<p>In this paper, first we introduce the notion of a weighted <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>-operator on Hom-Lie triple systems. Next, we construct an <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>-algebra whose Maurer-Cartan elements are weighted <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>-operators. Consequently, we obtain the twisted <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L_{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>-algebra that controls deformations of a given weighted <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>-operator on Hom-Lie triple systems. Subsequently, we introduce a cohomology theory for weighted <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>-operators on Hom-Lie triple systems. As applications of our cohomology, we use the first cohomology group to classify infinitesimal deformations and we examine the obstruction class of an extendable <i>n</i>-order deformation. We conclude by introducing a new algebraic structure associated with weighted <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>-operators, which we term the Hom-post-Lie triple system. Finally, we establish relationships between weighted <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>-operators on Hom-Lie algebras and the induced Hom-Lie triple systems.</p>

Weighted $\mathcal {O}$-Operators on Hom-Lie Triple Systems

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发表日期：

2026年2月14日

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Wen Teng

摘要

In this paper, first we introduce the notion of a weighted $\mathcal {O}$ $O$ -operator on Hom-Lie triple systems. Next, we construct an $L_{\infty }$ $L_{\infty}$ -algebra whose Maurer-Cartan elements are weighted $\mathcal {O}$ $O$ -operators. Consequently, we obtain the twisted $L_{\infty }$ $L_{\infty}$ -algebra that controls deformations of a given weighted $\mathcal {O}$ $O$ -operator on Hom-Lie triple systems. Subsequently, we introduce a cohomology theory for weighted $\mathcal {O}$ $O$ -operators on Hom-Lie triple systems. As applications of our cohomology, we use the first cohomology group to classify infinitesimal deformations and we examine the obstruction class of an extendable n-order deformation. We conclude by introducing a new algebraic structure associated with weighted $\mathcal {O}$ $O$ -operators, which we term the Hom-post-Lie triple system. Finally, we establish relationships between weighted $\mathcal {O}$ $O$ -operators on Hom-Lie algebras and the induced Hom-Lie triple systems.

Weighted \(\mathcal {O}\)-Operators on Hom-Lie Triple Systems

摘要