Finite-dimensional Lie algebras in bosonic quantum dynamics: The single-mode case

TL;DR

This work classifies finite-dimensional Lie subalgebras of the real single-mode skew-hermitian Weyl algebra $\\hat{A}_1$ that are generated by monomials, and extends the analysis to algebras containing a free Hamiltonian term. It proves a core constraint: any non-abelian finite-dimensional subalgebra of $\\hat{A}_1$ that includes a free Hamiltonian must embed in the Schrödinger algebra $\\mathcal{S}\\cong\\mathfrak{sl}(2,\\mathbb{R})\\ltimes\\mathfrak{h}_1$, with the complete list of non-abelian monomial-generated algebras consisting of eight distinct types including $\\mathcal{S}$, $\\mathfrak{sl}(2,\\mathbb{R})$, $\\mathfrak{sl}(2,\\mathbb{R})\\oplus\\mathbb{R}$, and $\\mathfrak{wh}_2$, among others. The authors also classify finite-dimensional nilpotent and non-solvable realizable algebras, and establish the abelian centralizer condition as a key tool. They extend the analysis to linear combinations of monomials via an Igusa-type framework and discuss implications for quantum control, notably in the context of factorization-based time evolution (Wei–Norman) and the design of finite control resources. The results provide a rigorous algebraic foundation for when single-mode bosonic dynamics admit closed-form, finite-factorization solutions and set the stage for multi-mode generalizations and control-theoretic applications in quantum technologies.

Abstract

We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra $\hat{A}_1$, defined as the real subalgebra of the Weyl algebra $A_1$ consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for $\hat{A}_1$, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in $\hat{A}_1$ generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in $\hat{A}_1$ that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in $\hat{A}_1$? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of $\hat{A}_1$ that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in $\hat{A}_1$, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.

Figures (6)

• Figure 1: Schematic overview of the classification strategy for finite-dimensional Lie subalgebras of $\hat{A}_1$. The study is divided into two main branches: the classification of solvable Lie algebras (left), and the classification of the non-solvable Lie algebras (right). Double arrows indicate causal relations, while single arrows represent other structural dependencies. A double arrow is crossed out when one classification does not imply the other. Green boxes indicated classifications that have been achieved in this work, while the red boxes highlights those that remain outstanding. The flow-chart immediately shows that a complete classification of all solvable Lie algebras would immediately imply a classification of all nilpotent Lie algebras, since every nilpotent algebra is solvable. The reverse does not hold. Furthermore, there is no exploitable causal relationship between the classification of solvable and non-solvable Lie algebras. In order to achieve the classification of all non-solvable Lie algebras we start by identifying all semisimple Lie algebras $\mathfrak{s}$ that can be faithfully realized in $\hat{A}_1$. This informs the second step, namely, classifying all $\operatorname{ad}$-invariant algebras for such semisimple Lie algebras $\mathfrak{s}$. These two results together yield a complete classification of all non-solvable finite-dimensional Lie algebras that can be faithfully realized in $\hat{A}_1$ via the Levi-Mal'tsev theorem [ Kuzmin:1977].
• Figure 2: Depiction of the Schrödinger algebra using a directed graphs as presented in Definition \ref{['directed:graphs']}. In (a) the chosen basis is the one found in Definition \ref{['def:Schroedinger:algebra']}; in (b) the chosen basis is $\{i,g_+^{\iota_1},g_-^{\iota_1},i(a^\textup{$\dagger$} a+1/2), g_+^{2\iota_1},g_-^{2\iota_1}\}$, inspired by the basis of $\hat{A}_1^0\oplus\hat{A}_1^1\oplus\hat{A}_1^2$. The red edges and vertices belong to the special linear algebra $\mathfrak{sl}(2,\mathbb{R})$, the yellow ones to the Heisenberg algebra $\mathfrak{h}_1$, and the blue dotted edges represent commutation relations between elements from $\mathfrak{sl}(2,\mathbb{R})$ and $\mathfrak{h}_1$. This highlights the structure of the Schrödinger algebra as a semidirect sum of the special linear algebra and the Heisenberg algebra, further emphasized by the dotted rectangles encircling the basis elements of the respective subalgebras.
• Figure 3: Depiction of three different generic Commutator Chains $C^{\mathrm{gen}}$. The red chain utilizes the same auxiliary element at each step, while the blue chain employs a different auxiliary element at every step. The green chain illustrates a chain that eventually terminates.
• Figure 4: Schematic overview of the generic Commutator Chains used in the proof of Theorem \ref{['thm:andreea:sona']}. The central chain (Case 1) corresponds to the situation where the two initial monomials $g_{\hat{\sigma}}^{\hat{\gamma}}\in\hat{A}_1^1\oplus\hat{A}_1^2\oplus\hat{A}_1^\perp$ and $g_\sigma^\gamma\in\hat{A}_1^\perp$ can be directly used to construct a chain whose elements have a simple and easily verifiable property: in the expansion in terms of monomials of every chain element $u^{(\ell)}$ there exists a unique non-vanishing monomial with highest degree. This makes it straightforward to prove that the degrees of the elements in the chain increase strictly with each step. In contrast, the remaining Cases 2 and 3 do not have this property. However, each of them can be reduced to Case 1 by constructing at most two auxiliary elements (indicated by the end of the light-blue lines) that can in turn be used so that the resulting chain generated by such two elements can be treated analogously to Case 1.
• Figure 5: Schematic visualization of Algorithm \ref{['alg:generating:all:subalgebras:from:subset:of:basis']} applied to the basis $\mathcal{B}=\{s_1=i,s_2=ia^\textup{$\dagger$} a, s_3=g_+^{\iota_1},s_4=g_-^{\iota_1},s_5=g_+^{2\iota_1},s_6=g_-^{2\iota_1}\}$. Here we omit to include the trivial Lie subalgebras $\langle s_1 \rangle_{\mathrm{Lie}}{},\ldots,\langle s_6 \rangle_{\mathrm{Lie}}{}$, and $\langle \mathcal{B} \rangle_{\mathrm{Lie}}{}$ generated by subsets of $\mathcal{B}$ in the initial $\mathcal{L}$-set of all such subalgebras. Instead, we initialize $\mathcal{L}$ as the empty set and generate these subalgebras explicitly. This is achieved by performing the initial for-loop over the indices $\{0,1,\ldots,5\}$, defining $\{s_0\}:=\emptyset$, and not skipping the generation step when $\langle \mathcal{S}\cup \{s_k\} \rangle_{\mathrm{Lie}}{}=\langle \mathcal{B} \rangle_{\mathrm{Lie}}{}$ occurs the first time. Only steps in which a new realization of a Lie subalgebra is generated and added to $\mathcal{L}$ are visualized. Horizontal lines indicate points at which the function Fct1 is executed, while vertical lines represent for-loops initiated by the recursive procedure Proc2, where new elements are added to the basis of the previously generated subalgebra. For each subalgebra generated via Fct1, we also provide its isomorphism class.

Theorems & Definitions (189)

• Definition 1: Lie closure
• Example 2
• Example 3
• Proposition 4
• proof
• Definition 5
• Definition 6: Monomial generators
• Definition 7
• Definition 8
• Definition 9: Partioning monomial generators