Explicit Super-Linear Algebra over K[{\theta}1, {\theta}2]: A General Berezinian Correction Formula and Multiplicativity Theorem for Arbitrary Block Size

Abstract

We study super-matrices $M_{m|n}(R)$ over the rank-two exterior algebra $R=K[\theta_1,\theta_2]/(\theta_1^2,\theta_2^2,\theta_1\theta_2+\theta_2\theta_1)$, the smallest supercommutative ring on which the Berezinian's odd-by-odd correction term $BD^{-1}C$ need not vanish. We recall that the supertrace satisfies graded cyclicity, $\mathrm{str}(XY)=(-1)^{|X||Y|}\mathrm{str}(YX)$, so $\mathrm{str}([A,B])=0$ for even $A,B$, and that an explicit $1|1$ matrix over $R$ has $BD^{-1}C\neq0$ surviving to second order in the odd generators. We then prove the extension anticipated in the $1|1$ note: a closed-form Berezinian formula for arbitrary even $X\in M_{m|n}(R)$, $m,n\geq1$ (Theorem 1), given through a single $m\times m$ matrix pairing $\Delta$ built from the odd blocks of $X$; and a fully general multiplicativity theorem $\mathrm{Ber}(XY)=\mathrm{Ber}(X)\mathrm{Ber}(Y)$ for even invertible $X,Y\in M_{m|n}(R)$ (Theorem 2), proved by an explicit trace-cyclicity cancellation. Both results specialize exactly to the known $1|1$ formulas when $m=n=1$, and we verify them on a worked $m=2,n=1$ example. We extend the framework from two to an arbitrary number $r\geq2$ of odd generators, replacing the scalar pairing $\Delta$ by a $\Lambda^2(V)$-valued pairing on the $r$-dimensional space $V$ of odd generators, and identify this pairing, in the scalar case, with a Pl\"ucker coordinate of the odd off-diagonal data. We close with a discussion of related work of Khudaverdian--Voronov, and a remark identifying $\Delta$ as a matrix-valued contraction against the natural alternating pairing on the space of odd generators.