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ACAT 2002 Computation of Cohomology of Lie (Sup er)Algebra: Algorithms, Implementation and New Results
Vladimir V. Kornyak LIT, JINR, Dubna kornyak@jinr.ru June 25, 2002

Kornyak V. V. Computation of Cohomology of Lie (Super)Algebra

1

Homology and cohomology
Chain and cochain complexes: 0 C0 0 · · · - Ck - 0 C 0 - · · · - C
trivial cycle (b oundary)
d0 dk
-2

k

-2

k -1

- Ck k Ck -
-1

-1

+1

- · · ·
+1

k+1

k k-1 d

- C k - C
M

dk

k k+1 d

- · · · . Stokes' theorem:
- - F · d Stokes; s

Duality d
b a S V

d =

M

f (x)dx = F (b) - F (a) NewtonLeibniz; ( ( - в F ) · d- = a - · F ) · dV =
S V

!

chain

s d

'
p eculiarity

- - F · d a GaussOstrogradski.

c r
nontrivial cycle

Cocycles Coboundaries Cohomology

Z k = Ker dk = {C k | dC k = 0}. B k = Im dk-1 = {C k | C k = dC H k = Z k /B
k

k-1

}.

Kornyak V. V. Computation of Cohomology of Lie (Super)Algebra

2

Cohomology of Lie (sup er)algebra A in the mo dule X
C k = C k (A; X ) is a space of super skew-symmetric k -linear mappings A в · · · в A X, (C 0 = X by definition). Differential: (dk c)(a0 , . . . , ak ) =-
0i
(-1)s (-1)s
0ik

(ai )+s(aj )+p(ai )p(aj )

c([ai , aj ], ao , . . . , ai , . . . , aj , . . . , ak )

-

(ai )

ai c(ao , . . . , ai , . . . , ak ). p(ai ) Z 2 parity of ai ; Z i, p(ai ) = 0 (ai a0 , . . . , ai
-1

c(. . .) C k ; s(ai ) = nu

ai A;

is even is odd

)

mber of even elements in

, p(ai ) = 1 (ai

)

Kornyak V. V. Computation of Cohomology of Lie (Super)Algebra

3

Some interpretations in low dimensions (degrees) k
Trivial module H 1 (A) (A/[A, A]) describes "deviation of A from simplicity" H 2 (A) describes nontrivial central extensions of A Adjoint module H 1 (A; A) DerA/adA describes "external derivations of A", i.e., quotient space of all derivations w.r.t. internal derivations H 2 (A; A) describes infinitesimal deformations of A

Kornyak V. V. Computation of Cohomology of Lie (Super)Algebra

4

Main cause of computational difficulties
Basis of C k for Lie superalgebra C (ei1 , . . . , eik ; a ) C (ei1 ) · · · C (eik ) a ei1 · · · eik a . For n-dimensional ordinary Lie algebra acting in p-dimensional module dim C k = p for (n|m)-dimensional Lie superalgebra
k

n , k

dim C k = p
i=0

n k-i

m+i-1 i

n p +p k

k

i=1

n k-i

m+i-1 . i

Kornyak V. V. Computation of Cohomology of Lie (Super)Algebra

5

Scheme of splitting algorithm
C
k k-1 d -1 k k-1 d -1

- C - C =
g G

k

dk

k+1

C

- C - C

k

dk

k+1

k Cg -1

dk g

-1

-

k Cg

k - Cg

dk g

+1

G Z integer grading Z
k Cg -1 dk g
-1

-

k Cg

-

dk g

k Cg +1

=
sS

k- Cg,s 1

k- dg,s1

-

k Cg,s

k+1 - Cg,s

dk,s g

S finite or infinite set of subcomplexes
i Cg = sS i Cg ,s

di = g
sS

d

i g ,s

di - block matrices g
k g ,s

H

k g ,s

= Ker d

/Im d

k-1 g ,s

.

Kornyak V. V. Computation of Cohomology of Lie (Super)Algebra

6

Lie sup eralgebras with antibrackets (o dd Poisson structure)
Odd symplectic structure: x1 , . . . , xn ; 1 , . . . , n even and odd (grassmann) variables; n i i=1 dx di invariant 2-form; f (x1 , . . . , n ), g (x1 , . . . , n ) generating functions (hamiltonians). {f , g }
Bb

=

n i=1

f g xi i

+ (-1)p(

f ) f g i xi

Buttin brackets (antibrackets, odd Poisson brackets).

Buttin algebra B(n). =
n 2 i=1 xi i

operator of master equation in BatalinVilkovisky method.

Special Buttin algebra SB(n) satisfies to the divergence free condition f = 0. Special Leites algebra is the qoutient of SB(n) w.r.t. center Z , i. e. SLe(n) = SB(n)/Z. The case of even dimension n = 2m is more interesting for physics.

Kornyak V. V. Computation of Cohomology of Lie (Super)Algebra

7

Lie sup eralgebra SLe(2)
Variables: even x, y , gr(x) = gr(y ) = 1; odd , , gr() = gr( ) = -1. Basis elements and nonzero commutators:
| - 2| | - 1| | - 1| |0 | |0 | |0 | | | | | | | 1 1 1 1 1 1 | | | | | | O1 E2 E3 E4 E5 E6 O7 O8 E9 E10 E11 E12 = = = = = = = = = = = = . . . y y - x x y x y2 y 2 - 2xy xy - 1 x2 2 x2 (1 (2 (3 (4 (5 (6 (7 ) ) ) ) ) ) ) [E2 [E2 [E3 [E3 [E5 [E6 [E5 [ [ [ [ [ [ O1 E5 E6 O1 E4 E5 , , , , , , , , , , , , , E5 E6 E4 E5 E4 E4 E6 O7 O7 O7 O8 O8 O8 ] ] ] ] ] ] ] ] ] ] ] ] ] = = = = = = = = = = = = = . . . E2 -E3 -E2 -E3 -2E4 E5 2E6 -E2 -O7 -O8 E3 -O7 O8

(8) (9) (10) (11) (12) (13)

|2 |

Important subalgebras: A0 = span{O1 , E2 , . . . , E6 } non-positive subalgebra, A<0 = span{O1 , E2 , E3 } negative subalgebra, A0 = span{E4 , E5 , E6 } so(3) sl(2) sp(2) zero subalgebra, A0 = A<0 + A0 semidirect sum of commutative ideal and simple algebra.

Kornyak V. V. Computation of Cohomology of Lie (Super)Algebra

8

k Computing Hg (SLe(2)) (k, g ) [1, . . . , 10] [-2k, . . . , -2k + 16] by the C program LieCohomology
k\g+2k 1 0 1 1a 1 1 1 a2 1 1 1 a3 1 1 4 1 a4 1 1 1 a5 1 1 6 1 a6 1 1 1 a7 1 1 8 1 a8 1 1 1 a9 1 1 10 1 a10 1 12 23 23 11 3 6 6 1 4 8 8 1 23 13 b 4 26 13 4 26 13 4 26 13 4 26 13 4 26 13 4 26 13 4 26 13 4 26 13 4 5 10 10 1 44 16 6 6 12 12 1 73 22 8 7 14 14 1 8 16 16 1 9 18 18 1 10 20 20 1 333 38 23 1182 54 80 1780 65 124 1867 65 c 124 1867 65 124 1867 65 124 1867 65 124 1867 65 124 1867 65 124 11 22 22 1 444 42 28 12 24 24 1 575 46 34 13 26 26 1 732 50 40 4224 72 241 9398 92 489 14 28 28 1 913 54 47 6082 78 330 15 30 30 1 1124 58 54 8552 84 434 16 32 32 1 1363 62 62 11766 90 570 37649 116 1776 61884 140 2556 70110 153 d 2802 70817 153 ad 2802 70817 153 2802 70817 153 2802 70817 153 2802

2

2 4 12 24 8 11 3 2 4 12 24 8 11 3 2 4 12 24 8 11 3 2 4 12 24 8 11 3 2 4 12 24 8 11 3 2 4 12 24 8 11 3 2 4 12 24 8 11 3 2 4 12 24 8 11 3 2 4 12 24 8 11 3

116 171 244 26 30 34 10 14 18

3

56 118 226 414 718 18 26 34 39 48 7 15 23 36 52 56 121 246 491 952 18 26 34 42 52 7 15 23 41 71 56 121 246 492 970 18 26 34 42 52 7 15 23 41 71 56 121 246 492 970 18 26 34 42 52 7 15 23 41 71 56 121 246 492 970 18 26 34 42 52 7 15 23 41 71 56 121 246 492 970 18 26 34 42 52 7 15 23 41 71 56 121 246 492 970 18 26 34 42 52 7 15 23 41 71 56 121 246 492 970 18 26 34 42 52 7 15 23 41 71

1870 2858 60 66 119 176 3204 5584 72 84 197 311

15343 24348 100 108 787 1187

5

3528 6546 11878 21073 36540 76 88 104 113 130 197 358 606 1009 1578 3534 6605 12162 22102 39652 76 88 104 118 134 197 358 606 1009 1598 3534 6605 12162 22119 39796 76 88 104 118 134 197 358 606 1009 1598 3534 6605 12162 22119 39796 76 88 104 118 134 197 358 606 1009 1598 3534 6605 12162 22119 39796 76 88 104 118 134 197 358 606 1009 1598 3534 6605 12162 22119 39796 76 88 104 118 134 197 358 606 1009 1598

7

9

a = c( ), dim C
5 4,s

b = c(x, ) = c(y , );
6 4,s

ab = 0,
7 4,s

ac = 0,
6 4,s

a2 d = 0.
6 4,s 6 = 285 dim H4,s = 1

= 387 dim C

= 912 dim C

= 1847 dim Z

= 286 dim B

For (k,g) = (6,4) = (6, -2*6+16) time = 1 h 32 min 20.07 sec on PC Pentium 3, 667MHz

Kornyak V. V. Computation of Cohomology of Lie (Super)Algebra

9

Comp etitive pro jects
1. Homology Package, N. van den Hijligenberg, G. Post, in Reduce, 2. SuperLie, P. Grozman, D. Leites, in Mathematica.
k Comparison with SuperLie for Hg (SH(0|4))

k \g 1 2 3 4 5 6 7 8

-8

-6

-4

-2

0

2

4

6

8

SuperLie

LieCohomology, all tasks within several seconds on Pentium 3, 667 MHz