Lie Groups and Lie Algebras Kumar Balasubramanian Remark: These are a few important definitions, examples and results from the Lie Theory Course.

1. GLn (C) - Group of all n × n invertible matrices with complex entries. 2. Mn (C) - Set of all n × n matrices with entries in C. 3. Let G be any subgroup of GLn (C). G is called a matrix Lie Group if it satisfies the following. If (Am ) is any sequence of matrices in G and Am → A then A ∈ G or A is not invertible. 4. A function A : R → GLn (C) is called a one parameter subgroup if i) A is continuous ii) A(0) = 1 iii) A(t + s) = A(t) + A(s) ∀ t, s ∈ R 5. Let G be a matrix Lie group. The Lie Algebra of G, denoted by g, is the set of all matrices X such that etX is in G for all real numbers t 6. Let G a matrix Lie Group with lie algebra g. Let X ∈ g and A ∈ g, then AXA−1 ∈ g 7. Let G be a matrix Lie group and g be the lie algebra of G. Then g is always a vector space over R. 8. Let G and H be matrix lie groups, with lie algebras g and h respectively. Suppose Φ : G → H is a lie group homomorphism Then there exists a unique real linear map φ : g → h such that i) φ(AXA−1 ) = Φ(A)φ(X)Φ(A)−1

∀A ∈ G 1

ii) φ([X, Y ]) = [φ(X), φ(Y )] iii) dtd t=0 Φ(etX ) = φ(X) Remark:The above theorem says, every Lie Group homomorphism gives a Lie Algebra Homomorphism. Given a Lie algebra homomorphism φ : g → gl(V ) we can extend it to a lie group homomorphism Φ : G → GL(V ) if G is simply connected. 9. Adjoint mapping: G be a matrix Lie group with lie algebra g. For each A ∈ G, define a linear map AdA : g → g by AdA (X) = AXA−1 . Then the map AdA satisfies the following. i) AdA ∈ GL(g) A ∈ G. ii) AdA ([X, Y ]) = [AdA (X), AdA (Y )] 10. Ad : G → GL(g) (A → AdA ) is a lie group homomorphism. Let ad : g → gl(g) be the corresponding lie algebra homomorphism. Then eadX = Ad(eX ) ∀X ∈ g 11. If G is a connected matrix lie group, then every element A in G can be written as A = eX1 . . . eXm , X1 , X2 , ..., Xm ∈ g 12. Every continuous homomorphism between any two matrix lie groups is smooth. 13. A finite dimensional real or complex lie algebra is a real or complex vector space g, together with a map i) [, ]is bilinear ii) [X, Y ] = −[Y, X] iii) [X[Y, Z]]+[Y [Z, X]]+[Z[X, Y ]] = 0,

∀X, Y, Z ∈ g (Jacobi Identity)

14. Ado’s Theorem: Every finite dimensional real or complex lie algebra is isomorphic to a real or complex subalgebra of gl(n, R) or gl(n, C)

2

15. Exponential mapping and it’s properties: Let G be a lie group and g be the lie algebra of G. For X ∈ g we define the exponential as eX =

∞ X Xm m! m=0

16. Properties of the exponential mapping i) e0 = I ii) (eX )−1 = e−X iii) det(eX )= etrace(X) iv) e(α+β)X = eαX eβX v) eX+Y = eX + eY vi) eCDC

−1

∀α, β ∈ C if

XY = Y X

= CeD C −1

17. Lie Product Formula X

Y

eX+Y = lim e m e m


m

m→∞

18. Let G be a matrix lie group with lie algebra g. H ⊂ G is called an analytic subgroup or connected lie subgroup of G if i) H is a subgroup of G ii) Lie(H)=h is a subspace of g=Lie(G) iii) Every element of H can be written in the form eX1 . . . eXm , h

X1 , . . . , Xm ∈

19. Let G be a matrix lie group with lie algebra g and let Π be a finite dimensional real or complex representation of G, acting on the space V . Then there is a unique representation π of g acting on the same space V satisfying i) Π(eX ) = eπ(X) ∀ X ∈ g ii) π(X) = dtd t=0 Π(etX ) 3

iii) π(AXA−1 ) = Π(A)π(X)Π(A)−1

∀ X ∈ g,

∀ A∈G

20. A finite dimensional representation of a group or lie algebra is said to be completely reducible if it is isomorphic to a direct sum of irreducible irreducibles. 21. A group G is said to have the complete reducibility property, if every representation of G is completely reducible. Remark: Some things to remember on complete reducibility of representations. i) (Π, V ) be a unitary representation of a group G (lie algebra g). Then (π, V ) is always completely reducible. ii) Some examples of compact groups O(n), SO(n), U (n), SU (n), Sp(n) 22. Universal Property of Tensor Products: If U and V are finite dimensional real or complex Nvector spaces, then a tensor product of U with V is a vector space W = U V , together with a bilinear map φ : U ×V → W with the following property. If ψ is any other bilinear map from U × V → X, then there exists a unique linear map ψe : W → X such that the diagram commutes. φ

/W U ×VP PPP PPP P ψ˜ ψ PPPP P(  X.

23. Theorem of highest weight for sl(3, C) i) Every irreducible representation π of sl(3, C) is direct sum of its weight spaces. ii) Every irreducible representation of sl(3, C) has a unique highest weight ν0 and two equivalent irreducible representations have the same highest weight. iii) Two irreducible representations of sl(3, C) with the same highest weight are equivalent. 4

iv) If π is an irreducible representation of sl(3, C), then the highest weight µ0 of π is of the form µo = (m1 , m2 ), m1 , m2 are non-negative integers. v) If m1 , m2 are non-negative integers, then there exists an irreducible representation of sl(3, C) with the highest weight µ0 = (m1 , m2 ). vi) The dimension of the irreducible representation with highest weight µ0 = (m1 , m2 ) is 21 (m1 + 1)(m2 + 1)(m1 + m2 + 2). 24. A complex lie algebra g is called indecomposable if the only ideals in g are 0 and g. 25. A complex lie algebra g is called simple if the only ideals in g are 0 and g and dimg ≥ 2. 26. A complex lie algebra g is called reductive if g is isomorphic to a direct sum of indecomposable lie algebras. 27. A complex lie algebra g is called semisimple if g is isomorphic to a direct sum of simple lie algebras. 28. An important characterization of semisimple Lie Algebras: A complex lie algebra is semisimple iff it is isomorphic to the complexification of the lie algebra of a simply connected compact matrix lie group. i.e g∼ = kC where k = Lie(K) is the lie algebra of the simply connected compact matrix lie group K 29. An Example of a complex Semisimple Lie Algebra sl(3, C) ∼ = su(3)C

and su(3) = Lie(SU (3))

Remark: SU (3) is compact and simply connected. 30. Weyl Group of SU(3): Let h = CH1 + CH2  Z = A ∈ SU (3) | AdA (H) = H ∀ H ∈ h  N = A ∈ SU (3) | AdA (H) ∈ h ∀ H ∈ h

5

Z and N are subgroups of SU (3) and Z is a normal subgroup of N . The Z . Weyl Group W is defined to be the quotient group N 31. Action of the Weyl group on h: For each element w ∈ W , choose an element A of the corresponding equivalence class of N . Then for H ∈ h define the action of w.H of w on H by w.H = AdA (H) 32. Different notions of weights i) An ordered pair (m1 , m2 ) ∈ C2 is called a weight for π if there existes v 6= 0 ∈ V such that π(H1 )v = m1 v and π(H2 )v = m2 v ii) Weights as elements of h∗ . Let h = CH1 + CH2 be the Cartan Subalgebra. Let µ ∈ h∗ . µ is called a weight for π if there exists v 6= 0 ∈ V such that π(H)v = µ(H)v ∀ H ∈ h. iii) Weights as elements of h. We choose an inner product on h which is invariant under the action of the Weyl Group. Innerproduct on h < A, B >= trace(A∗ B) For each α ∈ h, define α ∈ h∗ as α(H) =< α, H > . We use this inner product to identify h with h∗ . Now we can look at the weights as elements of h. Let α ∈ h. α is called a weight for π if there exists v 6= 0 ∈ V such that π(H)v =< α, H > ∀ H ∈ h 34. If π is any finite dimensional representation of sl(3, C) and µ ∈ h∗ is a weight for π then for any w ∈ W , w.µ is also a weight for π, and the multiplicity of w.µ is the same as the multiplicity of µ. i.e. The Weyl Group leave the weights and their multiplicities invariant. 35. Suppose that π is an irreducible representation of sl(3, C) with highest weight µ0 . Then, an element µ of h is a weight for π iff the following two conditions are satisfied: i) µ is contained in the convex hull of the orbit of µ0 under the weyl group. 6

ii) µ0 −µ is expressible as a linear combination of the positive simple roots α1 and α2 with integer coefficients. 35. Compact Real Forms for Complex Semisimple Lie Algebras: Let g be a complex semisimple Lie Algebra. A compact real form of g is a real subalgebra l of g with the property that every element X ∈ g can be uniquely written as X = X1 + iX2 (i.e lC = g) and such that there exists a compact simply connected matrix Lie group K1 such that the Lie algebra l1 of K1 is isomorphic to l. Remark: For a complex semisimple Lie algebra g, a compact real form always exists. 36. Some examples of compact real forms for complex semisimple Lie algebras i) Let g = sl(n, C) and l = su(n). l = su(n) is clearly a real subalgebra of g = sl(n, C) = su(n)C and su(n) is the Lie algebra of the compact simply connected matrix Lie group SU(n) ii) Let g = so(3, C) and l = so(3). l = so(3) is clearly a real subalgebra of so(3, C) = su(2)C and so(3) ∼ = su(2) which is the Lie algebra of the compact simply connected matrix Lie group SU(2)

7