Embedding $K$-algebras into Leavitt algebra $L_K(1, 2)$
Boris Bilich, Roozbeh Hazrat, Tran Giang Nam
TL;DR
The paper develops an algebraic framework to determine which algebras with a countable basis embed into the Leavitt algebra $L_k(1,2)$. Central to the approach is Bergman’s universal realization of finitely generated conical monoids as $ u$-monoids of hereditary $k$-algebras, enabling unital embeddings of Bergman algebras into $L_k(1,2)$ and yielding embeddings for Leavitt path algebras of finite graphs with condition (L). A key negative result shows that the first Weyl algebra cannot embed into $L_k(1,2)$ (or into any $L_k(m,n)$ with $m<n$), answering Brownlowe and Sørensen’s question in the affirmative. The paper also develops a graded embedding theory, proving a graded embedding of $L_k(E)$ into $L_k(1,2) ens L_k(1,2)$ for row-finite graphs, thereby placing Leavitt path algebras inside a graded setting and illustrating a broader embedding framework via Bergman-type constructions.
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 $L_K(1,2)\otimes_K L_K(1,2)$.