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Lecture 8 - Universal Enveloping Algebras and Related Concepts, II February 8, 2013 1 Filtrations and Graded Algebras 1.1 The basics An algebra U has a filtration if there are subsets {U (i) }i∈Z of U so that i) · · · ⊆ U (i) ⊆ U (i+1) . . . ii) U(i) U (j) ⊆ U (i+j) S iii) U = i U (i) . A stronger notion is that of a grading. An algebra U is called a graded algebra if it has subsets {U i } so that i) U i U j ⊆ U i+j L ii) U = i U (i) . We have the following main points: Homogeneous Elements. An element x of a graded algebra U is called homogeneous if x ∈ U i for some i. The decomposition of a general element x ∈ U is X x = xi (1) i where xi = π i (x) is the vector-space projection onto the ith summand. 1 Canonical Filtration. Given a grading on U , its canonical filtration is given by M U (i) = Uk (2) k≤i Associated Graded Algebra. Given a filtered algebra U , we can define its associated graded algebra (which is usually a different algebra), by first setting i U = U (i) / U (i−1) . i j (3) i+j After noting that U U ⊆ U , we see that the product on U passes to a new product L i on i U , which we take to be the associated graded algebra: M i (4) U , U . i Graded Ideals. If U is graded and I is an ideal, then I isLcalled a graded ideal provided that if x ∈ I, then also each π i (x) ∈ I. That is, I = i I ∩ U i . If U is graded and I is a graded ideal, then U/I is graded. Inherited Filtrations. If π : A → B is a surjection of algebras and A is filtered by {A(i) }, then B is filtered by {B (i) , π(A(i) )}. 1.2 1.2.1 Examples Tensor, symmetric, and antisymmetric algebras Given a vector space V the tensor algebra T V is a graded algebra T V = T i V = V ⊗i . The ideal IS generated by elements of the form L i≥0 T i V with x⊕y − y ⊕x (5) x⊕y + y ⊕x (6) and IA generated by elements are both graded ideals, so, as we have seen, generate the graded algebras M SV , T V / IS = SiV i ^ M ^i V , T V / IA = V (7) i where S i V = spanF x1 · · · xi xl ∈ V ^i V = spanF x1 ∧ · · · ∧ xi xl ∈ V 2 (8) 1.2.2 The Clifford Algebras The Clifford algebras, on the other hand, are quotients by ideas generated by the nonhomogeneous elements x ⊗ y + y ⊗ x + 2 hx, yi, and so do not inherit a grading in the usual sense. Nevertheless they inherit what is called a Z2 -grading. We can decompose the tensor algebra into two parts T V = T 0V ⊕ T 1V (9) where T 0V = M T 1V = M n≥0 n≥0 T 2n V (10) T 2n+1 V are termed the even and odd tensors. Note that T 0 V is itself a subalgebra. The observation that xa ∈ T a V , xb ∈ T b V implies xa ⊗ xb ∈ T a+b (mod 2) V justifies calling this a Z2 -grading. One observes that the ideal generated by x ⊗ y + y ⊗ x + 2 hx, yi s.t. x, y ∈ V (11) is a Z2 -graded ideal. Therefore the algebras Clp,q and Cln are Z2 -graded, where we have 0 1 Clp,q = Clp,q ⊕ Clp,q , Cln = Cln0 ⊕ Cln1 . (12) Further, the Cl0 components are subalgebras. We call the subalgebras 0 Clp,q and Cln0 (13) the even Clifford algebras. We note one further important fact. Let Vp,q be the vector space with inner product of signature (p, q), where we take x1 , . . . , x p , y 1 , . . . , y q (14) to be an orthonormal basis, meaning hxi , xj i = δij hyi , yj i = −δij (15) hxi , yj i = 0. 0 (and Cln → Cln0 ) induced on Proposition 1.1 There is an isomorphism Clr,s → Clr+1,s elements v ∈ Vp,q by v 7→ v · xp+1 . 3 (16) Pf. Essentially trivial: one easily sees the kernel is zero, and the map is surjective. For a concrete example, consider the algebra H ≈ Cl2,0 . (17) We have two canonical structures on this algebra, a filtration and a Z2 -grading, both inherited from T R2 , where we take {i, j} to be an orthonormal basis of R2 . As for the filtration, we have (Cl2,0 ) (0) ≈ R (Cl2,0 ) (1) ≈ spanR { 1, i, j } (Cl2,0 ) (2) = Cl2,0 ≈ spanR { 1, i, j, ij }. (18) As for the Z2 -grading, we have 0 Cl2,0 ≈ spanR { 1, ij } 1 Cl2,0 ≈ spanR { i, j } (19) and we see immediately the isomorphism Cl1,0 ≈ span{1, i} 0 Cl2,0 ≈ span{1, ij} 7−→ (20) induced precisely as specified above. Finally note that a generic quaternion a + bk + cj + di (where k = ij) can be written, in its Z2 -graded form, as (a + bk) + (c − dk) j (21) so that we should regard H as C ⊕ C with a twisted product. 2 Statement of Poincare-Birkhoff-Witt and Its Corollaries Given a Lie algebra g, we have seen the construction of its universal enveloping algebra U (g) ≈ T g / Ig (22) where Ig ⊂ T g is the ideal generated by x ⊗ y − y ⊗ x − [x, y]. We know that U (g) inherits a filtration from T g; let U (g) (23) be the associated graded algebra. As usual let Sg is the symmetric algebra on the underlying vector space g. 4 We have vector space homomorphisms Tg πT π U (g) U (g) (24) to which we give the name π T : T g → U (g) (25) Lemma 2.1 The map The map π T it factors through the canonical homomorphism T g → Sg. Thus we have a commutative diagram of vector space homomorphisms Tg πT can. U (g) Sg πT (26) πS π U (g) and a commutative diagram of graded algebra homomorphisms Tg can. Sg πT (27) πS U (g) Pf. Given x ∈ T m g, y ∈ T s g be homogeneous tensors, and let x̄ = π T (x), ȳ = π T (y) be their images in U (g). Their multiplication is given by π T (x)π T (y) = πT (x)πT (y) + U (m+s−1) (g) = πT (x ⊗ y) + U (m+s−1) (g) (28) = π T (x ⊗ y) so that π T is a homomorphism. Further, if x, y ∈ T 1 g ≈ g, then x̄ȳ − ȳx̄ = πT (x)πT (y) − πT (y)πT (x) + U (1) (g) = πT ([x, y]) + U (1) (g) (29) = 0 + U (1) (g) Therefore IS ⊆ Ig so the diagram of algebras (27) commutes. 5 Theorem 2.2 (Poincaré-Birkhoff-Witt, highbrow version.) π S : Sg ≈ (30) U (g). Following Humphreys, we give a few fundamental corollaries along with their proofs. Corollary 2.3 Assume W ⊆ T m g is a vector subspace, and that the canonical projection T g → Sg sends W isomorphically onto S m g. Then πT (W ) is a compliment to U (m−1) (g) in U (m) (g). Pf. The following diagram of vector spaces commutes: U (m) (g) πT π T mg U can. m (31) πS m S g Using can.(W ) = S m g and the theorem, the composition of the maps on the bottom line is an isomorphism from W onto its image, so therefore too the top line. Since π factors U (m−1) (g) out of U (m) (g), the image of W must be complimentary to it. Corollary 2.4 The canonical map g → U (g) is injective. Pf. This is the case m = 1 in the previous lemma. Corollary 2.5 (Poincare-Birkhoff-Witt, blue collar version.) If {x1 , . . . , xn } is an ordered basis of g, the monomials of the form xi1 xi2 . . . xik , i1 ≤ i2 ≤ · · · ≤ ik where (32) constitute a vector space basis of U (g). Pf. Such monomials form a vector space basis of Sg. Corollary 2.6 (Functorial properties) If π T : T g → U (g) is the canonical map and if l ⊆ g is a subalgebra, then we have a natural embedding U (l) ⊆ U (g), and U (l) is generated by π T (l). Further, U (g) is naturally a U (l)-module in the algebra sense. 6 3 Proof of Poincare-Birkhoff-Witt We begin by constructing an action of g on Sg, which we then prove gives Sg a module structure. Let {x1 , . . . , xn } be a basis of g, and let zi be their images in Sg under g → T g → Sg. If Λ = (λ1 , . . . , λk ) is any multi-index, define zΣ = zλ1 . . . zλk ∈ Sg (33) We write |Σ| = k, and if λ < λi (or λ ≤ λi ) for all λi ∈ Σ, we write λ < Σ (of λ ≤ Σ). The following lemma defines the desired action. Lemma 3.1 The symmetric space Sg has a unique g-module structure under the conditions that if Σ is a multi-index with |Σ| = m then Am ) xi .zσ = zi zΣ when i ≤ Σ 0 Bm ) xi .zΣ0 − zi zΣ0 ∈ S (|Σ |−1) whenever |Σ0 | ≤ |Σ| Cm ) xi .xj .zT − xj .xi .zT = [xi , xj ].zT whenever |T | = |Σ| − 1. Pf. By induction we can assume the action has been defined on S (m−1) ⊂ Sg satisfying (Am−1 ), (Bm−1 ), and (Cm−1 ). Assuming that |T | = |Σ| − 1 we have that [xi , xj ].zT is already defined. By (Am ) we have that xi .zΣ (34) is defined when i < Σ. Let λ ∈ Σ be a member of Σ with λ < i and λ ≤ Σ, and let Σ0 be Σ with λ missing. Since xλ .zΣ0 = zλ zΣ0 = zΣ is defined, we have by (Bm−1 ) that xi .zΣ = xi .zλ zΣ0 = xi .xλ .zΣ0 = xλ .xi .zΣ0 + [xi , xλ ].zΣ0 (35) Since xi .zΣ0 is defined and since λ ≤ (i, Σ0 ), the left side is now defined as well, and so the action is entirely defined. It remains to check that (Cm ) holds. Going down one further step, let µ ∈ Σ0 be so that µ ≤ Σ0 and let Σ00 be Σ0 without µ. Then by (Am−2 ) we have 7 zΣ0 = xµ .zΣ00 , so xi .xλ .zΣ0 − xλ .xi .zΣ0 = xi .xλ .xµ .zΣ00 − xλ .xi .xµ .zΣ00 = xi .xµ .xλ .zΣ00 − xλ .xµ .xi .zΣ00 + xi .[xλ , xµ ].zΣ00 − xλ .[xi , xµ ].zΣ00 = xµ .xi .xλ .zΣ00 − xµ .xλ .xi .zΣ00 + [xi , xµ ].xλ .zΣ00 − [xλ , xµ ].xi .zΣ00 + xi .[xλ , xµ ].zΣ00 − xλ .[xi , xµ ].zΣ00 = xµ .[xi , xλ ].zΣ00 + [xi , xµ ].xλ .zΣ00 − [xλ , xµ ].xi .zΣ00 (36) + xi .[xλ , xµ ].zΣ00 − xλ .[xi , xµ ].zΣ00 = xµ .[xi , xλ ].zΣ00 + [[xi , xµ ].xλ ].zΣ00 − [[xλ , xµ ], xi ].zΣ00 = [xi , xλ ].xµ .zΣ00 + [xµ , [xi , xλ ]].zΣ00 + [xλ , [xµ , xi ]].zΣ00 + [xi , [xλ , xµ ]].zΣ00 = [xi , xλ ].zΣ0 Now we conclude the proof of the Poincare-Birkhoff-Witt theorem. The various canonical maps give us the following commutative diagram: g iU iT U (g) (37) Tg πT The previous lemma provided a Lie algebra homomorphism g → gl(Sg), which extends via the various universal properties to the maps g iU U (g) θ iT gl(Sg) ψ (38) Tg πT which, by uniqueness, implies ψ = θ ◦ πT . (39) Now, lets consider an element t ∈ T g that maps under π T to 0 in U (g). By diagram (27) we will have proven π S is an isomorphism if and only if ker can = ker π T 8 (40) Clearly ker can ⊆ ker π T , so we must prove the opposite inclusion. Assume t ∈ ker π T . Because π T is a graded homomorphism, we can assume t ∈ T m g is homogeneous. What π T (t) = 0 means is that πT (t) ∈ U (m−1) = πT (T m−1 g), so that there is some t ∈ T m−1 g with πT (t) = πT (t0 ). But then 0 ψ(t − t0 ) = θ(πT (t − t0 )) = 0. (41) However we know how ψ works: it is symmetrization followed by multiplication in Sg. Thus the symmetrization of t−t0 is zero, so t−t0 is purely antisymmetric, meaning t−t0 ∈ ker can. However ker can is graded, and since t is the m-homogeneous component of t − t0 , we have t ∈ ker can. (42) 9