Linear Odd Poisson Bracket
on Grassmann Algebra
Vyacheslav A. Soroka"
Kharkov Institute of Physics and Technology
310108 Kharkov, Ukraine
Abstract
A linear odd Poisson bracket realized solely in terms of Grassmann
variables is suggested. It is revealed that with the bracket, correspond-
ing to a semi-simple Lie group, both a Grassmann-odd Casimir function
and invariant (with respect to this group) nilpotent differential opera-
tors of the first, second and third orders are naturally related and enter
into a finite-dimensional Lie superalgebra.
1. Recently a linear degenerate odd Poisson bracket built only of Grassmann
variables has been introduced [1]. It was constructed for this bracket, in contrast
with the non-degenerate odd bracket having the only Grassmann-odd nilpotent dif-
ferential "-operator of the second order, at once three Grassmann-odd nilpotent
"-like differential operators of the first, the second and the third orders with re-
spect to Grassmann derivatives. It was also shown that these "-like operators
together with a Grassmann-odd nilpotent Casimir function of this degenerate odd
bracket form a finite-dimensional Lie superalgebra. Following to [2], in the present
report we extend the above-mentioned results to the case of an arbitrary linear odd
Poisson bracket, which is also realized solely in terms of the Grassmann variables
and corresponds to a semi-simple Lie group.
2. There is a well-known linear even Poisson bracket given in terms of the
commuting (Grassmann-even) variables Xą (g(Xą) = 0)
{Xą, X}0 = cą łXł , (ą, , ł = 1, ..., N), (1)
where cą ł are Grassmann-even (g(cą ł) = 0) constants which, because of the main
properties of the even Poisson bracket:
{A, B + C}0 = {A, B}0 + {A, C}0 , (2)
g({A, B}0) = g(A) + g(B) (mod 2) , (3)
"
E-mail: vsoroka@kipt.kharkov.ua
1
arXiv:math-ph/0002031 v1 10 Feb 2000
{A, B}0 = -(-1)g(A)g(B){B, A}0 , (4)
(-1)g(A)g(C){A, {B, C}0}0 = 0 , (5)
(ABC)
{A, BC}0 = {A, B}0 C + (-1)g(A)g(B) B{A, C}0 , (6)
are antisymmetric in the two lower indices
cą ł = -cął (7)
and obey the conditions
cącł + ccłą + cłcą = 0 . (8)
A sum with the symbol (ABC) in (5) means a summation over cyclic permutations of
the quantities A, B, C. In relations (2)-(6) A, B, C are functions of the variables Xą
and g(A) is a Grassmann parity of the quantity A. The linear even bracket (1) plays
a very important role in the theory of Lie groups, Lie algebras, their representations
and applications (see, for example, [3, 4]). The bracket (1) can be realized in a
canonical even Poisson bracket
N
! !
{A, B}0 = A B ,
" " - " "
qą pą pą qą
ą=1
!
"
where and are the right and left derivatives and "xA a" , on the following
" "
"xA
bilinear functions of coordinates qą and momenta pą
Xą = cą łqpł
if cą ł satisfy the conditions (7), (8) for the structure constants of a Lie group.
As in the Lie algebra case, we can define a symmetric Cartan-Killing tensor
gą = gą = cął cł (9)
and verify with the use of relations (8) an anti-symmetry property of a tensor
cął = cą gł = -cął . (10)
By assuming that the Cartan-Killing metric tensor is non-degenerate det(gą) = 0
(this case corresponds to the semi-simple Lie group), we can define an inverse tensor
gą
ą
gągł = ł , (11)
with the help of which we are able to build a quantity
C = XąXgą ,
2
that, in consequence of relation (10), is for the bracket (1) a Casimir function which
annihilates the bracket (1) and is an invariant of the Lie group with the structure
constants cął and the generators Tą
{Xą, C}0 = cą łXł"XC = TąC = 0 .
3. Now let us replace in expression (1) the commuting variables Xą by Grass-
mann variables Śą (g(Śą) = 1). Then we obtain a binary composition
{Śą, Ś}1 = cą łŚł , (12)
which, due to relations (7) and (8), meets all the properties of the odd Poisson
brackets:
{A, B + C}1 = {A, B}1 + {A, C}1 , (13)
g({A, B}1) = g(A) + g(B) + 1 (mod 2) , (14)
{A, B}1 = -(-1)(g(A)+1)(g(B)+1){B, A}1 , (15)
(-1)(g(A)+1)(g(C)+1){A, {B, C}1}1 = 0 , (16)
(ABC)
{A, BC}1 = {A, B}1 C + (-1)(g(A)+1)g(B) B{A, C}1 . (17)
It is surprising enough that the odd bracket can be defined solely in terms of the
Grassmann variables as well as an even Martin bracket [5]. On the following bilinear
functions of canonical variables commuting qą and Grassmann ą
Śą = cą łqł
a canonical odd Poisson bracket
N
! !
{A, B}1 = A B
" " - " "
qą ą ą qą
ą=1
is reduced to the bracket (12) providing that cą ł obey the conditions (7), (8).
On functions A, B of Grassmann variables Śą the bracket (12) has the form
!
{A, B}1 = A cą łŚł Ś B ,
" "
Śą
The bracket (12) can be either degenerate or non-degenerate in the dependence on
whether the matrix cą łŚł in the indices ą, is degenerate or not. Raising and
lowering of the indices ą, , the non-degenerate metric tensors (9), (11) relate with
each other the adjoint and co-adjoint representations which are equivalent for a
semi-simple Lie group
Śą = gąŚ , "Śą = gą"Ś .
3
Hereafter only the non-degenerate metric tensors (11) will be considered.
4. By contracting the indices in a product of the Grassmann variables with the
upper indices and of the successive Grassmann derivatives, respectively, with the
lower indices in (8), we obtain the relations
ŚąŚ(cą cł + 2cłąc) = 0 , ŚąŚŚłcą cł = 0 , (18a, b)
(cą cł + 2cłąc )"Śą"Ś = 0 , cącł "Śą"Ś "Śł = 0 , (19a, b)
which will be used later on many times. In particular, taking into account relation
(18b), we can verify that the linear odd bracket (12) has the following Grassmann-
odd nilpotent Casimir function
1
"
"+3 = ŚąŚŚłcął , ("+3)2 = 0 , (20)
3!
which is an invariant of the Lie group
{Śą, "+3}1 = Śłcą ł"Ś "+3 = Są"+3 = 0 (21)
with the generators Są obeying the Lie algebra permutation relations1
[Są, S] = cą łSł . (22)
It is a well-known fact that, in contrast with the even Poisson bracket, the
non-degenerate odd Poisson bracket has one Grassmann-odd nilpotent differential
"-operator of the second order, in terms of which the main equation has been
formulated in the Batalin-Vilkovisky scheme [6, 7, 8, 9, 10, 11] for the quantization
of gauge theories in the Lagrangian approach. In a formulation of Hamiltonian
dynamics by means of the odd Poisson bracket with the help of a Grassmann-odd
Ż Ż
Hamiltonian H (g(H) = 1) [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] this "-
operator plays also a very important role being used to distinguish the Hamiltonian
Ż
dynamical systems, for which the Liouville theorem is valid "H = 0, from those
Ż
ones, for which this theorem takes no place "H = 0 [1]2.
Now let us try to build the "-operator for the linear odd bracket (12). It is
remarkable that, in contrast with the canonical odd Poisson bracket having the
only "-operator of the second order, we are able to construct at once three "-like
Grassmann-odd nilpotent operators which are differential operators of the first, the
second and the third orders respectively
1
" ł
"+1 = ŚąŚcą ł"Ś , ("+1)2 = 0 ; (23)
2
1
Note that below [A, B] = AB - BA and {A, B} = AB + BA.
2
Note also applications of the odd bracket to the integrability problem [24, 25, 26]
4
1
"
"-1 = Śłcął"Śą"Ś , ("-1)2 = 0 ; (24)
2
1
"
"-3 = cął"Śą"Ś "Śł , ("-3)2 = 0 . (25)
3!
The nilpotency of the operators "+1 and "-1 is a consequence of relations (18b)
and (19b). The operator "+1 is proportional to the second term in a BRST charge
1
ł
Q = ŚąGą - ŚąŚcął"Ś ,
2
where Śą and "Śą represent the operators for ghosts and antighosts respectively. Q
itself will be proportional to the operator "+1 if we take the representation Są (21)
for group generators Gą. The operator "-1, related with the divergence of a vector
field {Śą, A}1
"
"Śą{Śą, A}1 = "ŚąSąA = - 2"-1A ,
is proportional to the true "-operator for the bracket (12).
It is also interesting to reveal that these "-like operators together with the
Casimir function "+3 (20) are closed into the finite-dimensional Lie superalgebra, in
which the anticommuting relations between the quantities " ( = -3, -1, +1, +3)
(20), (23)-(25) with the nonzero right-hand side are
{"-1, "+1} = Z , (26)
{"-3, "+3} = N - 3Z , (27)
where
N = -cąłcął
is a number of values for the indices ą, , ł (ą, , ł = 1, ..., N) and
Z = D - K (28)
is a central element of this superalgebra
[Z, "] = 0 , ( = -3, -1, +1, +3) . (29)
In (28)
D = Śą"Śą
(30)
is a dilatation operator for the Grassmann variables Śą, which distinguishes the
"-operators with respect to their uniformity degrees in Ś
[D, "] = " , ( = -3, -1, +1, +3) (31)
5
and is in fact a representation for a ghost number operator, and the quantity K has
the form
1
K = ŚąŚcą cł"Śł "Ś . (32)
2
The operator Z is also a central element of the Lie superalgebra which contains both
the operators " (20), (23)-(25), Z (28) and the operator D (30)
[Z, D] = 0 . (33)
We can add to this superalgebra the generators Są (21) with the following com-
mutation relations:
[Są, "] = 0 , [Są, Z] = 0 , [Są, D] = 0 , (34)
which indicate that both the Casimir function "+3 and the operators " ( =
-3, -1, +1), Z and D are invariants of the Lie group with the generators Są. In
order to prove the permutation relations for the Lie superalgebra (20)-(34), we have
to use relations (18) and (19). Note that the central element Z (28) coincides
with the expression for a quadratic Casimir operator of the Lie algebra (22) for the
generators Są given in the representation (21)
SąSgą = Z . (35)
5. Thus, we see that both the even and odd linear Poisson brackets are internally
inherent in the Lie group with the structure constants subjected to conditions (7)
and (8). However, only for the linear odd Poisson bracket realized in terms of the
Grassmann variables and only in the case when this bracket corresponds to the semi-
simple Lie group, there exists the Lie superalgebra (20)-(34) for the "-like operators
of this bracket.
Note that in the case of the degenerate Cartan-Killing metric tensor (9), relation
(10) remains valid and we can construct only two "-like Grassmann-odd nilpotent
operators: "-1 (24) and "-3 (25), which satisfy the trivial anticommuting relation
{"-1, "-3} = 0 .
Note also that anticommuting relations for the operators
i
i i
1
"
= Śł cą ł "Śą"Ś , ( )2 = 0 ,
"-1
"-1
2
i
corresponding to the Lie algebras with structure constants cął (i = 1, 2), vanish
i
provided that cą ł satisfy compatibility conditions [27]
{i k}
cą cł = 0 ,
(ął)
6
where {ik} denotes the symmetrization of the indices i and k.
The Lie superalgebra (20)-(34), naturally connected with the linear odd Poisson
bracket (12), may be useful for the subsequent development of the Batalin-Vilkovisky
formalism for the quantization of gauge theories. Indeed, very similar to (12) odd
Poisson brackets on the Grassmann algebra are used in a generalization [28] of
the triplectic formalism [29] which is a covariant version of the Sp(2)-symmetric
quantization [8] of general gauge theories. We should therefore expect that the Lie
superalgebra (20)-(34), closely related with the linear odd bracket (12), will also find
the application for the further development of the above-mentioned generalization
of the triplectic formalism. Let us note that the superalgebra (20)-(34) can also be
used in the theory of representations of the semi-simple Lie groups.
The author is sincerely thankful to V.D. Gershun, D.A. Leites and S.L. Lyakhovich
for useful discussions and is indebted to J.D. Stasheff for illuminating remarks. The
author wishes to thank J. Wess for kind hospitality at the University of Munich
where this work was completed.
This work was supported in part by the Ukrainian State Foundation of Funda-
mental Researches, Grant No 2.5.1/54 and by Grant INTAS No 93-127 (Extension).
References
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7
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