Explicit Super-Linear Algebra over K[θ1, θ2]: A General Berezinian Correction Formula and Multiplicativity Theorem for Arbitrary Block Size
Abstract
We study super-matrices Mm|n(R) over the rank-two exterior algebra R=K[θ1,θ2]/(θ12,θ22,θ1θ2+θ2θ1), the smallest supercommutative ring on which the Berezinian's odd-by-odd correction term BD−1C need not vanish. We recall that the supertrace satisfies graded cyclicity, str(XY)=(−1)|X||Y|str(YX), so str([A,B])=0 for even A,B, and that an explicit 1|1 matrix over R has BD−1C≠0 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∈ Mm|n(R), m,n≥1 (Theorem 1), given through a single m× m matrix pairing Δ built from the odd blocks of X; and a fully general multiplicativity theorem Ber(XY)=Ber(X)Ber(Y) for even invertible X,Y∈ Mm|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≥2 of odd generators, replacing the scalar pairing Δ by a Λ2(V)-valued pairing on the r-dimensional space V of odd generators, and identify this pairing, in the scalar case, with a Plücker coordinate of the odd off-diagonal data. We close with a discussion of related work of Khudaverdian--Voronov, and a remark identifying Δ as a matrix-valued contraction against the natural alternating pairing on the space of odd generators.
Versions
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BibTeX
@misc{devichandrika2026explicit,
title = {{Explicit Super-Linear Algebra over K[θ1, θ2]: A General Berezinian Correction Formula and Multiplicativity Theorem for Arbitrary Block Size}},
author = {Devichandrika V},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05148},
eprint = {2609.05148},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05148},
}
RIS
TY - UNPB
TI - Explicit Super-Linear Algebra over K[θ1, θ2]: A General Berezinian Correction Formula and Multiplicativity Theorem for Arbitrary Block Size
AU - Devichandrika V
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05148
UR - https://arxiv.org/abs/2609.05148
AB - We study super-matrices M_m|n(R) over the rank-two exterior algebra R=K[θ_1,θ_2]/(θ_1^2,θ_2^2,θ_1θ_2+θ_2θ_1), the smallest supercommutative ring on which the Berezinian's odd-by-odd correction term BD^−1C need not vanish. We recall that the supertrace satisfies graded cyclicity, str(XY)=(−1)^|X||Y|str(YX), so str([A,B])=0 for even A,B, and that an explicit 1|1 matrix over R has BD^−1C≠0 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∈ M_m|n(R), m,n≥1 (Theorem 1), given through a single m× m matrix pairing Δ built from the odd blocks of X; and a fully general multiplicativity theorem Ber(XY)=Ber(X)Ber(Y) for even invertible X,Y∈ 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≥2 of odd generators, replacing the scalar pairing Δ by a Λ^2(V)-valued pairing on the r-dimensional space V of odd generators, and identify this pairing, in the scalar case, with a Plücker coordinate of the odd off-diagonal data. We close with a discussion of related work of Khudaverdian--Voronov, and a remark identifying Δ as a matrix-valued contraction against the natural alternating pairing on the space of odd generators.
ER -