Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Many efforts have been devoted to the compre-

hension of the scaling laws in aging (Bouchaud

et al.). Among the theories and models of aging that

have been proposed, one can cite the phenomeno-

logical model known as ‘‘trap model,’’ devel oped by

Bouchaud and collaborators that assimilates aging

to a random walk between ‘‘traps’’ characterized by

a broad distribution of trapping times. Suitable

choices of the trapping-time dist ribution allow to

derive scaling laws similar to the ones characteristic

of aging systems . A different theory, the ‘‘droplet

model’’ for spin glasses assimilates aging phenomena

to the competition between slowly growing domains

of equilibrium phases, in analogy with the dynamics

of phase separation in first-order phase transitions.

The approach that has led to the most detailed and

spectacular predictions has been the study of

microscopic mean-field models.

Mean-Field Models of Disordered

Systems

Mean-field theory starts from the analysis of the

relaxation dynamics of disordered systems with

weak long-range forces (Bouchaud et al.). The

reference model of spin glass mean-field theory is

the so called p-spin model, which considers N spins

with random p-body interactions with each other

and is described by the Hamiltonian

ðSÞ¼

1;N

<<i

;...;i

S

½6

where the quenched coupling constants J

,..., i

are

assumed to be i.i.d. Gaussian variables with zero

average and N dependent variance E(J

,..., i

) =

p!=2N

p1

. The case p = 2 coincides with the SK

model defined in the introduction. The reason for

considering the p-spin generalization is that the

order of the transition passes from the second one

for p = 2 to the first one for p  3 and that this last

case has been suggested to provide a mean-field limit

for the structural glass transition. It is also useful to

define Hamiltonians

H½S¼

p1

½S½7

that mix p-spin Hamiltonians for different p. These

are random Gaussian functions of the spin variables,

with covariance induced by the coupling distribution

E½HðSÞHðS

Þ ¼ Nf ðqðS; S

ÞÞ

p1

qðS; S

½8

where the function

qðS; S

Þ¼

i¼1

is the overlap between configurations. A crucial

hypothesis in the study of relaxation in spin systems

is that any local spin update rule verifying the detailed

balance condition with respect to the Boltzmann–

Gibbs measure gives rise to the same long-time

properties. In this perspective, in Monte Carlo simula-

tions, it is convenient to use Ising spins with

Metropolis or Glauber dynamics. Much theoretical

progress has been achieved considering spherical

models where the spin variables are real numbers

subject to a global spherical constraint

= N and

evolve according to the following Langevin dynamics:

ðtÞ

¼

@HðSðtÞÞ

 ðtÞS

ðtÞþ

ðtÞ½9

where (t) is a time-dependent multiplier that at

each instant of time insures that the spherical

constraint is respected, and 

(t) is a thermal white

noise with variance

Eð

ðtÞ

ðsÞÞ ¼ 2 T

ðt  sÞ½10

In order to model the quench from high temperature

performed in experiments, the initial conditions are

randomly chosen with uniform probability. To

describe long but finite-time dynamics, it is neces-

sary to consi der the limit of large volume N !1for

finite time, which is the only case where one can

have infinite relaxation times. Application of func-

tional Martin–Siggia–Rose techniques has allowed

the derivation of closed integro-differential equa-

tions for the spin autocorrelatio n function

Cðt; t

Þ¼ lim

N!1

i¼1

ðtÞS

ðt

Þi

and the response to an impulsive external field

ðtÞ; Rðt; t

Þ¼ lim

N!1

i¼1

S

ðtÞ

h

ðt



where the average has to be intended on quenched

disordered couplings, initial conditions, and realiza-

tion of thermal noise. Unfortunately, in the case

p = 2 relevant for spin glass phenomenology, the

spherical constraints reduce the model to a linear

system where different eigenmodes of the interaction

matrix J

evolve independently. This oversimplifica-

tion renders the model similar to systems apt to

describe phase separa tion rather then freezing

phenomena. Many of the glassy features of the SK

model however are captured by a mixture of p = 2

556 Glassy Disordered Systems: Dynamical Evolution

with p = 4 Hamiltonians and f (q) = (1=2)(q

þ aq

For the general Hamiltonian [7] one gets the

coupled equations

@Cðt; t

¼ðtÞCðt; t

ðCðt; t

ÞÞRðt; t

ÞCðt

; t

ðCðt; t

ÞÞRðt

; t

@Rðt; t

¼ðtÞRðt; t

ðCðt; t

ÞÞRðt; t

ÞRðt

; t

Þ½11

(t) is a multiplier that at each instant of time

insures the spherical constraint C(t, t) = 1, and is

determined by

ðtÞ¼

Cðt; t

ÞRðt; t

ÞþT ½12

In the next sections, we will discuss how these

equations describe dynamical freezing at low tem-

perature. The gross features are determined by the

form of the function f (q). Two main behaviors can

be identified:

1. Systems of type I. This behavior is found if

000

(q)=(f

(q))

3=2

is a monotonically decreasing

function of q. To this family belongs the pure

spherical p-spin model for p  3, and one finds a

dynamical transition not corresponding to a

point of singularity in the free energy where the

Edwards–Anderson parameter jumps discontinu-

ously to a nonzero value. Models of this family

have been proposed as appropriate mean-field

limits for structural glass behavior.

2. Systems of type II. This behavior is found if

000

(q)=(f

(q))

3=2

is a monotonically increasing

function of q . This family mimics the behavior

of the SK model. An example of function f

verifying the condition for type II behavior is

f (q) = 1=2(q

þ aq

) for sufficiently small but

positive values of a. I n t his case, the dynamical

transition is found at a point of second-order

singularity of the free energy and the Edwards–

Anderson parameter is continuous at the transi-

tion. Models of this family provide a mean-field

limit for spin glass type behavior.

Equilibrium Dynamics at High Temperature

At high temperature, after a finite transient, eqn [11]

describes equilibrium behavior. In these conditions,

time-translation invariance holds C( t, t

) = C(t  t

R(t, t

) = R(t  t

) while the Lagrange multiplier 

becomes time independent. In addition, correlation

and response are related by the fluctuation–

dissipation theorem (FDT) relation

RðtÞ¼

dCðtÞ

½13

Ergodic behavior is manifest in the fact that the

dynamics decorrelate completely; lim

t !1

C(t) = 0.

Then from [11] one gets the equilibrium equation:

dCðtÞ

¼TCðtÞ

dsf

ðCðt  sÞÞ

dCðsÞ

½14

It is worth noticing that this equation, apart from an

irrelevant inertial term, coincides for type I systems

with the schematic mode-coupling theory (MCT)

equation which has been succe ssfully used to

describe moderate supercooled liquids (Goetze

1989). In the context of liquid theory, mode-

coupling equations stem from an approximate

treatment for the dynamical evolution of the

density–density space and time-dependent correla-

tion function. The schematic MCT equations con-

sider an equation for a single mode, neglectin g any

space dependence of the correlator.

Both in type I and in type II systems, eqn [14]

displays a dynamical transition at a finite tempera-

ture T

where the relaxati on time diverges as a

power law  jT  T

and the asymptotic value of

the correlation acquires a nonz ero value.

This behavior in type I systems represents a failure

of MCT to describe the temperature dependence of

the relaxation time in supercooled liquids, which, as

previously observed, empirically follows the Vogel–

Fulcher law. The MCT temperature is interpreted as

a singularity which is avoided in supercooled

liquids, thanks to relaxation mechanisms specific of

short-range systems. It has been noticed that this

singularity at T

can be associated to the growth of

spatial heterogeneities and dynamical correlations,

as exemplified in the behavior of the four-point

function



ðtÞ¼

i;j

ðtÞS

ð0ÞS

ð0Þi

and its associate correlation length (Franz and Parisi

2000, Biroli and Bouchaud 2004).

Off-Equilibrium Dynamics Below T

: Aging

and Slow Dynamics

Type I systems Below the transition temperature

slow dynamics and aging set in. In 1993,

Cugliandolo and Kurchan found a long-time

Glassy Disordered Systems: Dynamical Evolution 557

solution to the equations of motion [11] for type I

systems describing an asymptotic off-equilibrium

state that follows from high-temperature quench.

Soon after, type II systems were also analyzed

(Bouchaud et al.).

The equations can be analyzed in the limit in

which both times tend to infinity t, t

!1. In this

regime all ‘‘one-time quantities,’’ that is, state

functions like energy, magnetization, etc., reach

asymptotic time-independent limit. Though the

decay to the asymptotic value cannot read directly

from the analysis of the equations in that limit,

numerical and theoretical evidence sugges ts that the

final values are approached as power laws in time.

The study of correlation and response functions

displays an asymptotic scaling behavior similar to

the one observed in glassy systems in laboratory and

numerical experiments.

Two different interesting regimes are found , first of

all there is a stationary regime: the limit t, t

!1is

performed keeping the difference t  t

= s finite. In

this regime, equilibrium behavior is observed, with

correlation and response related by the FDT relation

(s) = @C

(s)=@s. The stationary regime is fol-

lowed by an aging regime, where correlations decay

below the value q

= lim

s !1

(s) down to zero.

One of the most striking features of aging evolution is

that the system – though at a decreasing speed –

constantly move far apart from any visited region of

configuration space. The decay of correlations is

nonstationary and takes place on a timescale (t

)

diverging for large t

. While the theory can infer the

existence of the timescale (t

), its precise form

remains undetermined. This is a consequence of an

asymptotic invariance under monotonous time repar-

ametrizations t !g(t) appearing for large times.

Coherently with nonstationary behavior, other equi-

librium propertie s break down in the aging regime.

Correlation and response which do not verify the

FDT are rather asymptotically related by a genera l-

ized form of the fluctuation–dissipation relation

ðt; t

Þ¼

ððt  t

Þ=ðt

ÞÞ

½15

This relation, despite predicting the vanishing of

the instantaneous response, implies a finite contribu-

tion of the aging dynamics to the value of the

integrated ZFC and TRM responses. The constant

X, called fluctuation–dissipation ratio (FDR), is a

temperature-dependent factor monotonically vary-

ing between the values 1 and 0 as the temperature is

decreased from T

down to zero. Violations of the

FDT have to be expected in any off-equilibrium

regime; however, a constant ratio between response

and derivative of the correlation is very nongeneric.

It is of great theoretical importance that the same

constant that governs the FDR among spin auto-

correlation and magnetic response, also appears in

the relation of any other conceivable couple of

correlation and conjugated response in the system.

Slow dynamics can be interpreted as motion

between finite-life metastable states with well-

defined free energy f and exponential multiplicity

exp(N(f )). The FDR verifies the generalized

thermodynamic relation

@

½16

This relation is in turn intimately related to the

possibility of considering the ratio T

eff

= T=X as

an effective temperature, that governs the

heat exchanges among slow degrees of f reedom

(Cugliandolo et al. 1997). Slow degrees of freedom

do not exchange heat with the fast ones, but they

are in equilibrium between themselves at the

temperature T

eff

. The validity of relation [16 ] has

been put at the basis of a detailed statistical

description of the glassy state (Franz and Virasoro

2000, Biroli and Kurchan 2001, Nieuwenhuizen

2000) which assumes that metastable states with

equal free energy are encountered with equal

probability during the descent to equilibrium.

Modified thermodynamic relations follow, that

condensate all the dependence on the thermal

history in the value of the effective temperature.

Given the interest of a thermodynamic description

of the glassy state, many numerical studies have

addressed the problem of the identification and

determination of effective temperatures from the

fluctuation–dissipation relations, and its relation

with configurational entropy. In Figure 3 the result

of a numerical study on a realistic system is

presented, verifying relation [15]. Experimental

verifications are at the moment starting and new

results are waite d in the future.

Type II syst ems In these systems the dynamic

transition occurs at the point of thermodynamic

singularity, where the Edwards–Anderson parameter

becomes nonzero in a second-order fashion. The

magnetic susceptibility exhibits a cusp singularity

similar to the one found in spin glass materials.

Differently from type I systems, one-time quantities

tend to their equilibrium values for long times. The

off-equilibrium nature of the relaxation shows up in

the behavior of correlations and responses, which

display aging behavior.

Their behavior generalizes the one found in type I

systems, with a more complex pattern of violation

558 Glassy Disordered Systems: Dynamical Evolution

of time-translation invariance and FDT. Also in this

case a short-time equilibrium behavior can be

identified where the correlation decreases from 1 to

and a long-time inhomogeneous aging behavior

where correlations decrease to zero. Differently from

type I system it is impossible to characterize aging

through a unique timescale (t

). One finds instead

a continuum of timescales hierarchically organized.

The analysis of the equations at the

reparametrization-invariant level reveals the existence

of a continuum of separate timescales (t

, q)asso-

ciated to each value of C(t, t

) = q < q

and that

lim

(t

, q)=(t

, q

) = 0forq > q

,meaningthat

for finite t

thetimetodecaytoq

is much larger than

thetimetodecaytoq. For large times, 1 << t

<< t

, the correlations verify the ultrametric prop-

erty C(t

, t

) = min[C(t

, t

), C(t

, t

)]. To each time-

scale corresponds in this case a different effective

temperature, and correlation and response are related

by the equation

XðqÞ¼ lim

t;t

Cðt;t

Þ¼q

Rðt; t

@Cðt; t

Þ=@t

½17

where the function X(q) is an increasing function of

q with the properties of a cumulative probability

distribution. In fact it can be seen (Franz et al. 1999)

that this is related to the Parisi overlap probability

function describing the correlations among ergodic

components at equilibrium, in a generalization of

relation [14]. Figure 4 shows the result of a

numerical experiment in a three-dimensional spin

glass, where X(q) is not piecewise constant.

The ideas presented in this article, fruits of mean-field

theory of disordered systems, are objects of intense

debate in their application to the physics of short-range

systems. Many of the relations derived have stimulated

a lot of numerical, experimental, and theoretical work.

Some of the predictions of the theory are very well

verified in many short-range glassy systems, at least on

the accessible timescales. Notably, the violations of

FDR, and the possibility to associate the values of the

FDR to effective temperatures is very well verified both

in structural glass models, and in finite-range spin

glasses. Since finite aging times imply finite length scales

over which the dynamic variables can exhibit correlated

behavior, this indicates that the mean-field theory is at

least good at describing glassy phenomena on a local

scale. The question if the mean-field theory also gives a

good description on the infinite time limit and the

anomalous response persists forever is at present an

open theoretical problem. It relates to the possibility of

having mean-field type of equilibrium ergodicity break-

ing, which is an open question, object of active research.

See also: Interacting Stochastic Particle Systems;

Short-Range Spin Glasses: The Metastate Approach;

Spin Glasses.

Further Reading

Barrat J-L and Kob W (1999) Europhysics Letters 46: 637.

Barrat J-L and Kob W Fluctuation–dissipation ratio in an aging

Lennard–Jones glass.

0.2 0.4 0.6 0.8 1.0

C(τ

+ t

)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

Tχ(τ + t

)

= 0.3

Figure 3 Fluctuation–dissipation plot; 

ZFC

(t, t

) vs. C(t, t

)in

a model of Lennard-Jones glass for different values of the

waiting time t

. The slope of the curves is equal to the finite-time

FDR divided by the temperature. One observes the characteristic

shape of type-I systems with an FDR equal to 1 in the stationary

regime for high correlations and equal to a constant smaller then

1 in the low-correlations aging regime. Reproduced from Kob W

and Barrat J-L (1999) Europhysics Letters 46: 637, with

permission from EDP Sciences.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

Figure 4 Fluctuation–dissipation plot in a three-dimensional

spin glass at low temperature. As predicted for type-II systems,

the FDR is an increasing function of the correlation, constant

only in the stationary part of the relaxation. Reproduced from

Marinari E et al. (1998) Violation of the fluctuation dissipation

theorem in finite dimensional spin glasses. Journal of Physics A:

Mathematical and General 31: 2611, with permission from

Institute of Physics Publishing Ltd.

Glassy Disordered Systems: Dynamical Evolution 559

Biroli G and Bouchaud J-P (2004) Diverging length scale and

upper critical dimension in the mode-coupling theory of the

glass transition. Europhysics Letters 67: 21.

Biroli G and Kurchan J (2001) Metastable states in glassy systems.

Physical Review E 64: 16101.

Bouchaud J-P, Cugliandolo L, Kurchan J, and Mezard M Out of

equilibrium dynamics of spin-glasses and other glassy systems.

In: Young AP (ed.) Spin Glasses and Random Fields.

Singapore: World Scientific.

Cugliandolo LF, Kurchan J, and Peliti L (1997) Energy flow,

partial equilibration and effective temperatures in systems

with slow dynamics. Physical Review E 55: 3898.

Debenedetti P (1996) Metastable Liquids, Concepts and Princi-

ples. New Jersey: Princeton University Press.

Fischer KH and Hertz JA (1991) Spin Glasses. Cambridge:

Cambridge University Press.

Franz S, Mezard M, Parisi G, and Peliti L (1999) The response of

glassy systems to random perturbations: a bridge between

equilibrium and off-equilibrium. Journal of Statistical Physics

97: 459.

Franz S and Parisi G (2000) On nonlinear susceptibility in

supercooled liquids. In: Franz S, Glotzer SC, and Sastry S

(eds.) Proceedings of the Conference ‘‘Unifying Concepts in

Glass Physics’’, Trieste, 5–18 September, 1999. Journal of

Physics: Condensed Matter 12: 6335.

Franz S and Virasoro M (2000) Quasi-equilibrium interpretation

of aging dynamics. Journal of Physics A 33: 891.

Goetze W (1991) Aspects of structural glass transitions. In:

Hansen JP, Levesque D, and Zinn-Justin J (eds.) Liquid,

Freezing and the Glass Transition. Amsterdam: Les Houches,

North-Holland.

Marinari E et al. (1998) Violation of the fluctuation dissipation

theorem in finite dimensional spin glasses. Journal of Physics

A: Mathematical and General 31: 2611.

Nieuwenhuizen ThM (2000) In: Franz S, Glotzer SC, and Sastry S

(eds.) Thermodynamic Picture of the Glassy State, Trieste,

5–18 September, 1999. Journal of Physics: Condensed Matter

12: 6335.

Norblad P and Svendlidh P (1997) Experiments in spin glasses. In:

Young AP (ed.) Spin Glasses and Random Fields. Singapore:

World Scientific.

Young AP (ed.) (1997) Spin Glasses and Random Fields.

Singapore: World Scientific.

Graded Poisson Algebras

A S Cattaneo, Universita

tZu

rich, Zu

rich, Switzerland

D Fiorenza and R Longoni, Universita

di Roma ‘‘La

Sapienza,’’ Rome, Italy

Definitions

Graded Vector Spaces

By a Z-graded vector space (or simply, graded

vector space) we mean a direct sum A = 

i 2Z

vector spaces over a field k of characteristic zero.

The A

are called the components of A of degree i

and the degree of a homogeneous element a 2A is

denoted by j a j. We also denote by A[n] the graded

vector space with degree shifted by n, namely

A[n] = 

i 2Z

(A[n])

with (A[n])

= A

iþn

. The tensor

product of two graded vector spaces A and B is

again a graded vector space whose degree r

component is given by (A B)

= 

pþq = r

B

The symmetric and exterior algebras of a graded

vector space A are defined, respectively, as S(A) =

T(A)=I

and

(A) = T(A)=I

,whereT(A) = 

n0

n

is the tensor algebra of A and I

(resp. I

)isthe

two-sided ideal generated by elements of the form

a b  (1)

jajjbj

b a (resp. a b þ (1)

jajjbj

b a),

with a and b homogeneous elements of A.Theimages

of A

n

in S(A)and

(A) are denoted by S

(A)and

(A), respectively. Notice that there is a canonical

decalage isomorphism S

(A[1]) ’

(A)[n].

Graded Algebras and Graded Lie Algebras

We say that A is a graded algebra (of degree zero) if

A is a graded vector space endowed with a degree

zero bilinear associative product: A A !A.A

graded algebra is graded commutative if the product

satisfies the condition

a b ¼ð1Þ

jajjbj

b a

for any two homogeneous elements a, b 2A of

degree jaj and jbj, respectivel y.

A graded Lie algebra of degree n is a graded

vector space A endowed with a graded Lie bracket

on A[n]. Such a bracket can be seen as a degree n

Lie bracket on A, that is, as a bilinear operation

{ , }:A A !A[n] satisfying graded antisym me-

try and graded Jacobi relations:

fa; bg ¼ ð1Þ

ðjajþnÞðjbjþnÞ

fb; ag

fa; fb; cgg ¼ ffa; bg; cgþð1Þ

ðjajþnÞðjbjþnÞ

fafb; cgg

Graded Poisson Algebras

We can now define the main objec t of interest of

this note:

Definition 1 A graded Poisson algebra of degree n,

or n-Poisson algebra, is a triple (A, , { , }) consisting

of a graded vector space A = 

i 2Z

endowed with

a degree zero graded commutative product an d with

560 Graded Poisson Algebras

a degree n Lie bracket. The bracket is required to

be a biderivation of the product, namely:

fa; b c g¼fa; bgc þð1Þ

jbjðjajþnÞ

b fa; cg

Notation. Graded Poisson algebras of degree zero

are called Poisson algebras, while for n = 1 one

speaks of Gerstenhaber (1963) algebras or of

Schouten algebras.

Sometimes a Z

-grading is used instead of a

Z-grading. In this case, one just speaks of even and

odd Poisson algebras.

Example 1 Any graded commutative algebra can

be seen as a Poisson algebra with the trivial Lie

structure, and any graded Lie algebra can be seen as

a Poisson algebra with the trivial product.

Example 2 The most classical example of a

Poisson algebra (already considered by Poisson

himself) is the algebra of smoot h functions on R

endowed with usual multiplication and with the

Poisson bracket {f , g} = @

i f @

g  @

i g@

f , where the

’s and the q

’s, for i = 1, ..., n, are coordinates on

. The bivector field @

i ^@

is induced by the

symplectic form ! = dp

^dq

. An immediate gener-

alization of this example is the algebra of smooth

functions on a symplectic manifold (R

, !) with the

Poisson bracket {f , g} = !

f @

g, where !

the bivector field defined by the inverse of the

symplectic form ! = !

^dx

; viz. !

= 

A further generalization is when the bracket on

) is defined by {f , g} = 

f @

g, with the

matrix function  not necessarily nondegenerate.

The bracket is Poisson if and only if  is skewsym-

metric and satisfies



þ 



þ 



¼0

An example of this, already considered by Lie

(1894), is 

(x) = f

, where the f

’s are the

structure constants of some Lie algebra.

Example 3 Example 2 can be generalized to any

symplectic manifold (M, !). To every function

h 2C

(M) one associates the Hamiltonian vector

field X

which is the unique vector field satisfying

! = dh. The Poisson bracket of two functions

f and g is then defined by

ff ; gg¼i

In local coordinates, the correspond ing Poisso n

bivector field is related to the symplectic form as in

Example 2.

A generalization is the algebra of smooth

functions on a manifold M with bracket

{f , g} = hjdf ^dgi, where  is a bivector field

(i.e., a section of

TM) such that {, }

= 0,

where { , }

is the Schouten–Nije nhuis bracket

(see the first subsection in the next section for

details, and Example 2 for the local coordinate

expression). Such a bivector field is called a Poisson

bivector field and the manifold M is called a Poisson

manifold. Observe that a Poisson algebra structure

on the algebra of smoot h functions on a smooth

manifold is necessarily defined this way. In the

symplectic case, the bivector field corresponding to

the Poisson bracket is the inverse of the symplectic

form (regarded as a bundle map TM !T



M).

The linear case described at the end of Example 2

corresponds to M = g



where g is a (finite-

dimensional) Lie algebra. The Lie bracket

g !g

is regarded as an element of g 



(



)

and reinterpreted as a Poisson bivector field on g.

The Poisson algebra structure restricted to polyno-

mial functions is described at the beginning of the

next section.

Batalin–Vilkovisky Algebras

When n is odd, a generator for the bracket of an

n-Poisson algebra A is a degree n linear map from

A to itself,

:A!A½n

such that

ða bÞ¼ðaÞb þð1Þ

jaj

a ðbÞþð1Þ

jaj

fa; bg

A generator  is called exact if and only if it satisfies

the condition 

= 0, and in this case  becom es a

derivation of the bracket:

ðfa; bgÞ ¼ fðaÞ; bgþð1Þ

jajþ1

fa; ðbÞg

Remark 1 Notice that not every odd Poisson

algebra A admits a genera tor. For inst ance, a

nontrivial odd Lie algebra seen as an odd Poisson

algebra with trivial multiplication admits no gen-

erator. Moreover, even if a generator  for an odd

Poisson algebra exists, it is far from being unique. In

fact, all different generators are obtained by adding

to  a derivation of A of degree n.

Definition 2 An n-Poisson algebra A is called an

n-Batalin–Vilkovisky algebra, if it is endowed with

an exact generator.

Notation.Whenn = 1 it is customary to speak

of Batalin–Vilkovisky algebras, or simply BV alge-

bras (see Batalin–Vilkovi sky Quantization; see also

Batalin and Vilkovisky (1963), Getzler (1994), and

Koszul (1985)).

Graded Poisson Algebras 561

There exists a characterization of n-Batalin–

Vilkovisky algebras in terms of only the product and

the generator (Getzler 1994, Koszul (1985). Suppose

in fact that a graded vector space A is endowed with a

degree zero graded commutative product and a linear

map  : A !A[n]suchthat

= 0, satisfying the

following ‘‘seven-term’’ relation:

ða b cÞþðaÞb c þð1Þ

jaj

a ðbÞc

þð1Þ

jajþjbj

a b ðcÞ

¼ ða bÞc þð1Þ

jaj

a ðb cÞ

þð1Þ

ðjajþ1Þjbj

b ða cÞ

In other words,  is a derivation of order 2.

Then, if we define the bilinear operation

{,}:A A !A[n]by

fa; bg¼ð1Þ

jaj



ða bÞðaÞb

ð1Þ

jaj

a ðbÞ



we have that the quadruple (A, ,{,},)isan

n-Batalin–Vilkovisky algebra. Conversely, one easily

checks that the product and the genera tor of an

n-Batalin–Vilkovisky algebra satisfy the above

‘‘seven-term’’ relation.

Examples

Schouten–Nijenhuis Bracket

Suppose g is a graded Lie algebra of degree zero.

Then A = S(g[n]) is a (n)-Poisson algebra with its

natural multiplication (the one induced from the

tensor algebra T(A)) and a degree n bracket

defined as follows (Koszul 1985 , Krasil’shchik

1988): the bracket on S

(g[n]) = g[n] is defined as

the suspension of the bracket on g, while on

(g[n]), for k > 1, the bracket, often called the

Schouten–Nijenhuis bracket, is defined inductively

by forcing the Leibniz rule

fa; b c g¼fa; bgc þð1Þ

jbjðjajþnÞ

b fa; cg

Moreover, when n is odd, there exists a generator

defined as

ða

a

a

i<j

ð1Þ



; a

ga



a

where a

, ..., a

2g and =ja

jþ(ja

jþ1)(ja

jþþ

i1

jþi1) þ(ja

jþ1)(ja

jþþ

jþþ ja

j1

jþ

j2). An easy check shows that 

=0, thus

S(g[n]) is an n-Batalin–Vilkovisky algebra for

every odd n2N. For n= 1 the -cohomology

g is the usual Cartan–Chevalley–Eilenber g

cohomology.

In particular, one can consider the Lie algebra

g = Der(B) = 

j 2Z

Der

(B) of derivations of a graded

commutative algebra B. More explicitly, Der

(B)con-

sists of linear maps : B !B of degree j such

that (ab) = (a)b þ (1)

j jaj

a(b) and the bracket

is {, } =    (1)

jjj j

 .Thespaceofmulti-

derivations S(Der(B)[1]), endowed with the Schouten–

Nijenhuis bracket, is a Gerstenhaber algebra.

We can further specialize to the case when B is the

algebra C

(M) of smooth functions on a smooth

manifold M; then X(M) = Der(C

(M)) is the space of

vector fields on M and V(M) = S (X(M)[1]) is the

space of multivector fields on M. It is a classical

result by Koszul (1985) that there is a bijective

correspondence between generators for V(M) and

connections on the highest exter ior power

dimM

TM of the tangent bundle of M. Moreover,

flat connections correspond to generators which

square to zero.

Lie Algebroids

A Lie algebroid E over a smooth manifold M is a

vector bundle E over M together with a Lie algebra

structure (over R) on the space (E) of smooth

sections of E, and a bundle map  : E !TM, called

the ancho r, extended to a map between sections of

these bundles, such that

fX; fYg¼ffX; YgþððXÞf ÞY

for any smooth sections X and Y of E and any

smooth function f on M. In particular, the anchor

map induces a morphism of Lie algebras





: (E) !X(M), namely 



({X, Y}) = {



(X), 



(Y)}.

The link between Lie algebroids and Gerstenhaber

algebras is given by the following Proposition

(Kosmann-Schwarzbach and Monterde 2002, Xu

1999):

Proposition 1 Given a vector bundle E over M,

there exists a one -to-one correspondence between

Gerstenhaber algebra structures on A =(

(E))

and Lie algebroid structures on E.

The key of the proposition is that one can extend

the Lie algebroid bracket to a unique graded

antisymmetric bracket on (

(E)) such that

{X, f } = (X)f for X 2(

(E)) and f 2(

(E)),

and that for Q 2(

qþ1

(E)), {Q, } is a derivation

of (

(E)) of degree q.

Example 4 A finite-dimensional Lie algebra g can

be seen as a Lie algebroid over a trivial base

manifold. The corresponding Gerstenhaber algebra

is the one of last subsection.

562 Graded Poisson Algebras

Example 5 The tangent bundle TM of a smooth

manifold M is a Lie algebroid with ancho r map

given by the identity and algebroid Lie bracket given

by the usual Lie bracket on vector fields. In this

case, we recover the Gerstenhaber algebra of multi-

vector fields on M described in the last subsection.

Example 6 If M is a Poisson manifold with Poisson

bivector field , then the cotangent bundle T



inherits a natural Lie algebroid structure where the

anchor map 

: T



M !T

M at the point p 2M is

given by 

()() = (, ), with ,  2T



M, and the

Lie bracket of the 1-forms !

and !

is given by

g¼L



ð!

 L



ð!

 dð!

The associated Gerstenhaber algebra is the de Rham

algebra of differential forms endowed with the

bracket defined by Koszul (1985) . As shown in

Kosmann-Schwarzbach (1995),  (



M)) is indeed

a BV algebra with an exact generator =[d, i



] given

by the commutator of the contraction i



with the

Poisson bivector  and the de Rham differential d.

Similar results hold if M is a Jacobi manifold.

It is natural to ask what additional structure on a

Lie algebroid E makes the Gerstenhaber algebra

(

(E)) into a BV algebra. The answer is given by

the following result, which is proved in Xu (1999).

Proposition 2 Given a Lie algebroid E, there is a

one-to-one correspondence between generators for the

Gerstenhaber algebra (

(E)) and E-connections on

rk E

E (where rk E denotes the rank of the vector

bundle E). Exact generators correspond to flat E-

connections , and in particular, since flat E-connections

always exist, (

(E)) is always a BV algebra.

Lie Algebroid Cohomology

A Lie algebroid structure on E !M defines a

differential  on (



)by

f :¼ 



df ; f 2C

ðMÞ¼ð



and

h;X ^Yi:¼hh; Xi; Yihh; Yi; Xi

h; fX; YgiX; Y 2ðEÞ;2ðE



where 



: 

(M) !(E



) is the transpose of





: (E) !X(M) and h, i is the canonical pairing of

sections of E



and E.On(



), with n  2, the

differential  is defined by forcing the Leibniz rule.

In Example 4 we get the Cartan–Chevalley–

Eilenberg differential on



;inExample 5 we

recover the de Rham differential on 



(M) =

(



M), while in Example 6 the differential on

V(M) =(

TM)is{,}

Lie–Rinehart Algebras

The algebraic generalization of a Lie algebroid is a

Lie–Rinehart algebra. Recall that given a commu-

tative associative algebra B (over some ring R) and a

B-module g, then a Lie–Rinehart algebra structure

on (B, g) is a Lie algebra structure (over R)ong and

an action of g on the left on B by derivations,

satisfying the following compatibility conditions:

f; ag¼ðaÞ þ af; g

ðaÞðbÞ¼aððbÞÞ

for every a, b 2B and ,  2g.

The Lie–Rinehart structures on the pair (B, g)

bijectively correspond to the Gerstenhaber algebra

structures on the exterior algebra

(g)ofg in

the category of B-modules. When g is of finite rank

over B, generators for these structures are in turn

in bijective correspondence with (B, g)-connections on

g, and flat connections correspond to exact

generators. For additional discussions, see Gerstenhaber

and Schack (1992) and Huebschmann (1998).

Lie algebroids are Lie–Ri nehart algebras in the

smooth setting. Namely, if E !M is a Lie algebroid,

then the pair (C

(M), (E)) is a Lie–Rinehart

algebra (with action induced by the anchor and the

given Lie bracket).

Lie–Rinehart Cohomology

Lie algebroid cohomology may be generalized to

every Lie–Rinehart algebra (B, g). Namely, on the

complex Alt

(g, B) of alternating multilinear func-

tions on g with values in B, one can define a

differential  by the rules

ha;i¼ðaÞ; a 2B ¼ Alt

ðg; BÞ;2g

ha;^i¼hha;i;ihha;i;iha; f; gi

;  2g; a 2Alt

ðg; BÞ

and forcing the Leibniz rule on elements of

Alt

(g, B), n  2.

Hochschild Cohomology

Let A be an associative algebra with product , and

consider the Hochschild cochain complex

Hoch(A) =

n0

Hom(A

n

, A)[n þ 1]. There are

two basic operations between two elements

f 2Hom(A

k

, A)[k þ 1] and g 2Hom (A

l

, A)[l þ

1], namely a degree zero product

f [ gða

a

kþl

¼ð1Þ

f ða

a

Þgða

lþ1

a

kþl

Graded Poisson Algebras 563

and a degree 1 bracket {f , g} = f  g  (1)

(k1)(l1)

g  f , where

f  g ða

a

kþl1

k1

i¼1

ð1Þ

iðl1Þ

f ða

a

gða

iþ1

a

Þa

kþl1

It is well known from Gerstenhaber (1963) that the

cohomology HHoch(A) of the Hochschild complex

with respect to the differential d

Hoch

= {, } has the

structure of a Gerstenhaber algebra. More generally,

there is a Gerstenhaber algebra structure on Hochs-

child cohomology of differential graded associative

algebras (Loday 1998).

Graded Symplectic Manifolds

The construction of Example 3 can be extended

to graded symplectic manifolds (see Supermanifolds;

see also Alexandrov et al. (1997), Getzler (1994),

and Schwarz (1993)). Recall that a symplectic

structure of degree n on a graded manifold N is a

closed nondegenerate 2-form ! such that L

! = n!

where L

is the Lie derivative with respect to the

Euler field of N (see Roytenberg (2002) for details).

Let us denote by X

the vector field associated to the

function h 2C

(N) by the formula i

! = dh. Then

the bracket

ff ; gg¼i

gives C

(N) the structure of a graded Poisson

algebra of degree n.

If the symplectic form has odd degree and the

graded manifold has a volume form, then it is

possible to construct an exact generator defined by

ðf Þ¼

divðX

where div is the divergence operator associated to

the given volume form (Getzler 1994, Kosmann-

Schwazbach and Monterde 2002).

An explicit characterization of graded symplectic

manifolds has been given in Roytenberg (2002).In

particular, it is proved there that every symplectic

form of degree n with n  1 is necessarily exact.

More precisely, one has ! = d(i

!=n).

Shifted Cotangent Bundles

The main examples of graded symplectic manifolds

are given by shifted cotangent bundles. If N is a

graded manifold then the shifted cotangent bundle



[n]N is the graded manifold obtained by shifting

by n the degrees of the fibers of the cotangent

bundle of N. This graded mani fold possesses a

nondegenerate closed 2-form of degree n, which can

be expressed in local coordinates as

! ¼

^dx

where {x

} are local coordinates on N and {x

} are

coordinate functions on the fibers of T



[n]N.In

local coordinates, the bracket between two homo-

geneous functions f and g is given by

ff ; gg¼ð1Þ

jjf j

ð1Þ

ðjf jþnÞðjgjþnÞþjx

jjgj

If in addition the graded manifold N is orientable,

then T



[n]N has a volume form too; when n is odd,

the exact generator (f ) = (1=2)divX

is written in

local coordinates as

 ¼

In the case n = 1, we have a natural identification

between functions on T



[1]N and multivector fields

V ( N )on N , and we recover again the Gerste nhaber

algeb ra of the subsec tion ‘‘Schou ten–Nijenhu is

bracket .’’ Moreove r, it is easy to see that, under

the above identification,  applied to a vector field

of N is the usual divergence operator.

Examples from Algebraic Topology

For any n > 1, the homology of the n-fold loop

space 

(M) of a topological space M has the

structure of an (n  1)-Poisson algebra (May 1972).

In particular, the homology of the double loop space



(M) is a Gerstenha ber algebra, and has an exact

generator defined using the natural circle action on

this space (Getzler 1994). The homology of the free

loop space L(M) of a closed oriented manifold M is

also a BV algebra when endowed with the ‘‘Chas–

Sullivan intersection product’’ and with a generator

defined again using the natural circle action on the

free loop space (Cohen and Jones 2002).

Applications

BRST Quantization in the Hamiltonian Formalism

The BRST procedure is a method for quantizing

classical mechanical systems or classical field the-

ories in the prese nce of symmetries (see BRST

Quantization). The starting point is a symplectic

manifold M (the ‘‘phase space’’), a function H (the

‘‘Hamiltonian’’ of the system) governing the evolu-

tion of the system, and the ‘‘constraints’’ given by

564 Graded Poisson Algebras

several functions g

which commute with H and

among each other up to a C

(M)-linear combina-

tion of the g

’s.

Then the dynamics is constrained on the locus V of

common zeros of the g

’s. When V is a submanifold,

the g

’s are a set of generators for the ideal I of

functions vanishing on V. Observe that I is closed

under the Poisson bracket. Functions in I are called

‘‘first class constraints.’’ The Hamiltonian vector fields

of first-class constraints, which by construction tan-

gential to V, are the ‘‘symmetries’’ of the system.

When V is smooth, then it is a coisotropic

submanifold of M and the Hamiltonian vector fields

determined by the constraints give a foliation F of

V. In the nicest case V is a principal bundle with F

its vertical foliation and the algebra of functions

(V=F) on the ‘‘reduced phase space’’ (see Poisson

Reduction, and Symmetry and Symplectic Reduction)

V=F is identified with the I-invariant subalgebra of

(M)=I.

From a physical point of view, the points of V=F are

the interesting states at a classical level, and a

quantization of this system means a quantization of

(V=F). The BRST procedure gives a method of

quantizing C

(V=F) starting from the (known) quan-

tization of C

(M). Notice that these notions immedi-

ately generalize to graded symplectic manifolds.

From an algebraic point of view, one starts with a

graded Poisson algebra P and a multiplicative ideal I

which is closed unde r the Poisson bracket. The

algebra of functions on the ‘‘reduced phase space’’ is

replaced by (P=I)

, the I-invariant subalgebra of P=I.

This subalgebra inherits a Poisson bracket even if

P=I does not. Moreover, the pair (B, g) = (P=I , I=I

)

inherits a graded Lie–Rinehart structure . The ‘‘Rine-

hart complex’’ Alt

P=I

(I=I

, P=I) of alternating multi-

linear functions on I=I

with values in P=I, endowed

with the differential described in the subsection

‘‘Lie-R inehart cohomol ogy,’’ plays the role of the de

Rham complex of vertical forms on V with respect

to the foliation F determined by the constraints.

In case V is a smooth submanifold, we also have

the following geometric interpretation: let N



denote the conorm al bundle of V (i.e., the annihi-

lator of TV in T

P). This is a Lie subalgebroid of



P if and only if V is coisotropic. Since we may

identify I=I

with sections of N



C (by the de Rham

differential), (P=I, I=I

) is the corresponding Lie–

Rinehart pair. The Rinehart complex is then the

corre sponding Lie a lgebroid complex (

( N



V )



)

with differenti al descr ibed in the subsecti on ‘‘Lie

algeb roid coh omolog y.’’ The image of the anchor

map N



V !TV is the distribution determining F,

so by duality we get an injective chain map from the

vertical de Rham complex to the Rinehart complex.

The main point of the BRST procedure is to define

a chain complex C



(



 ) P , where  is a

graded vector space, with a cobou ndary operator D

(the ‘‘BRST operator’’), and a quasi-isomorphism

(i.e., a chain map that induces an isomorphism in

cohomology)

 : ðC



; DÞ! Alt

P=I

ðI

=I; P=IÞ; d



This means in particular that the zeroth cohomology

of functions on the

‘‘reduced phase space.’’ Observe that there is a

natural symmetric inner product on 



  given by

the evaluation of 



on . This inner product, as an

element of S

(  



) ’S

()  ( 



)  S

(



is concentrated in the component  



, and so

it defines an element in

([1]  



[1]) ’

()[2]  ( 



)  S

(



)[2], that is, a degree

zero bivector field on [1]  



[1]. It is easy to see

that this bivector field induces a degree zero Poisson

structure on S(



[1]  [1]). From another view-

point this is the Poisson structure corresponding to

the canonical symplectic structure on T



[1].

Finally, we have that S(



[1]  [1]) P is a

degree zero Poisson algebra. Note that the super-

algebra underlying the graded algebra S(



[1] 

[1]) P is canonically isomorphic to the complex



(



 ) P. When P = C

(M), we can

think of S(



[1]  [1]) C

(M) as the algebra

of functions on the graded symplectic manifold

N = ([1]  



[1])  M (the ‘‘extended phase

space’’). In physical language, coordinate functions

on [1] are called ‘‘ghost fields’’ while coordinate

functions on 



[1] are called ‘‘ghost momenta’’ or,

by some authors, ‘‘antighost fields’’ (not to be

confused with the antighosts of the Lagrangian

functional-integral approach to quantization).

Suppose now that there exists an element

 2S(



[1]  [1]) P such that {, } = D, that

one can extend the ‘‘known’’ quantization of P to a

quantization of S(



[1]  [1]) P as operators

on some (graded) Hilbert space T and that the

operator Q which quantizes  has square zero.

Then one can consider the ‘‘true space of physical

states’’ H

(T ) on which the ad

-cohomology of

operators will act. This provides one with a

quantization of (P=I)

For further details on this procedure, and in

particular for the construction of D,wereferto

Henneaux and Teitelboim (1992), Kostant and

Sternberg (1987),andStasheff (1997), and references

therein. Observe that some authors refer to this

method as BVF (Batalin–Vilkovisky–Fradkin) and

reserve the name BRST for the case when the g

’s are

the components of an equivariant moment map.

Graded Poisson Algebras 565