The exterior graded Swiss-Cheese Operad ΛVS2

ABSTRACT

In this paper we introduce ${\Lambda }_V^{S^2}$ a generalization of the exterior algebra of a vector space V. We study some of its properties and give explicit computations for the case $\mathit{dim}(V_2)=2$. In particular we show the existence and uniqueness of a determinant-like function $de{t}^{S^2}:V_2^{\otimes 6}\to k$ with the property that $de{t}^{S^2}({\otimes }_{1\le i<j\le 4}(v_{i,j}))=0$ if there exists $1\le x<y<z\le 4$ such that $v_{x,y}=v_{x,z}=v_{y,z}$. We also introduce the notion of GSC-operads and show how ${\Lambda }_V^{S^2}$ fits in that context.

KEYWORDS:

• Determinant
• exterior algebra
• Swiss-Cheese operad

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 15A75
• Secondary: 18M60

Acknowledgments

We thank Tong Sun for some discussions and help with MATLAB. We thank Alin Stancu and Florin Panaite for comments and discussions. We thank anonymous referees for bringing to our attention connections with projective geometry, reference [21], and for suggestions that helped us improve the presentation.

Data availability statement

The data that support the findings of this study are available from the corresponding author, MS, upon reasonable request.

Disclosure statement

There is no conflict of interest.