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 respec- tively. 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 )]

d tX iii) dt t=0 Φ(e ) = 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 −1 each A ∈ G, deﬁne a linear map AdA : g → g by AdA(X) = AXA . 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 X1 Xm written as A = e . . . e ,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

∞ X Xm eX = 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 ∀α, β ∈ C v) eX+Y = eX + eY if XY = YX

vi) eCDC−1 = CeDC−1

17. Lie Product Formula

m X+Y X Y e = lim e m e 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)

X1 Xm iii) Every element of H can be written in the form e . . . e ,X1,...,Xm ∈ h

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

d tX ii) π(X) = dt t=0 Π(e )

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 vector spaces, then a tensor product of U with V is a vector space W = U N 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 U × V / W P PPP PPP ψ˜ ψ PPP PPP ( 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 1 µ0 = (m1, m2) is 2 (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 complexiﬁca- tion 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

2 i) An ordered pair (m1, m2) ∈ C is called a weight for π if there existes v 6= 0 ∈ V such that π(H1)v = m1v and π(H2)v = m2v

∗ 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)