DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

CONSERVATION LAWS IN QUANTUM SUPER PDEs
A. PRÁSTARO
Conservation laws are considered for PDEs built in the category Q S of quantum supermanifolds.
These are functions defined on the integral bordism groups of such equations
and belonging to suitable Hopf algebras (full quantum Hopf algebras). Inparticular,we
specialize our calculations on the quantum super Yang-Mills equations and quantum
black holes.
Copyright © 2006 A. Prástaro. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
In this section we will resume some of our fundamental definitions and results for PDEs
in the category of quantum supermanifolds, Q S , where the objects are just quantum supermanifolds,
and the morphisms are maps of class Q k w, k ∈{0,1,2,...,∞,ω} [1–8]. A small
subcategory is S M ⊂ Q S of supermanifolds as defined in [3].
Example 1.1. Let π : W → M be a fiber bundle, in the category Q S , such that dimW =
(m|n,r|s), over the quantum superalgebra B ≡ A × E and dimM = (m|n)overA and such
that E is a Z-module, with Z ≡ Z(A) ⊂ A, thecenterofA. Thequantum k-jet-derivative
space J ̂D k (W) ofπ : W → M is the k-jet-derivative space of sections of π, belonging
to the class Qw.Thek-jet-derivative k J ̂D k (W) is a quantum supermanifold modeled on
the quantum superalgebra B k ≡ ∏ 0≤s≤k( ∏ i 1+···+i s∈Z 2, i r ∈Z 2 Â s i 1···i s
(E)), with  s i 1···i s
(E) ≡
Hom Z (A i1 ⊗ Z ···⊗ Z A is ;E), Â 0 (E) ≡ A × E, Â 1 i (E) ≡ Â 0 (E) × Â 1 (E) ≡ Hom Z (A 0 ;E) ×
Hom Z (A 1 ;E). Each  s i 1···i s
(E) is a quantum superalgebra with Z 2 -gradiation induced by
E. Hence, s i 1···i s
(E) q ≡ Hom Z (A i1 ⊗ Z ···⊗ Z A is ;E p ), i r , p,q ∈ Z 2 , q ≡ i 1 + ···+ i s + p. If
(x A , y B ) 1≤A≤m+n,1≤B≤r+s are fibered quantum coordinates on the quantum supermanifold
W over M, then(x A , y B , yA,..., B yA B 1···A k
) are fibered quantum coordinates on J ̂D k (W)
over M, with the following gradiations: |x A |=|A|, |y B |=|B|, |yA B 1···A s
|=|B| + |A 1 | +
···|A s |. Note, also, that there is not symmetry in the indexes A i . J ̂D k (W) isanaffine
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 943–952