Theoretical and M a t h e m a t i c a l Physics, Vol. 118, No. 2, 1999
ON
DERIVATIONS
OF
THE
HEISENBERG
ALGEBRA
v . V. Z h a r i n o v 1 Derivations of the Heisenberg Mgebra 7/ and some related questions are studied. The ideas and the language of formal differential geometry are used. It is proved that all derivations of this algebra are inner. The main subalgebras of the Lie algebra ~(74) of all derivations of 74 are distinguished, and their properties are studied. It is shown that the algebra 74 interpreted as a Lie algebra (with the commutator as the Lie bracket) forms a one-dimensional central extension of ~(74).
Introduction The Heisenberg algebra (i.e., a polynomial algebra with generators satisfying the quantum commutation relation; e.g., see [1, 2l) is one of the fundamental objects in quantum theory and is involved in many physical and mathematical models and theories in one form or another. This algebra and its generalizations (deformations) have recently become of keen interest in both theoretical physics and mathematics, where it is regarded as a fundamental object and as a suitable model for checking various physical and mathematical ideas and constructions (e.g., see [3-8]). We study the Lie algebra of all derivations of the Heisenberg algebra, its main subalgebras, and the incidentally occurring structures. As a result of our desire for closure, this work is partly a survey and has a methodological character. For completeness, for example, we present the fundamentals of formal differential geometry (Sec. 0). To do so seems reasonable in view of the different approaches (e.g., see [915]) to this question--particularly, in relation to the notion of multiplicator and its role in constructing the theory (e.g., see [16]). Because of the multifaceted and widespread use of the Heisenberg algebra, its main properties are often known at the folklore level, which hampers exact identification of the original works. Therefore, in Sec. 1, we formally define the Heisenberg algebra and enumerate its elementary properties. Such a detailed introduction makes the further presentation clear and compact and also gives an idea of the place our results occupy in the context of the general theory. 0.
Elements
of formal
differential
geometry
0.1. N o t a t i o n . Let N be a fixed field of characteristic zero, and let ,4 be a topological algebra over F (an N-algebra). We use the following notation: cen,4 = {f E M: f g = g f , Vg E ,4} is the center of ,4, a n n a = {.f E A : f g = g f = O, Vg E A } is the a n n i h i l a t o r o f A , 12(.4) is the linear space of all continuous linear mappings from `4 into itself. The composition F, G ~-~ F o g defines the structure of a unital (i.e., containing a unit) associative algebra 12o(,4) in 12(,4), and the commutator F , G ~ [F,G] = F o G - G o F defines the structure of a Lie algebra s ] (,4). The action (more precisely, the left action) a : ,4 ~ s is defined by the rule f ~-~ a ( f ) , a ( f ) ( g ) = f g , V f , g E ,4.
ISteklov Ivlathematical Institute, Russian Academy of Sciences, Moscow, Russia, e-mail:
[email protected],
[email protected]. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118. No. 2. pp. 163--189, February. 1999. Original article sut)mitted July 21, 1998. {)041)-57.9/99/1182-0129522.00
@ 1999 l,(luwer Acadr
Publishers
12~J
0.2. M u l t i p l i c a t o r s . A mapping M E s M ( f ) g = f M ( g ) , g f , g E A.
is called a multiplicator (e.g., see [16]) if M ( f g ) =
The set fig(A) of all multiplicators of the algebra A possesses the following properties: 1. fig(A) is a unital subalgebra of go(A). 2. The kernel k e r M = { f E A: M ( f ) = 0} and the image im M = M ( A ) are two-sided ideals of A, V M E fig(A). 3. M : ann A ~-+ ann A and M : cen A ~-+ cen A, VM E fig(A). 4. [M, N ] ( f ) E a n n A , VM, N E fig(A) and Vf E A; in particular, the algebra fig(A) is commutative if a n n A = O. If A is an associative algebra, then fig(A) has the following additional properties: 5. a : cen A ~-r fig(A), i.e., a ( f ) E fig(A), Y f E cen A. 6. k e r a n c e n A = {f E c e n A : a ( f ) = 0} = a n n A ; in particular, the mapping a: c e n A ~ fig(A) is injective if ann A = O. 7. I[ there is an approximation of the unit {~n} C A (such that ~,~f -+ f and f~n -~ f , n ---r c~, V f E A), then ann A = 0, and the image a(cen A ) is dense in fig(A). Finally, if A is a unital associative algebra, then fig(A) has the following property: 8. a : cen A "" fig(A), and the inverse mapping a - 1 acts according to the rule M ~-+ a - l (M) = M (e), V M E fig(A), where e is the unit of the algebra A; in particular, the algebras fig(A) and cen A can be identified.
0.3. D e r i v a t i o n s . A mapping D E s D ( f g ) = D(.f)g + f D ( g ) , g f , g E A.
is called a derivation if the Leibniz rule holds, i.e., if
The set :D(A) of all derivations of the algebra A possesses the following properties: 1. 2. 3. 4.
D ( A ) is a left fig(A)-module, and M D = M o D, V M E D1(A) and VD E ~ ( A ) . ~ ( A ) is a subalgebra of the Lie algebra s .](A). D : ann A ~+ ann A and D : c e n A ~-+ cen A, VD E ~ ( A ) . [O, M] E fig(A), VD E ~ ( A ) and VM E 93/(A).
D(M)
In particular, for each D E ~ ( A ) , the rule = [D,M], M E fig(A), defines a mapping /9: 9"Yt(A) ~-+ ffJ/(A). Moreover, by the Jacobi identity,/~ has the following property:
5. D(M o N) = D(M) o N + M o D(N), VD E :D(A) and VM, N E fig(A). Thus, the rule D ~-~/~(D) = / 9 ,
D E ~ ( A ) , defines the action B: ~ ( A ) ~, ~(gYt(A)).
6. The action I3: ~ ( A ) ~-~ ~(fig(A)) is a morphism of Lie algebras. 7. The extended Leibniz rule holds: D ( M ( f ) ) = D ( M ) ( f ) + M ( D ( f ) ) , and V f E A.
VD E ~ ( A ) ,
V M E 93l(A),
If the algebra A is associative, then the actions have the following properties: 8. The actions a on c e n A and • on D(A) are compatible, i.e., / 9 ( o ( f ) ) = a ( D ( f ) ) , VD E D ( A ) and Vf E cen A. 9. V f E A, the action a _ ( f ) : A ~ A, a _ ( f ) ( g ) = [f,g], g E A, is a derivation of A (i.e., a _ ( f ) E
~(A)) A derivation of tile form a,_ (f) is termed imw.r it is obviously nontrivial if f ~ ten A. 130
0.4. C o n s t a n t s . An element f E A is called a constant if D(f) = 0, VD E :D(A). Let conA denote the set of all constants of the algebra A. Clearly, con.4 is a subalgebra of .4, and if the algebra A is unital. then its unit e belongs to con`4. A subset S C ~ ( A ) is said to be total if the condition D(f) = O, VD E S, implies f E con.4. 0.5. D i f f e r e n t i a l a l g e b r a s . A differentiM algebra is a pair (A, q:(A)), where A is an algebra and ~:(A) is a subalgebra of the Lie algebra :D(.4) (e.g., see [17, 18]). A differential algebra (A, ~:(A)) is said to be standard if ~:(A) is a free finite-dimensional submodule of a left 9~(`4)-module of ~(`4) with a pairwise commuting basis D l , . . . , Dm E ~)(A) ([D~,, D~] = 0, #, u E 1, m) such that
~(A) - [D] = { D = ~-'~ MuDu : Mu E gJI(A),# E I,rn} I.*= 1 m
(here and henceforth, k, l - {k, k + 1 , . . . , l }). Differential algebras occur in the algebraic-geometric approach to differential equations. In view of this, we use the terminology accepted in this approach (e.g., see [19, 20]). For example, the derivations belonging to the subalgebra ~:(A) are called Caftan derivations. A derivation X E ~ ( A ) is called a Lie-Bgcklund derivation if [X, D] E ~:(A), VD E ~:(A). The set of all these derivations forms a subalgebra ~ ( A ) of :D(A), and by construction, s is an ideal in ~ ( A ) . The quotient algebra Symr A = fB(A)/s is called the Lie algebra of symmetries of the differential algebra (.4, ~(.4)). 0.6. D i f f e r e n t i a l f o r m s . We assume that the algebra A under study satisfies the condition i. the multiplicator algebra 93l(.4) is commutative (e.g., ann A = 0). In this case, ~(`4) is an ffJ~(.4)-module, and its exterior Mgebra oo
A ' ~ ( . 4 ) = 1-I Ar~(.4)'
A0~(.4) = 9Jl(.4),
A1~(.4) = ~(.4),
...
r-w-O
is defined. In particular, for any D l , . . . , Dr E ~D(.4), the exterior product 1
D ~ A . - - A D r = ~. Z
sgnrD~(~l|
|
EAr~(.4)
~E6~
is defined, where 6 r is the group of all permutations of the numbers 1 , . . . , r. Let K: be an arbitrary F-algebra such that ii. the morphism of associative algebras ~o: 9J~(A) ~ 9JI(K:) is defined, iii. the morphism of Lie algebras r ~ ( A ) ~-~ ~(K:) is defined, iv. the compatibility conditions r
= ~(M)r
and
q~([M,D]) = [~o(M), ~b(D)]
hold VM E 9~(.4) and D E ~D(A). (We recall that M D E ~ ( A ) and [M, D] E 9~(A).) In particular, the algebra/(: is an 9Jl(A)-module. R e m a r k . The following algebras, for example, satisfy conditions ii-iv (we assume that condition i holds): 1. The algebra A in itself. In this case, ~p = idomA) and w = id~cA). 2. The multiplicator algebra 9J~(A). In this case, ~ = idomAi (we note that by i, 9N(gJ~(A)) -~ 9~(A)). and the morphism tp: D(A) ~-~ D(OY~(A)) is defined by the rule ~(D)(M) = [D,M], VD E D(A) and fiI E 9J~(A). 131
3. The Lie algebra of derivations ~ ( A ) on condition that [gX(A),:D(A)] = 0. Here, the morphism ~o: 9JI(A) ~ 9X(~(A)) is determined by the rule ~o(M)(X) = M X , and the morphism r ~D(A) ~ ( ~ ( A ) ) by the rule r = [D,X], VM E 9JI(A) and D , X E ~(A). We note that the condition [gX(A), fl3(A)] = 0 holds if 93l(A) = F-idA; clearly, in this case, condition i also holds. We set OO
fF (A; K) = Hom~(A ) (A*~(A);/C) = Z
W(A;/C).
r----0
In more detail, f/0 (A; K:) = Hom~(.a} (9~(A); K:)"~K:= (the isomorphism is defined by the rule fl~
]C) 3 w ~-~ k = w(idA) E /C), and for r > 0,
w(D1,...,D~) = w ( D I A . . . A D ~ ) ,
w6fl~(A;/C),
is a skew-symmetrical function 9X(A)-linear with respect to each of the arguments D 1 , . . . , D~ E ~(A). The elements of fl~(A;/C) are called differential r-forms. 0.7. O p e r a t i o n s on d i f f e r e n t i a l f o r m s . We enumerate the main structures defined on I~'(A;K:) and their properties. First, fl*(A;/C) is an ~(A)-module, and by definition,
( M w ) ( D , , . . . , D~) = (p(M)(w(D1,.. ., D~)) for all M E 9X(A), w E fir(A;/C), and D I , . . . , D ~ E ~ ( A ) . We define the following linear operations (we assume that fl~(A;/C = 0) for r < 0 and D E ~(A)): the convolution the Lie derivative the differential
LD: flr(A;/C) ~-+ flr-l(A;K:),
LD: flr(A;/C) ~-~ fY(A;K), d: fl~(A; E) ~-~ W+I(A; K:).
For this, we use the corresponding formulas (we assume that w E fF(A; K:))
(LDw)(DI,...,D~-I) =w(D, DI,...,D~_I),
D1,...,Dr-1 E ~(A)
(in particular, tDW = 0 for r = 0);
(LDw)(D,,...,D~) = r
- fiw(D,,...,[D,D,I,...,D~) i=l
for Dx,.. ,Dr E ~(A); and finally r
( ~ / ( D o , . . . , Dr) = ~ ( -
I/'~(D,)(~(Oo,..., D,,..., D~/) +
i=0
+
Z
(-I/'*J~(Io,,DA, Do .... , b , , . . . , b j .... ,Dr)
0 r or r < 0): 1. 2. 3. 4. 5. 6.
w . 0 E C S + q f ~ r + P ( A ; K), Vw 6 CS~r(A; K), 0 6 c q f } P ( . A ; K ) LD: C ~ f F ( A ; K ) ~ C ~ - ~ r - I ( A ; K ) , VD E ~ ( A ) Lo: C ~ f ~ r ( A ; K ) "-> C ~ f V - ~ ( A ; K ) , V D 6 r LD: C ~ f F ( A ; K ) ~ C ~ - ~ f F ( A ; / Q , VD 6 ~ ( A ) LD: Csf~r(A; K) ~-+ Csf~r(A; K), VD 6 ~ ( A ) d: C~fP'(A;IC) ~ C~fP'+'(A;IC)
0 . U . S p e c t r a l s e q u e n c e . We retain the notation in Sec. 0.10 and add the abbreviation C S f V = C ~ f P ( A ; IC). By construction, there is a filtration fV = C ~
D C*i~ r D C 2 f F D ... D C r D r D 0
compatible with the grading oo
r=O and with the differential d: 12r ~-~ f F +l, r = 0, 1 , . . . . According to the general theory (e.g., see [21]), the spectral sequence {ErP,q ,d P,q r ,. p , q , r = 0, 1 , . . . } with the terms E~ 'q - E~'q(A, ~(A); K) =
ZV~'q p,q ~pTl,q-I B r - I + ~r--I
is defined, where Z['q = {w 6 C P ~ p+q : dw 6 CP+rfl p+q+l }, BPr'q = {w = dO 6 C P ~ p+q : O 6 c p - r ~ p + q - 1 } ,
and the linear mappings dPr'q : EPr,q ~
EPr + r , q - r + l
are induced by the differential d (i.e., dPr'q[cz] -- [d~], V[w] 6 EPr'q). The terms of order r = co are given by the formulas E ;q = B~q + z~+l,q_l ,
p,q=O,1,...,
where Z ~ q : {ix) E CP~-~P+q: dim ~. 0},
B ~ q = {w = dO E CPI2 "+q : O 6 12P+q-l}.
We enumerate the following main properties of the spectral sequence: 1. dPr+r'q-r+l o d p'q = 0, Er+ P'ql = kerd~'q/im d p - ' ' q + r - I 2. E ~ q = E p,q, r > m a x { p , q + 1}
Let a symmetry [D] belong to Sym A (see Sec. 0.5). 3. the linear m a p p i n g t{D]: EPr'q ~4 EPr - r , q + r - I is defined by the rule I,[D][W] = [tDW], [w] @ EPr "q 4. the linear mapping L[D]" F-,Pr'q~4 E, Ur'q is defined by the similar rule LIDI[W] = [LDW], [w] 6 Eft "q O. LID ] o d~ "q = d p,q o LID I 6. Lit)l = I[~)! o d~q + d~ q o Irl)! 134
0.12. R e l a t i o n s h i p w i t h t h e d e R h a m c o h o m o l o g y . We retain the notation in Sec. 0.11 and add the abbreviation H k - Hk(.A;K). The filtration of differential forms by Cartan forms (see Sec. 0.11) generates the filtration of the de R h a m cohomologies H k =
H~
D H ~'k
D H 2'k D . . . D H k'k D O,
k=O, 1,...,
where the linear spaces are
{od E CP~ k : d.aJ = 0} {co = dO E CPf~ k : 0 E ilk-1
H p,k =
}'
p, k = 0, 1 , . . . .
1. The relations E ~ k-p = Hp'k / H p+l'k, p, k = O, 1, . . . , hold; in other words, the equality of graded linear spaces E k = G ( H k ) , k = O, 1 , . . . , holds, where
p+q=k
by definition and
G ( H k) = ~
H P ' k / H p+''k
p=O
is the graded space associated with the above filtration. .
Heisenberg
algebra
1.1. N o t a t i o n .
We use the following multi-index notation:
z = {0,+1,12,...}
z+ = {0,1,2,...}
N = {1,2,...}
i = ( i ~ , . . . , i d ) E Z d, where d E N is fixed liJ=il+...+ia,
i+j=(il+jl,...,id+ja),
i,jEZ
(#) = ( 0 , . . . , 1 , . . . , 0 ) E Z a, where unity occupies the # t h position, # E 1 , d i 0, form a basis in 7/,; 3. the bracket [., .], is determined by the relations
[pq,pk,]. =
Z
Cid'k' P "'" , --mn
i,j,k, l E
Za+. [i + j l > O,
Ik
+ tl > o,
Im+nl>0
where the structure constants C ~ t are defined in Sec. 1.10.
2.4. T h e o r e m .
The isomorphism of Lie algebras a _ : 74. ~_ ~ ( 7 4 )
is defined, where a _ ( f ) = {f, .} - F, f E 74,, and
,~_([f,g],)
= [o,_ (.f), ,~_ (g)],
c~_(f)' = - a _ ( y t ) ,
.f,g ~ 74.,
f ~ 74.
P r o o f . By Proposition 2.2 and the above argument, the proof reduces to a simple verification (the transposition is defined in assertion 7 in Sec. 1.11). Thus, the Lie algebra ~(74)
~F= U,-}: f = k
f i j p ij E 7 / } li+jl>o
has the basis R ii = (pij, .}, i , j E zd+, li + j] > O; the bracket and the action are given by the formulas
,vJ(p")= Z
[n'J,R"]= Im+nl>O
C''YPm';
Im+nl>O
and the structure constants are defined in Sec. 1.10.
In particular, [R~
i+(t'),j] = (i# + 1)R ij for any Iz E 1,d, i , j E Zd+, ]i + J] > 0, whence it follows
that, 1 the , ' o m n , , , a m is [~(74),~(7/)] = 9(74). 142
2.5. D e g r e e o f a d e r i v a t i o n a n d t h e g r a d i n g . Let
7/u =
{ E
}
f u p U E 7-/ ,
f =
li+jl=u
u E Z+:
be the linear space of all homogeneous normal polynomials of degree u. Then oo
v+w--2
7/.=E7/~
and
[7/~
C
E
u=l
v, w E N .
7/"'
u=-I
Let :D~(7/) = { F = {f,-}: f E 7/~} be the linear space of homogeneous derivations of degree u E N. By Theorem 2.4, the Lie algebra :D(7/) can be decomposed into the direct sum of linear spaces r
v+w-2
~(7/) = E ~ ' ( 7 / )
and
[~(77/),~'(7/)] C
1/,~1
E
~'(7/)'
v, w E N .
U=I
Similarly, we set 7/,k = 7-/, fq 7/k, k E g a (see assertion 6 in Sec. 1.11; we note that 7/,k = 7/k for k ~ 0). Let ~k(7/) = {F = {f, .}: f E 7/,k } be the linear spaces of 7-homogeneous derivations of order k E g a. It can readily be shown that the Lie algebra ~(7/) possesses a graduation,
[~k(7/),~(7/)] c ~k+~(7/),
= E
k,l e g d
kEZ d
2.6. C l a s s i c a l l i m i t . According to Sec. 2.4, the Lie algebra ~ ( 7 / ) has the basis ~ij = R 0, i , j E Zd+, li + j[ > 0, and the commutator
[~ij, ~kl] ---- ~'~ cij,kl cmn m Tl
.a
m~rt
where the structure constants are
vrnn(7"ij'kl= vrnnCO'kt(X)= ~
xlrl-l~(i,j;k,l~(~i+k-r(~j+l-r r , m n =
Irl>O
d ~ ( j _ . k . - I -.o.lvm ; ~ . ~ + k - ( . ) ~~.j + , - ( . ) + O ( x ) . Passing to the limit as x --+ 0 in the above relation, we obtain a Lie algebra Z with the basis ( 0 li + j[ > 0, and the bracket
i, j E Z~+,
d
[c
=
Z
Zg'ycm",
im+nl>O where the structure constants are
d Ztj,kl .....
=
Z
('i._t,k I ,
--
Io ~ , ~";1 , / ~u ,xni + k - ( I Q X j + l - ( uI J n)
-
u=l
1.13
We note that as x ~ 0, the algebra ~H passes into the algebra F[r] of commutative polynomials with generators r = (ql,.. 9 Pal) and the usual linear operations and multiplication. In this case, the basis Poisson brackets (see assertion 9 in Sec. 1.11) take the form d
=
Zm~ P
= E(j~,k t -ltit)pi+k-(tI,J+t-tt) =
m,n
#=I
f~=l \ OPt Oq.
Oqt
~p.,/' i , j , k , l E ZU+,
which implies
{]' g} ~ \ Opt aq.
(2.4)
Oq. O p t '
in complete agreement with the classical expressions (e.g., see [2]). In other words, 1. as g -4 0, the Poisson algebra ?t passes into the classical Poisson algebra F[r] of commutative polynomials with bracket (2.4). On the other hand, interpreting F[r] as a Lie algebra with bracket (2.4), we find that 2. the isomorphism of Lie algebras Z ~_ F[r]/{1} is defined, where {1} is the center of the Lie algebra Fir].
3. Main subalgebras 3.1. D e r i v a t i o n s o f t h e first d e g r e e . The derivations Q t = {q,,'} and P t = {Pt,'}, /z E 1,d, form a basis in the linear space fl31(7/) of derivations of the first degree. It is easy to verify the following assertions using formulas (1.1) and (1.2): 1. The derivations Q t and Pt act on 7-l according to the rules
Q~'(f)-
Of Opt '
P t ( f ) = Of Oq,'
lET"l,
I~ E 1,d.
2. The derivations Q~, and Pt, P, u E 1, d, pairwise commute, and ~1 (70 is therefore a commutative subalgebra of the Lie algebra ~(7-l). 3. The family Q l , . . . , P d is total, i.e., a normal polynomial f E 7t satisfies the system Q t ( f ) = P t ( f ) = O, p E 1,d, i f a n d o n l y i f f E H ~ = F. 4. The standard differential algebra (H, ~ l (H)) is defined (see Sec. 0.5). 3.2. E x p o n e n t i a l s . Let e = (a,b) = ( a l , . . . , ba) E F 2a. By exponentials (more precisely, c-exponentials; e.g., see [16]), we mean the solutions f of the system Qt(f) = atf,
P~,(f) = b t f ,
# E 1,d.
(3.1)
Because the operators Q , and Pt are derivations, the exponentials possess the characteristic property 1. a product f' 9f " is a (c' + c")-exponential if f ' is a d-exponential and f " is a c"-exponential. Simple calculations show that 2. every solution of system (3.1) in the class of formal power series has the form (up to ,mrmalization) bI
f(q,p;c) = E
144
-i!q' E
(s
( - 1 ) I J I ~ I/ =
~
(-1)ljIbiaJ ~ q p ~'. '
In particular, exponentials are formal (generalized) elements not belonging to the algebra 7/. Nevertheless, they can prove useful in applications. 3.3. D e r i v a t i o n s of t h e s e c o n d d e g r e e . We consider the derivations Q~,~ = {q~,q~, .} = Q~,, R~,~ = {qup~, .}, and P,~ = {pup~, .} = P~,, # , u ~ 1,d. It is easy to show the following using formulas (1.1): 1. T h e above derivations act according to the formulas Qu~'(f) = -
qu~p~ + q,,
0f R~,~(f) = q~' oq~
- ~
cOpucOp~.
,
Of c3p, p~'
( ofw
of
for all f E 7t, #, u E 1, d (we note that second derivatives are involved here). 2. T h e algebra ~2(71) is a subalgebra of ~(71) of dimension d(2d + 1) with the basis Q,~, R ~ , # , u E 1,d, and P ~ , p 0.
(4.1)
k,l,m,n
Furthermore, proceeding from the relation [~(7-t),~(7/)] = ~ ( 7 / ) (see assertion 1 in Sec. 2.4), we conclude that the differential is zero (dw = 0, w E fll(7/;F)) if and only if w = 0, i.e., kerdl = 0. In particular, H 1(7/; F) = ker d l / im do = 0. 4.2. Lie a l g e b r a 7t. Simultaneously with the Lie algebra ~ ( 7 / ) , we consider the Lie algebra 7 / w i t h the bracket {., .} (see assertion 9 in Sec. 1.11). We recall that the Lie algebra 7 / h a s the basis p~J, i,j E Ea+, and the commutation relations
{p~j, pkt} = ~ Cm;;'Jkl pm. . m~n
As in the case of the algebra ~(7/) (see assertion 1 in Sec. 2.4), the commutant is {7/,7-/} = 7/. We note that the center of the Lie algebra 7t is c e n T / = F (cf. assertion 3 in Sec. 1.11). Let A'7/ be the exterior algebra of the linear space 7/. According to the general theory (see [15]), the u-cochains ofT~ with a range in F are defined as the linear mappings from A~7/ into F, u E Z+, and the linear space of all cochains of 7-/with a range in F is therefore ~:'(7/; F) = HomF(A*7/; F) = (A'7/)'. In particular, r F) = F, and the dual linear space ~1 (7/; F) of 7 / h a s the dual basis v~O, O,j (pkt) = ~f~J, i , j , k , l E Zd+. The differential d = d,,: r ~ r u E Z+, is defined by the rule (d~)(fo, 99-, fu) =
~ (-1)v+ww({fv, f ~ } , f o .... , f - ~ , . . . , f ~ , . . . , f ~ ) oO
5r5060ij k~r),t
assertion 4 in Sec. 0.8, and a relation of the type Oij(f) = fii = O0 ( n ( f ) ) , we finally conclude that
2. one-dimensionM central extension (4.3) is associated with the 2-form
(4.5)
x = 2 ~ xlrl-~r!00r - 0,0 (5 fl2(74; F). Id>O
We consider the properties of the form X. By construction, it is closed. To verify this directly, we suppose that f t , f2, f3 E 74. Then - - -
-
=
= - O00({{/,,f'2},f.} + {{f2,f3} , f , } + {{f3,ft} ,:'2}) = O, where the relation [rr(f), rr(g)] = rr({f, g}) for f, g E 74 (see assertion 1 in Sec. 4.3) and the Jacobi identity are used.
149
The form X is not exact, however. Indeed, if we suppose that X = d~, where
= ~
g,'JO~ e fP(74;F)
Ii+jl>o is a 1-form with coefficients +ij E F, then
x(rr(f),~r(g)) = (d+)(~r(g),Tr(g)) = - + ( [ T r ( f ) , r r ( g ) ] ) = - + ( r r ( { f , g } ) ) = (dr
: -r for all f, g E 74, where
r = rr'g' =
~
++JO+j ~ C ( ~ ; F ) .
Ii+.ii>o On the other hand, we have X(n(f), r(g)) = -(dOoo)(f, g) by construction. Consequently, d(r + 0oo) = O, which means that r + tgoo = 0 because kerdl = 0 (see Sec. 4.2). More fully, ff2iiOij + 0oo = O, ti+jl>0 which is obviously impossible. This contradiction shows that the form X is not exact. We have thus proved the proposition below. 4.4. P r o p o s i t i o n .
The form X = 2 ~ xl~l-trtOo~ "Oro Irl>0
defines a nontrivial element [X] E H2(74; F). In particular, centrM extension (4.3) is nontrivial, and we have H2(74; F) # 0. 4.5. I n v a r i a n e e a l g e b r a o f t h e f o r m X. A form w E f~*(A;K) is said to be invariant with respect to a derivation D E D(A) (we use the notation in Sec. 0.7) if L o w = 0. By assertion 2 in Sec. 0.7, the set of all derivations with respect to which w is invariant forms a subalgebra (the invariance a/gebra w) of the Lie algebra ~ ( A ) . We calculate the invariance algebra of X. Let
F=
~ f i j R 0. Ii+jl>0
Then, according to assertion 4 in Sec. 0.7, we have LFX = LF(dX) + d(+FX ) = d(+FX), where the relation d X = 0 is taken into account. We have tFX E fP(74; F) and kerdl = 0 (see Sec. 4.1), and therefore LpX = 0 if and only if tFX = 0. This condition can be written more fully in the form of the equation
x ( F , G ) = ~ xlrl-'r!(/org~o - f~ogor), I~1>0
VG =
~ gktn k' 6 ~(74), Ik+tl>0
having the obvious solution for = fro = 0 for Irl > 0, the other coefficients fij being arbitrary. Therefore, 1. the invariance Mgebra of the form X consists of all derivations of the form
r=
Z Iil>0,[jl>0
A c k n o w l e d g m e n t s . This work was supported by the Russian Foundation for Basic Research (Grant No. 98-01-00640). LSO
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