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

Abstract

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 $$\mathfrak {su}(2^{n})$$ su ( 2 n ) , arising from the parameterization of a Hamiltonian for quantum simulation. Specifically, given an AI-type Cartan decomposition $${\mathfrak {g}} = {\mathfrak {k}} \oplus {\mathfrak {p}}$$ g = k ⊕ p of any Lie subalgebra $${\mathfrak {g}}$$ g in $$\mathfrak {su}(2^{n})$$ su ( 2 n ) , where $${\mathfrak {p}} = \widetilde{{\mathfrak {p}}} \oplus {\mathfrak {h}}$$ p = p ~ ⊕ h and $${\mathfrak {h}}$$ h is the maximal abelian subalgebra in $${\mathfrak {p}}$$ p , it is revealed that both the subalgebra $${\mathfrak {k}}$$ k and the corresponding $$\widetilde{{\mathfrak {p}}}$$ p ~ (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 $$\theta (g)=-g^{\top }$$ θ ( g ) = - g ⊤ , any Lie subalgebra $${\mathfrak {g}}$$ g in $$\mathfrak {su}(2^{n})$$ su ( 2 n ) with nondegenerate Cartan decomposition can thus be fully decomposed as the orthogonal direct sum of $${\mathfrak {h}}$$ h and pairs of commutative Lie subalgebras over $${\mathfrak {k}}$$ k and $$\widetilde{{\mathfrak {p}}}$$ p ~ , all of which, except $${\mathfrak {h}}$$ h , share the same dimension.