Embedding $K$-algebras into Leavitt algebra $L_K(1, 2)$

TL;DR

This paper demonstrates how certain $K$-algebras, including Leavitt path algebras of finite graphs, can embed into the Leavitt algebra $L_K(1,2)$, and explores limitations on embedding the Heisenberg and Weyl algebras.

Contribution

It provides new embedding results for $K$-algebras into $L_K(1,2)$ and establishes non-embeddability of the Weyl algebra, along with graded embedding conditions.

Findings

Bergman $K$-algebras embed into $L_K(1,2)$

Heisenberg equation cannot be realized in Steinberg algebra

Weyl algebra does not embed into $L_K(1,2)$

Abstract

Since the commutative monoid $T = (\{0, 1\}, \vee)$ is a weak terminal object in the category of conical monoids with order units, there is a unital homomorphism from every Bergman $K$-algebra corresponding to a conical finitely generated commutative monoid into the Leavitt algebra $L_K(1,2)$, where $K$ is a field. This fact will be used to give a short proof that Leavitt path algebras associated with finite graphs with condition $(L)$ embed into $L_K(1,2)$, as well as provide criteria for an embedding of $M_s(L_{K}(1, m))$ in $M_s(L_{K}(1, n))$. As our second main result, we show that the Heisenberg equation $xy-yx=1$ cannot be realized in any Steinberg algebra, implying that the first Weyl algebra cannot be embedded into $L_K(1,2)$, giving an affirmative answer to a question of Brownlowe and Sorensen on the embeddability of $K$-algebras with a countable basis inside $L_K(1,2)$. Whereas, $L_K(E)$ cannot be graded-embedded into $L_K(1,2)$ in general, in the final section we show that $L_K(E)$ does admit a graded embedding into $L_K(1,2)\otimes_K L_K(1,2)$.