An introduction to Lie algebra

Transcription

1 An introduction to Lie algebra cohomology and higher order Poisson structures Jose A. de Azcarraga Valencia University and IFIC (CSIC-UVEG) Argonne Workshop on Branes and Generalized Dynamics, October 20-24,

2 Plan of the talk: 1.Preliminaries: The three dierential operators d, i X, L X 2.Lie algebras and their dual aspects: invariant vector elds and forms 3.Lie algebra cohomology and its Chevalley-Eilenberg formulation 4.Invariant symmetric polynomials (Casimir-Racah) and higher order cocycles 5.Cohomology and topology for simple compact groups 6.Higher order simple Lie algebras and their Cartan-like classi- cation 7.Strongly homotopy (SH) and Batalin-Vilkovisky (BV) algebras 8.Generalizations of Poisson structures. Two ways: a) Nambu-Poisson structures (N-P) b) Generalized Poisson structures (GPS) 2

3 1.Preliminaries The Lie derivative The Lie derivative gives the innitesimal variation of tensor elds on M under a uniparametric group of dieomorphisms of M. Consider the transformations rules for (covariant) tensors as used in physics: 0 (x 0 )=(x) t 0 i(x 0 )=t 0 i t 0 i1i2 (x0 )=t j 1j2 1 Dene `local variations' j 0 0 i2 (x) 0 (x) ; (x) t i (x) t 0 i(x) ; t i (x) ::: 3

4 The denition of Lie derivative is then (X = j ) x i =( j (x)@ j )x i = i (x) =X:x = ; j (x)@ j (x) :=;L X ( t) i = ;( j t i +(@ i j )t j ):=;(L X t) i ( t) i 1i2 = j t i 1i2 +(@ i1 j )t ji 2 +(@ i2 j )t i 1j) := ;(L X t) i 1i2 : For an arbitrary (covariant) tensor: (x) =(x) i 1:::i q dx i 1 ::: dx i q X(x) k (L Y )(X i 1 ::: X i q )=Y (X i 1 ::: X i q ) ; qx i=1 (X i 1 ::: [Y X i] ::: X iq ) L X is a derivation, L X (t t 0 )=(L X t) t 0 + t L X t 0 aproperty which may beusedtodenel X on general tensors from the action of L X on scalars, vector elds (L X Y =[X Y ]) and on one-forms. 4

5 The exterior derivative d The exterior derivative is an (anti)derivation of degree +1, d : q (M) 7! (q+1) (M). It satises (anti)leibniz's rule if 2 q, 2 r, d( ^ ) =d ^ +(;1) q ^ d and is nilpotent, d 2 =0. Locally, onaq-form (x) = 1 q! (x) i1:::i q dx i 1 ^:::^dx i q d = i 1:::i q j dx j ^ dx i 1 ^ :::^ dx i q The coordinate free expression for d is given by Palais formula, (d)(x 1 ::: X q X q+1 )= X (;1) i+j ([X i X j ] X 1 ::: ^Xi ::: ^Xj ::: X q+1 ) i<j + X q+1 i=1 (;1) i+1 X i (X 1 ::: ^Xi ::: X q+1 ) : 5

6 If is a one-form: d(x 1 X 2 )=X 1 (X 2 ) ; X 2 (X 1 ) ; ([X 1 X 2 ]) : The inner product i X with respect to the vector eld X is the (anti)derivation (i X (^) =(i X )^ + (;1) q ^ i X ) of degree ;1 dened by (i X )(X 1 ::: X q;1 )=(X X 1 ::: X q;1 ) : The Lie derivative L X is given in terms of i X and d by the Cartan decomposition formula It follows that L X = i X d + di X [L X d]=0 : Other useful identities are and [L X i Y ]=i [X Y ] [L X L Y ]=L [X Y ] : 6

7 Left and right invariant vector elds on G Let G be a Lie group then we may dene the set of vector elds that are invariant under the left (right) translations of the group, i.e. L T g0x L (g) =X L (g 0 g) (R T g X R (g 0 )=X R (g 0 g)) where L T g X L (g) X R (g) 2 X(G) (R T g ) stands for the left (right) translation. The set of left- (right-) invariant [LI (RI)] vector elds span a Lie algebra, [X L (i) (g) XL (j) (g)] = Ck ij XL (k) (g) C [i1i2 C i3] =0 (Jacobi identity) where the square bracket means antisymmetrization. For the RIVF we nd (notice the ; sign) [X R (i) (g) XR (j) (g)] = ;Ck ij XR (k) (g) : The left, right invariance conditions read in terms of L X R L X L L X R (j) (g) XL (i) (g) =[XR (j) (g) XL (i) (g)] = 0 L X L (i) (g) XR (j) (g) =[XL (i) (g) XR (j) (g)]=0 7

8 Now let us dene the basis of LI one-forms! L(i) (g)! (i) (g) 2 1 (G) that is dual to the LI vector elds X L (g) X(g), i.e., such that! (i) (g)(x (j) (g)) = i j d! (i) (g)(x (j) (g) X (k) (g)) = ;! (i) (g)([x (j) (g) X (k) (g)]) or, in coordinate form (Maurer-Cartan equations), d! (i) (g) =; 1 2 Ci jk!(j) (g) ^! (k) (g) : The action d, where is a LI q-form (g) = 1 q! i1:::i q! (i1) (g) ^ :::^! (iq) (g) on LI vector elds reduces to d(x 1 ::: X q+1 )= X i<j (;1) i+j ([X i X j ] X 1 ::: ^Xi ::: ^Xj ::: X q+1 ) The transformation properties of! (i) (g) are given by L X(i) (g)!(j) (g) =;C j ik!(k) (g) (note that L X R (i)! R(j) = C j ik!r(k) and that L X R (i)! L(j) = 0=L X L (i)! R(j).) 8

9 Eilenberg formulation Lie algebra cohomology A V -valued n-cochain n on G is a skewsymmetric mapping n : G^ n ^G!V A n = 1 n! A i1:::i n! i 1 ^ :::^! i n : where V is a vector space, A =1 ::: dimv, and! i 2 G. The group of n-cochains is denoted by C n (G V). The coboundary operator s is dened by (s n ) A (X 1 ::: X n+1 ):= X n+1 j k=1 j<k + (;) j+k A ([X j X k ] X 1 ::: ^Xj ::: ^Xk ::: X n+1 ) X n+1 i=1 (;) i+1 (X i ) A :B (B (X 1 ::: ^Xi ::: X n+1 )) where is a representation of the Lie algebra G on V. s is nilpotent, i.e. s 2 =0 The n-th cohomology group H n (G V) is dened by H n (G V)=Z n (G V)=B n (G V) 9

10 ness of the sequence C n-1 s C n s C n+1 Z n B n s: Coboundary operator 2 C n : is a n ; cochain 2 Z n : is a n ; cocycle 2 Z n ) s =0 2 B n : is a n ; coboundary 2 B n )9=d= 2 C (n;1) : H n = Z n =B n :n; th () cohomology group s = d C n =n; forms on a manifold : de Rham cohomology s as before C n = skewsym: mappings on G : Lie algebra cohomology s = d C n =LIn ; forms on G : Lie alg: coho: (Chevalley ; Eilenberg f ormulation): 10

11 Whitehead's lemma for vector valued cohomology Let G be a nite-dimensional semi-simple Lie algebra over a eld of characteristic zero and let V be a nite-dimensional irreducible (G)-module such that (G)V 6= 0( non-trivial). Then, H q (G V)=0 8q 0 : If q = 0, the non-triviality of and the irreducibility imply that (G) v =0(v 2 V ) holds only for v =0. 11

12 Let V R (X) =0i.e., H n! H0 n. Then the rst term in the denition of the coboundary operator is not present and, for left invariant forms, d = s Let us identify X i with X (i) L (g) and!i with! (i)l (g) via the `localization' mechanism: From the basis X i of G T e (G) we may construct LIVF or LIF on the group manifold G by left translation, i.e. X i X L (i) (e) ) XL (i) (g) =LT g X L (i) (e)! i! (i)l (e) )! (i)l (g) =(L g ;1)! (i)l (e) Thus, the action of d on LI forms on G is the same as the action of s on the corresponding Lie algebra G-cochains. Notice, however, that the Lie algebra (CE) cohomology on G and the de Rham cohomology on G are in general dierent: a LI form on G may be de Rham exact, = d, but not a CE coboundary if the potential form is not LI (the case of the WZ terms for super-p-branes). Thus, H DR (G) H 0 (G R) : Nevertheless, for G compact H DR (G) =H 0 (G R) : 12

13 Introduce anticommuting, `odd' objects (the ghosts) c i c j = ;c j c i i j =1 ::: dim G and the ( = 0, trivial action) coboundary operator Then, s 0 = 1 2 Ck ij cj k : s 0 c k = ; 1 2 Ck ij ci c j (MC) and s 2 0 =0 (Jacobi) For the cohomology associated with a non-trivial action of G on V we introduce ~s and one checks that ~s = c i (X i )+ 1 2 Ck ij cj k ~s 2 =0 The `BRST'-cochains are dened by ~ A n = 1 n! A i1:::i n c i 1 :::c i n the action of ~s is equivalent to the action of s. As a result, ~s and s generate the same (Lie algebra) cohomology. 13

14 der cocycles (G simple) Symmetric invariant polynomials and higher order Casimir-Racah operators A (symmetric) LI polynomial on G k = k i 1:::i m! i 1 :::! i m! 2 LI 1 (G) is said to be invariant i L Xl k =0,X 2 X LI 1 (G), i.e., k([x l X i 1 ] ::: X i m )+k(x i 1 [X l X i 2 ] ::: X i m )+::: +k(x i 1 ::: [X l X im ]) = 0 ) C s li1 k si2:::i m + C s li2 k i1s:::i m + C s li m k i 1:::i m;1s =0 (ad-invariance of k. Note that `invariant' actually means bi-invariant i.e., also RI, since k was LI already). Let k i 1:::i m be a symm. invariant polynomial. Then C (m) = k i 1:::i m X i 1 :::X i m is a Casimir-Racah operator, i.e. [k i 1:::i m X i 1 :::X i m X s ]=0 8X s 2G A way to nd symmetric invariant polynomials (often used in the construction of characteristic classes), is by computing the symmetric trace in a representation of G. This means that k i 1:::i m =str(x i 1 :::X i m ) denes a symmetric invariant polynomial. 14

15 Let us consider the q-form = 1 q! Tr( ^ q ^) d = ; ^ (MC eqs:) where is the LI G-valued canonical one-form on G, =! i X i. (Note that q has to be odd if q is even is zero by virtue of the cyclic property of the trace). The form is closed, d = 0, and cannot be de Rham exact because q ; 1 is even. Thus, denes a non-trivial CE cocycle. In coordinates, is expressed as (q =2m ; 1) = 1 q! Tr(X i1 ::: X i2m;2 X )! i 1 ^ :::^! i 2m;2 ^! Thus, in the (2m-1)-cocycle general expression = 1 q! i1:::i2m;2! i 1 ^ :::^! i 2m;2 ^! i 1:::i2m;2 = k l 1:::l m;1[ Cl 1 i 1i2 :::Cl m;1 i2m;3i2m;2] where k is the invariant symm. polynomial given by k l 1:::l m;1 =str(x l 1 :::X l m;1 X ) Tr above becomes str because the antisymmetrization [i 1 i 2 :::i (2m;2) ]produces symmetry in l 1 :::l (m;1). Thus, to compute the (l) cocycles of G we need to know the symm. inv. polynomials on G or, equivalently, the (l) dierent Casimir-Racah operators. 15

16 ishes because for any symmetric G-invariant polynomial k, j 1:::j2m i1:::i2m Cl 1 j 1j2 :::Cl m j 2m;1j2m k l1:::l m =0 (this follows from the LI invariance of k) and 2(p+q);1 i1:::i 2(p+q);1 = 1 (2(p + q) ; 1)! j 1:::j 2(p+q);1 i1:::i 2(p+q);1 C l 1 j 1j2 :::Cl p j2p;1j2p Cm 1 j2p+1j2p+2 :::Cm q;1 j 2(p+q);3 j 2(p+q);2 k (p) (l1:::l p k (q) m1:::m q;1j 2(p+q);1 ) =0 : Invariant polynomials from cocycles (G simple) Now let us suppose that we have a(2m;1)-cocycle. Then t i 1:::i m =[ (2m;1) ] j 1:::j2m;2i m C i 1 j 1j2 :::Ci m;1 j2m;3j2m;2 is invariant and symmetric. This exhibits the one-to-one relation between the l classes of non-trivial cocycles of the Lie algebra cohomology and the l (primitive) Casimirs in the enveloping algebra U(G). For a review, details and references see physics/

17 (from J.A. de A., A.J. Macfarlane, A.J. Mountain and J.C. Perez Bueno, NP B510, 657 (1998),physics/ J.A. de A., A.J. Macfarlane, Int. J. Mod. Phys. A16, (2001),math-ph/ ) f 123 =1 f 147 =1=2 f 156 = ;1=2 f 246 =1=2 f 257 =1=2 f 345 =1=2 f 367 = ;1=2 f 458 = p 3=2 f 678 = p 3=2 Non-zero structure constants for su(3) (3-cocycle). d 118 =1= p 3 d 228 =1= p 3 d 338 =1= p 3 d 888 = ;1= p 3 d 448 = ;1=(2 p 3) d 355 =1=2 d 668 = ;1=(2 p 3) d 778 = ;1=(2 p 3) d 146 =1=2 d 157 =1=2 d 247 = ;1=2 d 256 =1=2 d 344 =1=2 d 558 = ;1=(2 p 3) d 366 = ;1=2 d 377 = ;1=2 3rd-order invariant symmetric polynomial for su(3) = 1= = 1= = p 3= = ; p 3= = ; p 3= = ; p 3= = p 3= = ; p 3= = ; p 3=6: Non-zero coordinates of the su(3) ve-cocycle. 17

18 f =1 f =1=2 f = ;1=2 f =1=2 f = ;1=2 f =1=2 f =1=2 f =1=2 f =1=2 f =1=2 f = ;1=2 f =1=2 f = ;1=2 f = p 3=2 f =1=2 f = ;1=2 f =1=2 f =1=2 f = p 3=2 f =1=2 f = ;1=2 f =1=2 f =1=2 f =1=(2 p 3) f =1=(2 p 3) f = ;1= p 3 f = p 2= p 3 f = p 2= p 3 f = p 2= p 3 Non-zero structure constants for su(4) (3-cocycle) =1= =1= =1= =1= = p 3= =1= = ;1= =1= =1= = ; p 3= = ;1= =1= = ;1= = ;1= = p 3= = ; p 3= = p 6= = ; p 6= = ; p 3= =1= =1= = ; p 3= = ;1= =1= = ;1= = ;1= =1= = ;1= = ; p 3= = ; p 3= = ; p 6= = ; p 6= =1= = ;1= =1= =1= =1= = ; p 3= = ; p 3= = p 6= = p 6= = ;1= = ;1= = ;1= = p 3= = ; p 3= = ; p 6= = p 6= =1= = ;1= = ; p 3= = ; p 3= = p 6= = p 6= = p 3= = ; p 3= = ; p 6= = p 6= = ;1= =1= = p 3= =1= = ;1= = ; p 3= =1= =1= =1= = ;1= =1= =1= = p 3= = ; p 3= = p 6= = ; p 6= =1= =1= =1= =1= =1= = ; p 3= = p 3= = p 6= = ; p 6= =1= =1= =1= = ; p 3= = ; p 3= = p 6= = p 6= =1= =1= = p 3=36 18

19 = ; p 3= = ; p 6= = p 6= = p 3= = p 3= = ; p 6= = ; p 6= = ;1= = ;1= =1= =1= = ; p 3= = ; p 3= = p 6= = p 6= = p 3= = ; p 3= = ; p 6= = p 6= = ;1= = ;1= = p 3= = p 3= = ; p 6= = ; p 6= = ; p 3= = p 3= = p 6= = ; p 6= = ;1= =1= = ; p 3= = ;1= =1= = ;1= = ;1= = p 3= =5 p 3= = p 6= = ; p 6= = ;1= =1= = p 3= = p 3= = p 6= = p 6= = p 3= = ; p 3= = p 6= = ; p 6= = ; p 2= = ; p 2= = ;1= =1= = ;1= = ;1= = ; p 3= = p 3= = ; p 6= = p 6= = p 3= = p 3= = p 6= = p 6= = p 2= = ; p 2= = ;1= = ;1= = p 3= =5 p 3= = p 6= = ; p 6= = ; p 2= = ; p 2= = ;1= =1= = p 2= = ; p 2= = ;1= = ;1= = ; p 3= = p 3= = p 3= = ; p 6= = ; p 6= = ; p 6=9 Non-zero coordinates of the su(4) ve-cocycle. 19

20 5.Cohomology and topology for simple compact groups For compact groups H 0 (G) H DR (G) In general, the n-th Betti number is dened by b n (G) = dim ; H n DR (G) : The Poincare polynomial of G is the polynomial P (G) :R! R given by P (G t) = dimg X p=0 b p (G)t p P (G ;1) = (G) : Consider now e.g. SU(2) S 3. The only nontrivial de Rham cocycle is proportional to the volume element on S 3, i.e. dim[hdr 3 (G)] = 1 the Lie algebra cocycle is given by i 1i2i3 k([x i1 X i2 ] X i3 ), i 1i2i3 = Cl 1 i 1i2 k l1i3 = C i1i2i3 = " i1i2i3 : This indeed gives the volume form on S 3. Therefore we see here that the CE and DR cohomologies on SU(2) can be identied. 20

21 SU(n)=SU(n ; 1) = S 2n;1 so that (from the point of view of de Rham cohomology) SU(l +1) [SU(l +1)=SU(l)][SU(l)=SU(l ; 1)]::::[SU(2)=1] and SU(l +1) S 2l+1 S 2l;1 ::: S 3 P (SU(l +1) t) = (1 + t 3 )(1 + t 5 ) :::(1 + t 2l+1 ) : For the C l series (Sp(l) in dimension n =2l) wemay use which gives Sp(l)=Sp(l ; 1) = S 4l;1 P (Sp(l) t) = (1 + t 3 )(1 + t 7 ) :::(1 + t 4l;1 ) : For the SO(n) groups (n odd, n = 2l +1, B l n even n = 2l, D l ) the situation is more involved and the results for B l and D l, P (SO(2l +1) t) = (1 + t 3 )(1 + t 7 ) :::(1 + t 4l;1 ) P (SO(2l) t) = (1+t 2l;1 )(1+t 3 )(1+t 7 ) :::(1+t 4l;5 ) cannot be naively deduced from SO(n)=SO(n ; 1) S n;1. In particular, the l-th order invariant giving the (2l ; 1)-cocycle of SO(2l) comes from the Pfaan. Aside remark: as a result, the Euler characteristic of any simple G is (G) =P (G t = ;1) = 0. 21

22 dim G Order m i of Casimirs Order (2m i ; 1) of G-co G l (l +1) 2 ; 1 [l >1] 2 3 ::: l ::: 2l +1 A l l(2l +1) [l >2] 2 4 ::: 2l 3 7 ::: 4l ; 1 B l l(2l +1) [l >3] 2 4 ::: 2l 3 7 ::: 4l ; 1 C l l(2l ; 1) [l >4] 2 4 ::: 2l ; 2 l 3 7 ::: 4l ; 5 2l ; D ,6 3,11 G ,6,8,12 3,11,15,23 F ,5,6,8,9,12 3,9,11,15,17,23 E ,6,8,10,12,14,18 3,11,15,19,23,27,3 E ,8,12,14,18,20,24,30 3,15,23,27,35,39,47 E lx (order (2m i ; 1) of cocycle) = dimg The l Casimirs and l cocycles for all compact simple Lie algebras i=1 22

23 their Cartan-like classication For ordinary simple Lie algebras, we have [X i X j ]=C k ij! ij X where! ij is the three-cocycle associated with the Cartan-Killing metric (this is why always H 3 (G) 6= 0). The elements X i can be realized in terms of matrices (operators) and the Lie bracket as [X i X j ] = X i X j ; X j X i, which implies Jacobi's identity because the product of operators is associative. If we dene a skew-symmetric `four-bracket' [X i 1 X i2 X i3 X i4 ]:="j 1j2j3j4 i1i2i3i4 X j1 X j2 X j3 X j4 we have for su(n), n 3 (and with c = 1 n below) [X i 1 X i2 X i3 X i4 ]= 1 2 2"j 1j2j3j4 i1i2i3i4 [X j1 X j2 ][X j3 X j4 ] = 1 1j2j3j4 2 2"j i1i2i3i4 Cl 1 j 1j2 Cl 2 j 3j4 X l1 X l2 = 1 1j2j3j4 2 2"j i1i2i3i4 Cl 1 j 1j2 Cl 2 1 : X + c l 1l2 ) j 3j4 = 1 2 3"j 1j2j3j4 i1i2i3i4 Cl 1 j 1j2 Cl 2 2 (d l1l2 j 3j4 d l1l2 : X =! i 1:::i4 : X since C j 1j2lCj3j4 l (skew in j's) is zero by JI. Thus, we have obtained a 4-bracket the structure constants of which are given by the 5-cocycle of su(n). 23

24 X 2S n (;1) () [X (1) ::: X (n) ] and check that for n even, the generalised Jacobi identity (GJI) [J.A. de A., A. Perelomov and J.C. Perez Bueno, J. Phys. A29, L151-L157 (1996), q-alg/ ibid. 29, (1996), hep-th/ P. Hanlon and M. Wachs, Adv. Math. 113, (1995) ] X 2S2n;1 (;1) () h [X(1) ::: X (n) ] X (n+1) ::: X (2n;1) i =0 holds true. This motivates the following Denition An order n (n even) higher order Lie algebra is a vector space G endowed with a G-valued totally skewsymmetric n-bracket such that the generalised Jacobi identity is fullled. Note: for n odd, the GJI / [X 1 ::: X 2n;1 ] (2n ; 1 bracket) rather than zero. Arguing similarly for the series A l B l C l ::: the associated higher order Lie algebras are given by [X i 1 ::: X i2m;2 ]= i1:::i2m;2 X where are the l (2m ; 1)-cocycles that play i1:::i2m;2 the r^ole of generalised structure constants. 24

25 The generalised Jacobi identity reads, in terms of the (2m ; 1)-cocycle coordinates " j 1:::j4p;1 i1:::i4p;1 j1:::j2p j =0 2p+1:::j4p;1 This equation holds true due by virtue of being a cocycle (and for p = 1reduces to the ordinary Jacobi condition). Theorem. Classication theorem for higher-order simple Lie algebras Given a simple algebra G of rank l, there are l ; 1 (2m i ; 2)-higher-order simple Lie algebras associated with G. They are given by the l ; 1 Lie algebra cocycles of order 2m i ; 1 > 3 which may be obtained from the l ; 1 symmetric invariant polynomials k on G of order m i >m 1 =2. The m 1 =2case (for which k is the Killing metric) reproduces the original simple Lie algebra G for the other l ; 1 cases, the skewsymmetric (2m i ; 2)-commutators dene an element of G by means of the (2m i ; 1)-cocycles. The higher-order structure constants of the algebra (as the ordinary structure constants with all the indices written down) are the fully antisymmetric Lie algebra cohomology cocycles and satisfy the generalised Jacobi condition. [J. A. de A. and J.C. Perez Bueno, Commun. Math. Phys. 184, (1997), hep-th/ ] 25

26 algebra Here we shall introduce the complete BRST operator associated with a Lie algebra. For ordinary G we have d! = ; 1 2 C ij!i ^! j d = ; ^ and, in terms of the anticommuting ghosts, s = ; 1 2 ci c j C ij Now we introduce ~ dm (~ : ~d m! 1 = ; (2m ; 2)! i1:::i2m;2!i 1 ^ :::^! i2m;2 : Using the canonical G-valued one-form and that [ 2m;2 ]:=! i 1^ ^! i 2m;2[X i 1 X i2m;2 ],theabove constitutes the generalised Maurer-Cartan (GMC) equation, 1 2m;2 ~d m = ; [ ] : (2m ; 2)! By using the GJI it follows that dm ~ is nilpotent, d ~ 2 m = 0, since ~d m dm ~ 1 1 2m;2 = ; [[ ] 2m;3 ] : (2m ; 2)! (2m ; 3)! 26

27 Each GMC equation can be expressed in terms of the ghost variables as a `generalised BRST operator' 1 ~s 2m;2 = ; (2m ; 2)! ci 1 :::c 2m;2 i By adding together all l generalised BRST operators, the complete BRST operator s c is obtained. Theorem Let G be a simple Lie algebra. Then, there exists a nilpotent associated operator s c given by the odd vector eld s c s 2 + :::+ s 2mi ;2 + :::+ s 2ml ; ::: ; 1 2 cj1 c j 2 j 1j2 1 ; (2m i ; 2)! cj 1 :::c j 2m i ;2 j 1:::j2m i ;2 1 ; (2m l ; 2)! cj 1 :::c j 2m l ;2 j 1:::j2m where i =1 ::: l, j 1j2 C j 1j2 and j 1:::j2m i ;2 are the l corresponding higher-order (2m i ; 1)-cocycles. The operator s will be called the complete BRST operator associated with G. Note. s 2 c =0by virtue of s 2mi ;2 2 =0(i =1 ::: l) crossed terms do not contribute (cf SH algebras). 27

28 Let (G) the exterior algebra of multivectors, X 1 ^^X q := i 1:::i q 1:::q X 11 X i q A coderivation is a : n! n;1 given 1 ^X n )= nx l=1 l<k (;1) l+k+1 [X l X k ] ^ X 1 ^ ^Xl ^ ^Xk 2 (@ 2 (X l ^ X k )=[X l X k 2 (X 1 ^X n )= 1 1 2! (n ; 2)! i 1:::i n 1:::n [X i1 X i2 ] ^ X i3 ^X i n We now 2 to (s s : n! s : s! 1 (lowers by s ; s (X 1 ^^X n )= 1 1 s! (n ; s)! i 1:::i n s(x i 1 ^^X i s ) ^ X is+1 ^X in (@ s (X i 1 ^^X i s )=[X i 1 ::: X i s s n =0 for s>n it follows s 2 0 The expression contains n s = n! dierent terms. (n;s)!s! For 4 : 7! 4 : 4! 1 4(X 2 i 1^ ^X i7 ) / [[X i1 ::: X i4 ] X i5 X i6 X i7 ] 0by GJI 28

29 Back to higher order derivations Recall that, with 2 n d(x i 1 ::: X i n+1 )= X s<t (;) s+t ([X is X it ] ::: ^Xis ::: ^Xit ::: X in+1 ) which contains n+1 2 terms. Let us introduce ~ d2 by (~ d2 )(X i 1 ::: X i n )= 1 1 (2 2 ; 2)! (n ; 1)! j 1j n+1 i1i ([X n+1 j 1 X j2 ] ::: X j n+1 ) ~d 2 : n! n+(22;3) = n+1 Generalize to ~d m : n! n+(2m;3) (~ dm )(X i 1 ::: X )= 1 1 in+(2m;3) (2m ; 2)! (n ; 1)! j 1j n+(2m;3) i1i n+(2m;3) ([X j1 ::: X j2m;2 ] X j2m;1 ::: X j n+2m;3 ) In particular, ~ d2 = ;d : 29

30 Then: a) On one-forms, ~ dm : 1! 2m;2 (raises order by 2m ; 3) i.e., (~ dm! )(X i 1 ::: X in+(2m;3) )= 1 (2m ; 2)! j 1j2m;2 i1i2m;2! ([X j 1 ::: X j2m;2 ])! ([X j 1 ::: X j2m;2 ]) = ~d m! = j1:::j2m;2 1 (2m ; 2)! j1j2m;2!j 1 ^^! j 2m;2 or generalized (higher order) Maurer-Cartan (GMC) equations b) Leibniz rule, ~ dm odd, ~d m ( ^ ) =~ dm ^ +(;1) n ^ ~ dm 30

31 c) ~ dm and are dual to each other with a n-form, on [n +(2m ; 3)]-multivector, ~d m / (~ dm is a [n +(2m ; 3)]-form and, on a [n +(2m ; 3)]-multivector, gives a [n+(2m;3);(2m; 2 ; 1)] = n-multivector) ~d 2 ;d (MC) is dual 2 d) The usual Leibniz rule is equivalent to m (d 1+ 1 d). Then, 4@ =(@ 4 is a coderivation with respect to the coproduct 4. One nds ~d m = n! 2m;2 ( 2 n ) as a consequence of using the same denition (antisym. with no weight factor) for the wedge product of forms and multivectors. 31

32 Vilkovisky (BV) algebras (See T. Lada + J. Stashe, Int. J. Theor. Phys. 32, (1993), hep-th/ ) Recall that, for n =2s ; 1, (n ; s) =s ; 1 and (@ s )(X 1 ^X 2s;1 )= 1 1 s! (s ; 1)! i 1i2s;1 s(@ s (X i 1 ^X i s ) ^ X is+1 ^X 2s;1 ) 0 by the GJI s (@ s (X i 1 ^X i s ) ^ X is+1 ^X 2s;1 )= [[X i 1 ::: X i s ] X is+1 ::: X i 2s;1 ] 0 (GJI) Relax now the GJI condition by introducing a collection of skewsymmetric maps l n : n (V )! V such that X i+j=n (i ; 1)! j! i 1i n 1n (;1)i(j;1) l i [l j (X i 1 ^X i j ) ^ X ij+1 ^^X in ]=0 SH algebras The collection of the skewsymmetric maps l n dene a SH Lie structure. When a unique l n (n even) is dened, one recovers the denition of a higher order Lie algebra since, for i = j = s and n =2s;1, gives the GJI. 1 1 (i;1)! j! P 2S n = P j n;j(unsuffles) i.e., 2 S n sases (1) <:::< (j) and (j +1)<:::<(n) : 32

33 n=1: l 1 is a diferential, l 2 1 =0 n =2: Then, i j =1 2 and with l 2 (X 1 ^ X 2 ) [X 1 X 2 ] ) ; 1 2 l 1(l 2 (X 1 ^ X 2 ) ; l 2 (X 2 ^ X 1 ))+ + l 2 (l 1 (X 1 ) ^ X 2 ; l 1 (X 2 ) ^ X 1 )=0 l 1 [X 1 X 2 ]=[l 1 X 1 X 2 ]+[X 1 l 1 X 2 ] i:e: l 2 denes a bracket for which l 1 is a derivation l 1 l 2 = l 2 l 1 n =3:9 l 1 l 2 l 3 and ) [[X 1 X 2 ] X 3 ] + cyclic 6= 0 l 2 l 2 = ;l 1 l 3 ; l 3 l 1 `controlled violation' of Jacobi identity. D =(l 1 + l l i + + l j + ) D 2 =0 ) The above for its homogeneous components (i + j = n +1) 33

34 Now, consider the Batalin-Vilkoviski algebras (see E. Getzler, CMP '94, hep-th/ Alfaro + Damgaard, PLB 369, 289 '96, hep-th/ Koszul, Soc. Math. France Asterisque, hours serie, (1985) F. Akman, q-alg/ ) Consider formally, and ignoring all Grassmann parities and sign factors, and introduce Batalin-Vilkovisky operator l 1 l 1 () is nilpotent Usual antibracket l 2 (A B) :=l 1 (AB) ; (l 1 A)B ; Al 1 (B) (or(a B) =(AB) ; (A)B ; A(B) : the antibracket measures the failure of to be a derivation of the product Witten + Koszul) `Three ; bracket 0 l 3 (A B C) :=l 2 (A BC) ; l 2 (A B)C ; Bl 2 (A C) Then, 2 =0 l 2 = l 2 l 2 2 = ;l 3 ; l 3 (The 3-bracket measures the breaking of the JI for the 2-bracket l 2 satises JI up to homotopy) When there is an innite number of higher antibrackets, the structure of higher order antibrackets is that of a SH algebra L 1. If we restrict to (l! l 2 l 3 ) this is a homotopy Lie algebra if strongly refers to all l's. 34

35 This structure also appears in classical closed string theory, where one introduces (Witten + Zwiebach, NP B377, , hep-th/ ) V =(f b a1 a 1+fa1a2 a 1 a 2+f b a1a2a3 b a 1 a 2 3 The f's are related to the string amplitudes and V 2 = 0 shows that the algebraic structure of closed string theory is that of a SH algebra, as expressed by its quadratic relations (cf. the complete BRST operator). L 1 : The L-algebras are given by the nilpotent BV 4 = l 1 operator L 2 : l 1 and the usual antibracket l 2 (A B) (A B) L 3 : l 1, l 2 and the `three-antibracket' l 3 (A B C) etc. In Koszul notation: l k+1 (A 1 ::: A k+1 )= k+1 4 (A 1 ::: A k+1 ) 35

36 The Schouten-Nijenhuis (SN) Bracket A = 1 p! Ai 1:::i i 1 ^^@ i p A of order p A 2 p (M) B = 1 q! Bj 1:::j j 1 ^^@ j q B of order q A 2 q (M) The SN bracket is a bilinear mapping [ ] SN : p q! p+q;1 [ ] SN is the unique extension of the Lie bracket which makes a Z 2 -graded Lie algebra of the graded-commutator algebra of skewsymmetric contravariant tensor elds 1 [A B] SN = (p + q ; 1)! [A B]k 1k k 1 ^^@ k p+q;1 [A B] k 1:::k p+q;1 1 SN = (p ; 1)!q! k 1:::k p+q;1 i1:::i p;1j1:::j q A i 1:::i p;1@ B j 1:::j q + (;)p p!(q ; 1)! k 1:::k p+q;1 i1:::j p j1j q;1 Bj 1:::j q;1@ A i 1:::i p 36

37 Properties of [ ] SN : Graded conmutativity :[A B] SN =(;) pq [B A] SN ) [A A] SN 0 for p odd Graded Jacobi: A of order p, B of order q, C of order r (;1) pr [[A B] C]+(;1) pq [[B C] A]+(;1) rq [[C A] B]=0 The adjoint action is a derivation with respect the wedge product, [A B ^ C] =[A B] ^ C +(;1) (p;1)q B ^ [A C] Note: (M) (the multivector elds on M) is a Z- graded space (with homogeneous subspaces a (M), a being the grade). Endowed with [ ] SN has the above three properties )Gerstenhaber algebra. 37

38 8.Generalizations of Poisson structures (a) Standard Poisson structures APoisson bracket (PB) on F(M) is a bilinear mapping f g : F(M) F(M)!F(M) that satises a) Skew-symmetry, ff gg = ;fg fg b) Leibniz's rule, ff ghg = gff hg + ff ggh c) Jacobi identity (JI), ff fg hgg + fg fh fgg + fh ff ggg =0 A PB on M denes a Poisson structure (PS). In local coordinates fx i g, it is possible to write where! ij = ;! ji and ff(x) g(x)g =! i f@ j g! k! lm +! k! mj +! k! jl =0 38

39 Geometrically, if we dene the PB by means of a bivector = 1 j k ff gg =(df dg) the Jacobi identity is equivalent to the vanishing of the Schouten-Nijenhuis (SN) bracket [ ] = 0 : Hence, will dene a PS i [ ] = 0. i dh =X H is a Hamiltonian vector eld and we have [X f X H ]=X ff Hg If the manifold M is the dual G of a Lie algebra G there always exists a PS given by (x i 2G ) fx i x j g = C k ij x k = 1 2 Ck ij x k@ i j! ij = C k ij x k 39

40 (b) Nambu-Poisson structures [Y.Nambu, Phys. Rev. D7, 2405 (1973). For the fundamental identity see V.T. Filippov, Sib. Math. J. 26, (1985), D. Sahoo + M. C. Valsakumar, Phys. Rev. A46, 4410 (1992) + Pramana 40, 1 (1993) L. Takhtajan, CMP 160, 295 (1994)] They are dened by the tensor = 1 n! i1:::i i 1 ^ i n and the Nambu-Poisson bracket satisfying ff 1 ::: f n g = (df 1 ::: df n ) a) Skew-symmetry, ff 1 ::: f i ::: f j ::: f n g = ;ff 1 ::: f j ::: f i ::: f n g b) Leibniz's rule ff 1 ::: f n;1 ghg = gff 1 ::: f n;1 hg + ff 1 ::: f n;1 ggh c) Fundamental identity (FI) ff 1 ::: f n;1 fg 1 ::: g n gg = fff 1 ::: f n;1 g 1 g g 2 ::: g n g + ::: + fg 1 ::: g n;1 ff 1 ::: f n;1 g n gg : 40

41 If the time evolution is dened by the FI tells us that _g = fh 1 ::: H n;1 gg d dt fg 1 ::: g n g = f_g 1 ::: g n g + :::+ fg 1 ::: _g n g : In other words, it guarantees that the time derivative is a derivation of the Nambu-Poisson bracket. In local coordinates, the FI is veried i [L. Takhatajan, CMP 160, 295 (1994)] i 1:::i n;1@ j 1:::j n ; (dierential condition) and 1 (n ; 1)! l 1:::l n j1:::j n (@ i 1:::i n;1l1 ) l2:::l n =0 +P () = 0 (algebraic condition), where is the 2n-tensor i 1:::i n j1:::j n = i 1:::i n j 1:::j n ; i 1:::i n;1j1 i n j2:::j n ; i 1:::i n;1j2 j1i n j3:::j n ; i 1:::i n;1j3 j1j2i n j4:::j n ; ::: ; i 1:::i n;1j n j 1j2:::j n;1i n and P interchanges i 1 and j 1. 41

42 This last condition implies that is a decomposable tensor (that may be written as a ^-product of vector elds) [Alekseevski + Guha, Acta Math. Univ. Com. LXV, 1-9 (1996) Gautheron, LMP 37, 103 (1996)]. Is the FI really fundamental? See T.L. Curtright and C.K. Zachos, math-ph/021102, hep-th/ , hep-th/ and contributions to this workshop. 42

43 [J.A. de A.,A.M. Perelomov and J.C. Perez Bueno, J. Phys. A29, (1996), hep-th/ J.A. de A., J.M. Izquierdo and J.C. Perez Bueno, J. Phys. A30, L607-L616 (1997), hep-th/ ] Since for an ordinary PS, dened by a bivector the Jacobi identity is given by the vanishing of the SN bracket [ ] = 0, it is natural to dene generalised Poisson structures (GPS) by demanding [ (2p) (2p) ]=0 " j 1:::j4p;1 i1:::i4p;1! j1:::j2p;1@! j 2p:::j4p;1 =0 where (2p) is the even multivector = 1 (2p)!! 1 ^ j 2p : Dening the GP bracket of a GPS by ff 1 ::: f 2p g =(df 1 :::df 2p ) we have that GPS satisfy the following properties a) Skew-symmetry b) Leibniz's rule c) The generalised Jacobi identity (GJI) Altff 1 ::: f 2p;1 ff 2p ::: f 4p;1 gg =0 Time evolution: _ f = fh1 ::: H 2p;1 fg This GJI does not imply the fundamental identity, so the bracket of constants of the motion is not a constant of the motion in general. (They are if f 1 ::: f q q 2, are in involution) In contrast, the FI implies the GJI, so that the GPS can be viewed as a generalization of the Nambu- Poisson structures. 43

44 Consider a simple Lie algebra G of rank l. There are l ; 1 higher order Lie algebras which, by the identication (G ) G, may dene a bracket f 2m s;2 g : G 2m s;2 G!G given by fx i 1 ::: x i2ms;2 g = i1:::i2ms;2 x x i 2G, where is the (2m s ; 1)-cocycle. The generalised Jacobi identity for the GPB (! i 1:::i2m 2;2 = i 1:::i2ms;2 x ), is satised automatically because the structure constants/cocycles i 1:::i2ms;2 satisfy the GJI of the higher order algebra. Hence,all higher-order Lie algebras dene linear GPS. Conversely, given a linear GPS, we may dene a higher-order Lie algebra, the structure constants of which are dened by the tensor. f ::: g $ [ ::: ] Generalised Poisson bracket Multibracket If quantization implies the replacement of observables by associative operators and the GP brackets by multicommutators, a replacement a la Dirac is possible for GP brackets. However, the time derivative is not a derivation of the bracket. The Nambu-Poisson structures are free from this problem, but in contrast the fundamental identity cannot be realized in an algebra of associative operators. 44

45 Standard Poisson str. GPS p Nambu-Poisson p Skewsymmetry p Leibniz rule identity Jacobi identity GJI p Fundamental (consequence of fundamental identity) p { FI is not an associative algebra identity Time is a derivation Liouville theorem in Realization of terms associative operators p { GJI has more terms p p GJIisan associative identity algebra is decomposable Decomposability is not decomposable rigid body eqs. Euler isotropic oscill. SU(n) Kepler problem Examples Innitely many in the linear case Are standard Q.M. (and C.M.) not generalizable and rather unique ) previous refs. and Curtright + Zachos contributions to the worksho (see 45