On the Refinement of Cartan Decomposition: An Implicit Commutative Substructure in $$\mathfrak {su}(2^{n})$$

Abstract The Cartan decomposition of a semisimple Lie algebra is pivotal in both theoretical exploration and practical application. It generalizes the classical spectral, polar, and singular value decompositions found in linear algebra. This paper explores an advanced refinement of the Cartan decomposition of $$\mathfrak {su}(2^{n})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>su</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , arising from the parameterization of a Hamiltonian for quantum simulation. Specifically, given an AI -type Cartan decomposition $${\mathfrak {g}} = {\mathfrak {k}} \oplus {\mathfrak {p}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mo>⊕</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> of any Lie subalgebra $${\mathfrak {g}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> in $$\mathfrak {su}(2^{n})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>su</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $${\mathfrak {p}} = \widetilde{{\mathfrak {p}}} \oplus {\mathfrak {h}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mover> <mml:mi>p</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>⊕</mml:mo> <mml:mi>h</mml:mi> </mml:mrow> </mml:math> and $${\mathfrak {h}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> is the maximal abelian subalgebra in $${\mathfrak {p}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> , it is revealed that both the subalgebra $${\mathfrak {k}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> and the corresponding $$\widetilde{{\mathfrak {p}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>p</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> (which is generally not a subalgebra) can further be partitioned, respectively, as the direct sums of equal-dimensional commutative Lie subalgebras. With respect to the involution $$\theta (g)=-g^{\top }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>g</mml:mi> <mml:mi>⊤</mml:mi> </mml:msup> </mml:mrow> </mml:math> , any Lie subalgebra $${\mathfrak {g}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> in $$\mathfrak {su}(2^{n})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>su</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with nondegenerate Cartan decomposition can thus be fully decomposed as the orthogonal direct sum of $${\mathfrak {h}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> and pairs of commutative Lie subalgebras over $${\mathfrak {k}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> and $$\widetilde{{\mathfrak {p}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>p</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> , all of which, except $${\mathfrak {h}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> , share the same dimension.