<p>The purpose of this article is twofold. First, we prove that the 8-dimensional Lie group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\operatorname {SL}(3,{\mathbb {R}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>SL</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> does not admit a left-invariant hypercomplex structure. To accomplish this we revise the classification of left-invariant complex structures on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\operatorname {SL}(3,{\mathbb {R}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>SL</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> due to Sasaki. Second, we exhibit a left-invariant hypercomplex structure on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\operatorname {SL}(2n+1,{\mathbb {C}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>SL</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which arises from a complex product structure on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\operatorname {SL}(2n+1,{\mathbb {R}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>SL</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\in {\mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>. We then show that there are no left-invariant HKT metrics compatible with this hypercomplex structure. Additionally, we determine the associated Obata connection and we compute explicitly its holonomy group, thus providing a new example of an Obata holonomy group properly contained in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\operatorname {GL}(m,{\mathbb {H}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>GL</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi mathvariant="double-struck">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and not contained in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\operatorname {SL}(m,{\mathbb {H}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>SL</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi mathvariant="double-struck">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$4m=\dim _{\mathbb {R}}\operatorname {SL}(2n+1,{\mathbb {C}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mi>m</mi> <mo>=</mo> <msub> <mo>dim</mo> <mi mathvariant="double-struck">R</mi> </msub> <mo>SL</mo> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>,</p>

Hypercomplex structures on special linear groups

期刊论文
发表日期：

2025年4月16日

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Adrián Andrada,
Agustín Garrone,
Alejandro Tolcachier

摘要

The purpose of this article is twofold. First, we prove that the 8-dimensional Lie group $\operatorname {SL}(3,{\mathbb {R}})$ $SL (3, R)$ does not admit a left-invariant hypercomplex structure. To accomplish this we revise the classification of left-invariant complex structures on $\operatorname {SL}(3,{\mathbb {R}})$ $SL (3, R)$ due to Sasaki. Second, we exhibit a left-invariant hypercomplex structure on $\operatorname {SL}(2n+1,{\mathbb {C}})$ $SL (2 n + 1, C)$ , which arises from a complex product structure on $\operatorname {SL}(2n+1,{\mathbb {R}})$ $SL (2 n + 1, R)$ , for all $n\in {\mathbb {N}}$ $n \in N$ . We then show that there are no left-invariant HKT metrics compatible with this hypercomplex structure. Additionally, we determine the associated Obata connection and we compute explicitly its holonomy group, thus providing a new example of an Obata holonomy group properly contained in $\operatorname {GL}(m,{\mathbb {H}})$ $GL (m, H)$ and not contained in $\operatorname {SL}(m,{\mathbb {H}})$ $SL (m, H)$ , where $4m=\dim _{\mathbb {R}}\operatorname {SL}(2n+1,{\mathbb {C}})$ $4 m = {dim}_{R} SL (2 n + 1, C)$ ,