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Some Problems on universal mappings
R.S.Ismagilov
Resume
First we describe inductive limits for some families of Lie algebras and groups, li e also consider the linear mappings from the space C°°(S',R) (smooth functions on a circle) to Lie algebras such that the following Local Commutatinty Property is satisfied: for ony two functions with disjoint supports the corresponding elements of the Lie algebra commute; we describe a universal mapping with this property.
1 Introduction
Mapping with the universality Property arise in many ocasions: recalL for example, that for linear spaces A, B the canonical mapping A x B -> /1 ® B is universal with respect to bilinear mappings from /1 x B to linear spaces. In this paper we consider some universal mappings for Lie algebras and groups. First we discuss some examples related to inductive limits of families of the Lie algebras and groups. Then (in §2) we examine linear maPPinSs from C'X,{S\R) to Lie algebras such that the local commutativity property is tumlled. this question originates from the ’'quantum field theory on a circle" ([lj).
2 Inductive limits
First recall the definition ([2]). Suppose we have a family of groups {Ga} and for am Ga and Gß a family (possibly empty) of homomorphisms fa8 : Ga -> Gß. A representation o^ the family {Ga.Jap} in a group G consists (by definition) of homomorphisms ua : Ga -> G such that the diagrams
Ga G
Uß
faß I /
. Gß
are commutative and G is generated by all ua(Ga). The inductive limit of our family {GaJaß} is, by definition, a representation {£*,<} with the following universality Property: for any representation {6>a} there exists a homomorphism p : G’ -> G with
ua - p O u*a for all a. _
The group G” can be described in terms of generators and relations; but the explicite
description usually turns out to be a difficult Problem. ^
For the family'{La, fa0} where La are Lie algebras and faß : La -> Lß homomorphisms
the inductive limit is defined in the similar way.
Now we consider some examples.
1. Heisenberg group (and Lie algebra) as inductive limit of commutative groups (commutative Lie algebras). Consider a field K,charK ^ 2, a K - linear space V,dimV > 4. and a bilinear form r : V x V —> K antisymmetrical and nondegenerate (i.e. for any
x £ V, x ^ 0, there exists a vector y £ V with r(x,y) ^ 0). A linear subspace Vj C V'’ is called isotropical if r is zero on Vi x Vi. Denote by Is(V,t) the set of all the isotropical subspaces. Any such subspace is considered as an commutative group (with respect to the addition of vectors). For any two subspaces Vo, Vi from Is(V,t) such that Vo C V'i wc have a homomorphism (inclusion) Vo C Vi. Our goal is to describe the inductive limit of the family of groups Is(V,t) with these homomorphisms-inclusions. To do this consider the Heisenberg group II - the set V x K with the group operation {vi,ki){v2,k2) = (ui + i’2,^i + k2 + r(i?i, t’2)). For any V'0 £ Is(V,t) we have a group homomorphism Vo —»• if, v i—> (v, 0), v £ Vo-
Theorem 1 The inductive limit of the family Is(V,t) is the Heisenberg group H ( with homomorphisms indicated above).
Proof of Theorem 1 Consider an arbitary representation of our family Is(V, r) in some group G. Clearly this representation is the same thing as a mapping $ : V —> G with the following Property:
if v; £ V,i = l,2,r(v1, v2) = 0,then $(v! + v2) = $(vi)$(v2), (1)
$(0) = e (the neutral element oj G).
Examine closely the mapping $ with this Property.
Lemma 1 We have
$(a + b) = $(^)$(&)$(|), Va, b £ V, (2)
Proof of Lemma 1 Recall that dimV > 4. charK ^ 2. From these conditions it follows that there exist vectors a. 8 £ V with
r[a.a) = r(o,6) = 0, r(;3,a) = r(j3,b) — 0
r(a, 8) +-T(a.b) — 0, (3)
' 4 ' '
Write the vector a + b as a sum of four vectors as follows:
a + b = (~a + a) + (^6 + /3) + {^b - ¡3) + (|a - a)
It follows from (3) that the sum of the. first and the second vectors on the right side of the last equality is orthogonal (with respect to t) to the sum of the. third and jourth vectors. Moreover the first and the second vectors are orthogonal and so are also the third and the fourth vectors. Thus the Property (1) implies
$(a + b) = $((^fl + a) + (h + /3))$((^> - ,8) + {^a - a)) =
= $(^a + a)$(|& + £)$(^6-/?)$(^ii-a) =
= $( —a + a)$(6)<I>(—a — a) =
= $(^a)(f(a)i>(6)<5(-o;)$(^a) = $(^a)$(6)$(^a)
Define a mapping w : V x V —> G by
w{a, b) = $(fl + 6)$(-6)$(_a)j G) h 6 K (4)
Lemma 2 We W w(a, 6)$(c) = $(c)a-(a, b) /or a// a, b, c 6 V\
Proof of Lemma 2 Applying (2) repeatedly we obtain
$(-c)w(a,b)<S>(c) = $(-c)$(a + 6)$(-6)$(-a)$(c) =
= $(-a - 6)($(a + 6)$(-c)$(a + 6))$(-6)($(-a)$(c)$(-a))$(a) =
= $( —a - 6)$(2a + 2b — c)$( — 6)$(c - 2a)$(a) =
= $(-a - 6)$(2a + 2b - c)($(-6)$(c - 2a)$(-6))$(&)$(a) =
= $(-a - 6)$(2a + 2b - c)$(c - 2a - 26)$(6)$(a) =
= $(-a - 6)$(6)$(a).
77ms $(-c)«;(a,6)$(c) does no£ rfepenrf on c £ V. Putting c = 0 proves our ¿emma.
folio J°U0WS fVOm ^ ^ ^ ^ W^’ ^ remGmS unchan9ed if we transform (a, b) as
(a, 6) -> (a', b) where a-a'la;a-a'ib, (5j
or (a, b) -> (a. b') where b - b' _L a, b - b' ± b. (g)
ravKt 10 T>Clmrl!’T(a■b] aho ~mtact
Lemma s Let a b a', b’ e V and r{a,b) = r(a',V) ± 0. Then the pair {a\b') can be
obtained fiom (a,b) applying a sequence of transformations of the form (5) and (6).
Proof of Lemma 3 Let V(a,b) and V(„':b') be linear snbspaees spanned by o.b and
■ ■ Let n be a rank of the bilinear form t(x,V),x E V(a,b):ye V(a'. V): notice that n
is simply the rank of the matrix ' "
/ r(a, a') r(a. b') \
I, T(4, a') T(b,b') j
Consider the following eases. Case 1; n = 0. Thus V(a, b) ,s orthogonal to V(a\ V) In this case the Lemma follows if we consider the transformations (a,b) - („ + a' L -> (a. + „ , V) W, V). Case 2; Via, i) = V{a‘, b'). Take vectors a, 1 ortHoJaa^V a H) an(l r(ab). Applying the case 1 we can pass from (a, 6) to (a1.b1) (applying
CaTT^liraf°;mat^s °fthc form (5), (6)) and then pass from (,,l)hl) L («',&')
- ' ' n ■ Applying the case 2 we can assume that a' 1 a,a' ± b,b! ± b r(a br) =
T{a,b); The desired transformations are (a,b) -> {a.b') -> (o' b'). Case )• n = 9 applying the case 2 we can assume that a' _L a,b' _L b,T(a',b) = r(a,6). Then to prove our Lemma use the transformations (a, b) —s- (a',b) —» (a\b')
Now we can conclude the Proof of the Theorem 1 as follows. From Lemma 3 it follows that if r(a,b) = T(a'. b') then w(a,b) = w(a',bl). Thus w(a,b) = r(r(a,b)) for some mapping r : K —> G. From (4) and Lemma 2 it follows that w(a + 6, c)w(a, b) = w(a, b + c)w{b,c), iw(a,0) = e. This easily gives r(0) = e,r(x + y) = r(x)r(y) for any x,y £ I\.
It follows that the mapping FI —> G,(v,k) $(v)r(k) is a homomorphism. This clearly
proves our Theorem.
Now consider subspaces Vo £ Is(v,t) as commutative Lie algebras (over the field K). The corresponding inductive limit is the Heisenberg Lie algebra V ® K with the Lie bracket [(ui, ki), (v2. k2)] — (0, T(ki.k2)). We do not dwell on this in details.
2. Lie algebras of vector fields. Consider two examples.
a) Let (X,w2) be a compact connected symplectic manifold; (see, for example, the book [3], pp.123 for all the notions used in this section). By V(X,w2) denote the Lie algebra of vector fields preserving the form w2 and by V0(X,w2) the Lie subalgebra hamiltonian vector fields. So, for any £ £ Vo(X,w2) we have a function (hamiltonian)
£ C°°(X,w2) such that df$ — i(()w2 where z(£) denotes an inner product; is defined up to an additive constant.
Consider all the domains Y C X diffeomorphic to R2n,dimX = 2n (we write Y ~ R2n). For any such Y denote by V0(Y, w2) the Lie subalgebra of all vector fields £ £ V{X, w2) supported in Y (i.e. £ is zero outside of a compact subset of K). If Y\ C Y2 then we have an obvious inclusion homomorphism V0(Yi,w2) —► Vo(y2!^'2)- Now we describe the inductive limit of the family of Lie algebras { Vo(V'. u’2). Y ~ R2n}.
Consider the direct sum of Lie algebras Vq{X, w2) © R and for any Y C X, Y ~ R2". define an inclusion V0(V',u;2) —> V0(X, tv2) ® (£, Jx f^(x)w2n) where is the
hamiltonian of £ supported in Y.
Theorem 2 The Lie algebra V0(X,w2) © R with the inclusions indicated above is the inductive limit of {Vo{Y,il'2),Y C X,Y ~ R2n}.
The similar result for the corresponding family of diffeomorphism groups is also valid: we do not consider this case.
b) Let (X. rn) be a compact connected manifold, dimX = n.n > 3, equipped with a volume form vn (so X is oriented). For any domain Y C X, Y ~ Rn denote Vo(V,rn) the Lie algebra of vector fields preserving the form vn and supported in Y. By VQ{X,vn) denote the linear span of all {V0(F, un), Y C X,Y ~ Rn}. Now we describe the inductive limit of the family of Lie algebras {Vo(F, u"), Y' C X,Y ~ Rn} with obvious inclusion homomorphisms Vo(yi,t,n) C Vo(y2,t!n) for Y\ C 12-
Let Ek = Ek(X) be the space of exterior k-forms on.Y, Zk = Zk(X) and Bk — Bk(X) the subspaces of closed and exact forms; put Hk = IIh(X) = Zk/Bk (k- domensional cohomologies of Ar). Consider the linear space En~2/Bn~2 and introduce a bracket [ , ] in it as follows. If 0i and 02 are (n —2)-forins supported in a domain Y C X,Y ~ Rn. then put [0i + Bn~2, 02 + Bn~2] = 03 + B!l~2 where 03 C Y and vector fields £ V0(y, i,n), k = 1,2. 3. defined by i(£k)t'n = d0k satisfy the equality [£i,£2] = 6- The bracket [ , ] is uniqly determined by this rule and makes En~2/Bn~2 a Lie algebra. Moreover it is a central extension of Vo{X,vn) by the centre Hn~2(X) considered as acommutative Lie algebra. For any Y C X, Y — Rn we have an inclusion Vo(V, i,n) —»• En~2/Bn~2,£ i—> 0 + Bn~2, where ?(Ou" = d& and suppO C Y.
Theorem 3 The Lie algebra En~2 /Bn~2 is the inductive limit of the family of Lie algebras {V0{Y,vn)Y CX,Y ~Rn}.
Similar results for difFeomorphism groups were obtained in [4]; they are more complicated to formulate (and to prove).
3 Locally commutative mappings from the space C°°(Sl,R) to Lie algebras
Let C°°(Sl,R) be the spase of smooth real functions on the circle S’1 and L an arbitrary real Lie algebra. A linear mapping / : C°°(S1,R) —» L is called locally commutative if for any two functions u,v from C°°(S1,R) having disjoint supports we have [f(u)J(v)} = 0. Consider all the pairs (L,f) where L is aa Lie algebra and / a locally commutative mapping from C°°(S\R) to L. The pair (L*, f*) is calleduniversalif for any pair (LJ) there exists a Lie algebra homomorphism p : L’ —> L with / = pofr and /*(£*) generates L* as a Lie algebra. Now we describe (L*,/*).
Consider the associative algebra A = A\ © A2 ® • • • where A\ = C°°(S1,R)1 Ak is the space of jets of C°°-functions u{xu ..., xn), xt G S1 in the neighbourhood of the ’’diagonal”
xi = ... = xn. If it G Am,v G An then uv G Am+n is defined as a jet of the function u(xl:.. .,xm)v(xm+u.. .,xm+n). Consider A as a Lie algebra with [u,v] = uv- vu and denote by L* the Lie subalgebra generated by A\. We have a mapping / : CCC(S\ R) = Ai C L‘.
Theorem 4 The pair (L*,/*) is universal.
To conclude this section slightly modify the notion of a locally commutative mapping. Consider now only topological (locally convex) Lie algebras L and locally commutative mappings / : C'°°(5'1, R) -> L subjected to the continuity condition which we now formulate. The mapping / has a prolongation / to the free Lie algebra F = Fl®F2®. . over the linear space C°°{S\R). Any Fk is embedded into (Cco(51, R)f ' C , R):, this
last space consists of smooth functions u(x1,..., xn), xt G S1. Consider only the mappings / for which the mapping f is continious on any Fk with respect to Sobolev norm || • ||p
(maximum of modules of derivativer of order < p) restricted to Fk C C ((S ) , R). The corresponding universal pair (for fixed p) denote by (L*,fp). The explicite description of it can be deduced from Theorem 4. We only consider the case p= 1. It turns out the L\ can be realized as the space of polinomials u\T + . . . + umTm where ut G C (S , R) and T is a formal variable. The Lie bracket is defined by [ukT ,vmT ] — 0 if k > 2,m >
2, [uiT, v!nTm] = ulVmTm+1, if m > 2 and [UlT,VlT] = (ulV; -u'lVl)T2.
References
[1] W.Nahm, Quantium Field Theories in one and two dimensions, Duke mathematical Journal, 1987, v.54, No.2, pp.579-613.
[2] Serre J.P., Arbres, amalgames et SL2, 1977, Soc. Math. France, Asterisque, v.46, 20-43.
[3] C. Godbillon, Geometrie differentielle et Mecanique analytique, 1969, Herman, Paris.
[4] Ismagilov R.S., On a group of volume preserving diffeomorphisms, Izvestija Academii nauk SSSR, ser. math., 1980, v.44, No 4, pp. 831-867.
