# superTrace -- Super trace of a homogeneous super matrix.

## Synopsis

• Usage:
P = superTrace(SM, R, L)
• Inputs:
• Outputs:
• P, ,

## 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

## Ways to use superTrace :

• "superTrace(SuperMatrix,Ring,List)"

## For the programmer

The object superTrace is .