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SuperLinearAlgebra :: superTrace

superTrace -- Super trace of a homogeneous super matrix.

Synopsis

Description

Let $A^{p|q}=Ax_1 \oplus \cdots \oplus Ax_p \oplus Ae_1\oplus \cdots \oplus Ae_q$ be a free module over $A$, where $x_i$s are even and $e_j$s are odd generators. A (homogeneous) morphism $T:A^{p|q}\rightarrow A^{r|s}$ has a matrix representation. Denote the matrix by $T$ then we have $T=\begin{pmatrix} T1&T2\\ T3&T4\end{pmatrix}$.

The super trace of $T$ is defined by $superTrace(T)= Trace(T_1)-(-1)^{p(T)} Trace(T_4)$. The inputs of this function are a SuperMatrix, a ring, which should have skew-symmetric variables, and a list, which is the list of skew-symmetric variables that are used in the superMatrixGenerator. In case that the superMatrix is homogeneous, the output is the super trace of the superMatrix.

i1 : R1 = QQ[x_0..x_3];
i2 : R2 = QQ[z_0..z_2];
i3 : R = superRing(R1, R2);
i4 : P1 = matrix{{x_0, x_1}, {x_2, x_3}};

             2       2
o4 : Matrix R  <--- R
i5 : P2 = matrix{{0, 0}, {0, 0}};

              2        2
o5 : Matrix ZZ  <--- ZZ
i6 : P3 = matrix{{0, 0}, {0, 0}};

              2        2
o6 : Matrix ZZ  <--- ZZ
i7 : P4 = matrix{{x_1, x_2}, {x_0, x_1}};

             2       2
o7 : Matrix R  <--- R
i8 : SP = superMatrixGenerator(P1, P2, P3, P4);
i9 : superTrace(SP, R, {z_0, z_1})

o9 = x  - 2x  + x
      0     1    3

o9 : R

Caveat

Ways to use superTrace :

For the programmer

The object superTrace is a method function.