<p>In this paper, we introduce a new <i>h</i>-deformation of the superspace <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {C}^{2|1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">|</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> and show that the corresponding function superalgebra <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {F}(\mathbb {C}^{2|1}_h)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <msubsup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>h</mi> <mrow> <mn>2</mn> <mo stretchy="false">|</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> admits a Hopf superalgebra structure. Covariant first-order differential calculus on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {F}(\mathbb {C}^{2|1}_h)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <msubsup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>h</mi> <mrow> <mn>2</mn> <mo stretchy="false">|</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is constructed by extending Woronowicz’s and Wess-Zumino’s approaches to the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {Z}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-graded case. We also define a compatible <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\star $</EquationSource> <EquationSource Format="MATHML"><math> <mo>⋆</mo> </math></EquationSource> </InlineEquation>-structure and establish the higher-order differential calculus, inner derivations, and Lie derivatives to formulate a full Cartan calculus on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {F}(\mathbb {C}^{2|1}_h)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <msubsup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>h</mi> <mrow> <mn>2</mn> <mo stretchy="false">|</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The resulting differential geometry provides a consistent framework for studying quantum Lie superalgebras and graded covariant structures on noncommutative superspaces.</p>

Cartan Calculus On Hopf Superalgebra $\mathcal {F}({\mathbb C}_h^{2|1})$

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发表日期：

2026年3月5日

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Sultan A. Celik,
Salih Celik

摘要

In this paper, we introduce a new h-deformation of the superspace $\mathbb {C}^{2|1}$ $C^{2 | 1}$ and show that the corresponding function superalgebra $\mathcal {F}(\mathbb {C}^{2|1}_h)$ $F (C_{h}^{2 | 1})$ admits a Hopf superalgebra structure. Covariant first-order differential calculus on $\mathcal {F}(\mathbb {C}^{2|1}_h)$ $F (C_{h}^{2 | 1})$ is constructed by extending Woronowicz’s and Wess-Zumino’s approaches to the $\mathbb {Z}_2$ $Z_{2}$ -graded case. We also define a compatible $\star $ $⋆$ -structure and establish the higher-order differential calculus, inner derivations, and Lie derivatives to formulate a full Cartan calculus on $\mathcal {F}(\mathbb {C}^{2|1}_h)$ $F (C_{h}^{2 | 1})$ . The resulting differential geometry provides a consistent framework for studying quantum Lie superalgebras and graded covariant structures on noncommutative superspaces.

Cartan Calculus On Hopf Superalgebra \(\mathcal {F}({\mathbb C}_h^{2|1})\)

摘要