A full description of the material covered in the book is given later in the introduction. In Chapter 9 we have added a discussion of the Littlewood-Richardson rule (including the role of the branching law GL(n,C) to reduce the proof to a well-known combinatorial construction).
The Classical Groups
- General and Special Linear Groups
- Isometry Groups of Bilinear Forms
- Unitary Groups
- Quaternionic Groups
- Exercises
Claim 1.1.6. Let V be a 2n-dimensional vector space over F and let B be a nondegenerate skew-symmetric bilinear form on V. Note that if M is a Hermitian form, then M(v,v)∈R for allv∈V.) We define U(V,B) (also denoted by U(B)when V is understood) to be the group of all elements g∈GL(V)such that B(gv,gw) =B(v,w) for allv,w∈ V.
The Classical Lie Algebras
- General and Special Linear Lie Algebras
- Lie Algebras Associated with Bilinear Forms
- Unitary Lie Algebras
- Quaternionic Lie Algebras
- Lie Algebras Associated with Classical Groups
- Exercises
We will denote this Lie algebra by gl(n,H), treated as a Lie algebra over R (we have not defined Lie algebras over oblique fields). These Lie algebras will be the main subject of study in the rest of this book.
Closed Subgroups of GL(n, R )
- Topological Groups
- Exponential Map
- Lie Algebra of a Closed Subgroup of GL(n, R )
- Lie Algebras of the Classical Groups
- Exponential Coordinates on Closed Subgroups
- Differentials of Homomorphisms
- Lie Algebras and Vector Fields
- Exercises
From Lemma 1.3.6, we now obtain fundamental identities that connect the Lie algebra structure gl(n,R) with the group structure GL(n,R). Therefore, the Lie algebra of H is the same as the Lie algebra of the identity component of H.
Linear Algebraic Groups
- Definitions and Examples
- Regular Functions
- Lie Algebra of an Algebraic Group
- Algebraic Groups as Lie Groups
- Exercises
We now show that a linear algebraic group overCis is a Lie group and that the Lie algebra defined by continuous one-parameter subgroups coincides with the Lie algebra defined by left-invariant derivations of the algebra of using regular functions. Conclude that the Lie algebra of GΓ is the same whether GΓ is considered a Lie group or a linear algebraic group.
Rational Representations
- Definitions and Examples
- Differential of a Rational Representation
- The Adjoint Representation
- Exercises
We say that (ρ,V) is locally regular if every finite-dimensional subspaceE⊂V is contained in a finite-dimensional G-invariant subspaceFsuch that the restriction of ρtoFi is a regular representation. If (ρ,V) is a regular representation and W ⊂V is a linear subspace, then we say that W is G-invariant ρ(g)w∈W for allg∈Gandw∈W. In terms of representative functions, we have Eπ1⊗π2=Span(Eπ1·Eπ2). the sums of products of representative functions of π1 and π2).
If H⊂GL(n,C) is another algebraic group with Lie algebra, we denote the additional representations of G and H by AdG and AdH, respectively. 1.51) Theorem 1.5.7. The differential of the additive representation of G is the representationad :g //End(g) given by.
Jordan Decomposition
- Rational Representations of C
- Rational Representations of C ×
- Jordan–Chevalley Decomposition
- Exercises
From the argument above and the uniqueness of the Jordaan decomposition, we deduce that the constraints of R(s)andR(u)toWm provide the semi-simple and powerless factors for the constraint ofR(g). Starting with the addition Jordan decomposition A=S+NinMn(C), we likewise see that the constraints of XS and XN toWm provide the semisimple and nilpotent parts of the constraint of XA. With these properties of the Jordan decompositions established, we can complete the proof as follows.
By the uniqueness of the Jordaan decomposition, it is therefore sufficient to prove that dρ(S) and ρ(s) are semisimple.
Real Forms of Complex Algebraic Groups
Real Forms and Complex Conjugations
Therefore, the dimension of the Lie group GR is the same as the dimension of Gas a linear algebraic group over C (see Appendix A.1.6). By generalizing the notion of complex conjugation, we now obtain a useful criterion (not involving a specific matrix form of G) for G to be isomorphic to a linear algebraic group defined over R. Then there exists a linear algebraic group H ⊂GL(n,C) defined over Rand an isomorphismρ:G //H such thatρ(τ(g)) =σ(ρ(g)), whereσ is the conjugation of GL( n,C) ) given by complex conjugation of matrix entries.
Let be a linear algebraic group and let G◦ be a connected component of the identity G (as a real Lie group).
Real Forms of the Classical Groups
The equationτ(g) =g can be written as g∗Ip,qg=Ip,q, so the indefinite unit groupU(p,q)(resp.SU(p,q)) defined in Section 1.1.3 is the real form ofGdefined byτ. We leave it as an exercise to show that the corresponding real form is isomorphic to the group O(p,q)(resp.SO(p,q)) defined in Section 1.1.2. Of these, only SL(n,C) is an algebraic group over C, while the other two are real forms of SL(n,C) (resp.SL(2n,C)).
In a real vector space, a Hermitian (respectively anor-Hermitian) form is the same as a symmetric (respectively anor-symmetric) form.
Exercises
On a complex vector space, skew-Hermitian forms become Hermitian after multiplication by i, and vice versa. Taking these restrictions into account, we see that the possibilities for unimodular isometry groups are those in Table 1.1. Note that the group SU(p,q) is not an algebraic group over C, even though the field is C, since its defining equations involve complex conjugation.
Show that the map g7→(g∗)−1 defines an undesirable automorphism of Gas a real Lie group, and thatg7→τ(g) =Kp,q(g∗)−1Kp,qdefines a complex conjugation of G.
Notes
Using unipotent elements in G, we show that the groups GL(n,C),SL(n,C),SO(n,C) and Sp(n,C) are connected (as Lie groups and as algebraic groups). This group and its Lie algebra play a fundamental role in the structure of other classical groups and Lie algebras. We decompose the Lie algebra of the classical group under the adjoint action of the maximal torus and find the invariant subspaces (called root spaces) and the corresponding signs (called roots).
In the last part of the chapter, we develop some general Lie algebra methods (solvable Lie algebras, Killing form) and show that every semisimple Lie algebra has a root space decomposition with the same properties as classical Lie algebras.
Semisimple Elements
Toral Groups
The rank is uniquely determined by the structure of the algebraic group T (this follows from Lemma 2.1.2 below or Exercise 1.4.5 #1). Choose a set∈T such that its coordinates=xi(t) satisfy. 2.3) This is always possible; for example we can take 1,. Since f(tn) =0 for all n∈Z, the coefficients {aK} satisfy the equations. 2.4) We claim that the numbers {tK :K∈Nl} are all distinct.
Maximal Torus in a Classical Group
If we write the elements of Mn(C) in the same block form and do the matrix calculation from equation (1.8), we find that the Lie algebraso(C2l+1,B)ofSO(C2l+1,B) consists of all matrices . According to (2.12), the subspaces Vµi and V1/µii are ωisotropic and the restriction ω on Vµi×V1/µii is not degenerate. a) the subspaces V1, V−1 and Wi are mutually orthogonal with respect to the form ω and the restriction ω to each of these subspaces is not degenerate;. By Lemma 1.1.5, we can find canonical Symplectic bases in each of the subspaces in the decomposition (b); in the case of Wiwe, it can take the base v1,.
As in the complex case, we construct B-isotropic canonical bases in each of the subspaces in the decomposition (b) (see Section B.2.1); the union of these bases gives an isotropic basis for Cn.
Exercises
By Corollary 2.1.8, we see that integerl=dimH does not depend on the choice of a particular maximal torus in G.
Unipotent Elements
- Low-Rank Examples
- Unipotent Generation of Classical Groups
- Connected Groups
- Exercises
On nilpotent elements in Lie(G) the exponential map is algebraic and maps them to unipotent elements in G. This gives an algebraic connection from Lie algebra representations to group representations, provided that the unipotent elements generate G. Hence the new argument given for SO(3,C) applies in this case, and we conclude that SO(4,C) is created from its unipotent elements.
The diagonal subgroup of SO(C2l,B) is isomorphic to the diagonal subgroup of SO(C2l+1,B) via this embedding, so it suffices to prove that every diagonal element in SO(Cn,B) is a product of unipotent elements when is even.
- Irreducible Representations of sl(2, C )
- Irreducible Regular Representations of SL(2, C )
- Complete Reducibility of SL(2, C )
- Exercises
We can describe the operation of the subspace W in Lemma 2.3.2 in matrix form as follows: Fork∈N defines the (k+1)×(k+1) matrices. By Lemma 2.3.2, we know that λ is a nonnegative integer, and the space spanned by the set {w0,yw0,y2w0, extends. Henceλ=k, and by Lemma 2.3.1 the matrices of the actions of x,y,h with respect to the ordered basis {w0,yw0,.
Find regular irreducible representations of SO(3,C). c) Let Veven⊂V be the space of even polynomials and Vodd⊂V be the space of odd polynomials.
The Adjoint Representation
- Roots with Respect to a Maximal Torus
- Commutation Relations of Root Spaces
- Structure of Classical Root Systems
- Irreducibility of the Adjoint Representation
- Exercises
From this it is clear that the restriction of the trace form to its non-degenerate. We write the kernelHi in terms of the elementary diagonal matricesEi=ei,i, as in Section 2.4.1. This can be obtained from the Dynkin diagram of type Bl by reversing the arrow on the double bond and reversing the coefficients of the highest root.
For each type of classical group, write the roots in terms of theεi (after identifying hwithh∗as in Section 2.4.1).
Semisimple Lie Algebras
- Solvable Lie Algebras
- Root Space Decomposition
- Geometry of Root Systems
- Conjugacy of Cartan Subalgebras
- Exercises
It is clear from the definition that a Lie subalgebra of a solvable Lie algebra is also solvable. Moreover, if is a solvable nonzero Lie algebra and we choose such that Dk(g)6=0 and Dk+1(g) =0, then Dk(g) is an abelian ideal that is invariant under all derivations ofg. Then the bilinear form tr(XY)ong is nondegenerate, andg=g1⊕ ··· ⊕gr (Direct sum of Lie algebra), where each is a simple Lie algebra.
Since the additive representation of a simple Lie algebra is true, the same holds for a semisimple Lie algebra.
Notes
Summary In this chapter we study the regular representations of a classical group Gby the same method used for the adjoint representation. The maximum weight theorem asserts that among the weights appearing in the decomposition, there is a unique maximal element, relative to a partial order that comes from a choice of positive roots for G. We prove that every dominant integral weight of a semisimple Lie algebra is the highest weight of an irreducible finite-dimensional representation of g.
When is the Lie algebra of a classical group G, the corresponding regular Race representations are constructed in Chapters 5 and 6 and studied in more detail in Chapters 9 and 10.
Roots and Weights
- Weyl Group
- Root Reflections
- Weight Lattice
- Dominant Weights
- Exercises
From the proof of Theorem 3.1.1, we know that every coset inWG is of the form sσH for someσ ∈Sn. The set of dominant weights is contained in the positive Weyl chamber (a cone with opening 45◦ ), and the effect of the Weyl group is generated by the reflections in the dashed lines that bound the chamber. We now prove that any weight can be transformed into a unique dominant weight using the Weyl group.
This means that the dominant weights give a cross section for Weyl group orbits in the weight lattice.
Irreducible Representations
- Theorem of the Highest Weight
- Weights of Irreducible Representations
- Lowest Weights and Dual Representations
- Symplectic and Orthogonal Representations
- Exercises
From Corollary 3.2.3, we conclude that if V is an irreducible finite-dimensional g-module, then it has a unique minimum weight µ∈ −P++(g), denoted by the property µν for allν∈X(V). In Section 5.5.2, we will show that σ is the differential of the natural representation SL(n,C)on VrCn. Let (π,V) be an irreducible-module with maximum weightϖi. a) Show that π is a rectangular representation in the following cases:.
Let(π,VAN)be the irreducible module with the highest weightρ=ϖ1+···+ϖl (the smallest regular dominant weight). a) Show that π is an orthogonal representation in the following cases:
Reductivity of Classical Groups
- Reductive Groups
- Casimir Operator
- Algebraic Proof of Complete Reducibility
- The Unitarian Trick
- Exercises
Since λ is the maximum weight of V, Theorem 2.3.6 applied to the subalgebra s(α) for α ∈Φ+ shows that λ is dominant (as in the proof of Corollary 3.2.3). We prepare the proof of Theorem 3.3.12 by the following lemma, which is analogous to Lemma 2.3.7, but with the Casimir operator replacing the diagonal element hinsl(2,C). From Theorem 2.2.5 we know that the only non-connected classical groups are the groups O(n,C)forn≥3.
Notation of section 3.3.2) Let be a semisimple Lie algebra and the finite-dimensional g-module with highest weightλ.
Notes
Abstract In this chapter we develop some algebraic tools necessary for the general theory of representations and invariants. The duality between regular irreducible representations of irreducible Gand representations of the commutative algebra of G plays a fundamental role in classical invariant theory. We study the representations of a finite group through the algebra of the group and its characters, construct induced representations and compute their characters.
Representations of Associative Algebras
Definitions and Examples
We identify g∈G with the element δg∈A[G], and define multiplication in A[G] as the bilinear extension of group multiplication. As in the case of group algebras, there exists an associative algebra U(g) (the universal envelope algebraofg) and an injective linear map j:g //U(g) such that j(g) generates U(g)and. As in the case of group algebras, the properties of these maps can be described axiomatically using the notion of aHopf algebra (see Exercises 4.1.8).
Diek-fold tensor powers of ρ and the symmetric and skew-symmetric powers are defined by analogy with the case of group representations.
Schur’s Lemma
Jacobson Density Theorem
