lie groups. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 22 / 36

Why should we care about the Lie algebra of a matrix Lie group? Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 23 / 36

They are somehow related to their matrix Lie groups! Proposition Let G and H be two matrix Lie groups with Lie algebras g and h respectively. Suppose that Φ : G H is a group homomorphism. The there exists an (Lie) algebra homomorphism φ : g h such that Φ(e A ) = e φ(a) for every A g. The converse of This proposition is not correct in general. There are some conditions if we impose we will see the converse. For example, if both groups are simply connected, every homomorphism between Lie algebras ends to a homomorphism between matrix Lie groups Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 24 / 36

The Lie algebra of general linear group GL(n, F) A is an arbitrary matrix in M n (C), e ta is invertible for all t R. So e ta GL(n, C). Hence, gl(n, C) = M n (C). If A is a real matrix, e ta is real for all t R. On the other hand, if e ta is a real matrix for every t R, we have that A = d e ta I dt (eta ) t=0 = lim t 0 t should be also real, because e ta I t M n (R) and M n (R) is a closed topological space. Therefore, gl(n, R) = M n (R). This argument will work for all real valued matrix Lie groups. It shows that: matrix Lie group is REAL = its Lie algebra is REAL. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 26 / 36

The Lie algebra of special linear group SL(n, F) det(e ta ) = e t trace(a). So trace(a) = 0, if e ta SL(n, C). On the other hand, if trace(a) = 0, det(e ta ) = e t trace(a) = 1 and so e ta SL(n, C). sl(n, C) = {A M n (C) : trace(a) = 0}. Similarly for SL(n, R), sl(n, R) = {A M n (R) : trace(a) = 0}. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 27 / 36

Special Unitary groups. e ta SU(n) if for each t R, (e ta ) = (e ta ) 1 = e ta. Also (e ta ) = e ta. Since e ta = e ta, we have A = d dt (eta ) t=0 = d dt (e ta ) t=0 = A. On the other hand, if A = A (e ta ) = (e ta ) = e ta = (e ta ) 1 which implies that A belong to the Lie algebra of SU(n). So su(n) = {A M n (C) : A = A and trace(a) = 0}. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 28 / 36

Orthogonal and Special orthogonal group We can definitely predict that o(n) and so(n), the Lie algebras of O(n) and SO(n) respectively, are real matrices. If A tr = A, for all t R. So A o(n). (e ta ) tr = e tatr = e ta = (e ta ) 1 Conversely, if A o(n), for each t we have e ta = (e ta ) 1 = (e ta ) tr = e tatr. If we get derivative from this equation for t = 0 we have A = d dt (e ta ) t=0 = d dt (etatr ) t=0 = A tr. Also note that A tr = A forces the entries of the main diagonal to be zero, so automatically trace(a) = 0. Therefore, o(n) = so(n) = {A M n (R) : A tr = A}. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 29 / 36

Lie groups Manifold A manifold is a second countable Hausdorff topological space M that is locally homemorphic to an open subset of R n. What does it mean? It means that for each point x M there exists a neighborhood U of x and a homeomorphism φ from U onto some open set in R n. Definition of Lie group A Lie group G is a group as well as a manifold so that the group operation i.e. (x, y) xy and the mapping inverse of the group i.e. x x 1 both are smooth functions. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 31 / 36

Matrix Lie groups are Lie groups GL(n, C) is a Lie group We know that det : M n (C) C is a continuous map; therefore, GL(n, C) = det 1 (C \ {0}) is an open subset of C n ( R 2n ). Moreover, if we look at matrix product as a combination of coordinate wise multiplication, we can judge that this product is smooth. Consequently, the process of generating the inverse of a matrix is a smooth function. Hence, GL(n, C) is a Lie group. Every closed subgroup of Lie group G is a Lie group. But the definition of linear Lie group implies that every matrix Lie group indeed is a Lie group, since it forms a closed subgroup of GL(n, C) for some n. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 32 / 36

The tangent space of Lie groups SO(2) is nothing but the simplest manifold, a circle! For each matrix A so(2), we have that A = So so(2) is bijective with R [ 0 a a 0 ] for some a R. The tangent line at the identity of the group SO(2) is bijective with so(2). Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 33 / 36

Indeed, general theory of Lie groups shows that for each Lie group G, the tangent space of G at the identity of the group is bijective with a Lie algebra. Specially for matrix Lie group G, the tangent space of G at the identity of the group is bijective with g. Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 34 / 36

References. HALL, BRIAN C., Lie groups, Lie algebras, and representations. An elementary introduction. Graduate Texts in Mathematics, 222. Springer-Verlag, New York, 2003. SEPANSKI, MARK R., Compact Lie groups. Graduate Texts in Mathematics, 235. Springer, New York, 2007. STILLWELL, JOHN, Naive Lie theory. Undergraduate Texts in Mathematics. Springer, New York, 2008. WESTRA, D. B., SU(2) and SO(3), http://www.mat.univie.ac.at/ westra/so3su2.pdf. Cartoons from http://thenoisychannel.com Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 35 / 36

Thank You! Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 36 / 36