Steinberg Algebras Research Articles

Using Steinberg algebras to study decomposability of Leavitt path algebras

Abstract Given an arbitrary graph E we investigate the relationship between E and the groupoid G E {G_{E}} . We show that there is a lattice isomorphism between the lattice of pairs ( H , S ) {(H,S)} , where H is a hereditary and saturated set of vertices and S is a set of breaking vertices associated to H, onto the lattice of open invariant subsets of G E ( 0 ) {G_{E}^{(0)}} . We use this lattice isomorphism to characterise the decomposability of the Leavitt path algebra L K ⁢ ( E ) {L_{K}(E)} , where K is a field. First we find a graph condition to characterise when an open invariant subset of G E ( 0 ) {G_{E}^{(0)}} is closed. Then we give both a graph condition and a groupoid condition each of which is equivalent to L K ⁢ ( E ) {L_{K}(E)} being decomposable in the sense that it can be written as a direct sum of two nonzero ideals. We end by relating decomposability of a Leavitt path algebra with the existence of nontrivial central idempotents. In fact, all the nontrivial central idempotents can be described.

Reconstruction of graded groupoids from graded Steinberg algebras

Abstract We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies C * {C^{*}} -isomorphism of C * {C^{*}} -algebras for graphs E and F in which every cycle has an exit.

Uniqueness Theorems for Steinberg Algebras

We prove Cuntz-Krieger and graded uniqueness theorems for Steinberg algebras. We also show that a Steinberg algebra is basically simple if and only if its associated groupoid is both effective and minimal. Finally we use results of Steinberg to characterise the center of Steinberg algebras associated to minimal groupoids.

Equivalent groupoids have Morita equivalent Steinberg algebras

Let G and H be ample groupoids and let R be a commutative unital ring. We show that if G and H are equivalent in the sense of Muhly–Renault–Williams, then the associated Steinberg algebras are Morita equivalent. We deduce that collapsing a “collapsible subgraph” of a directed graph in the sense of Crisp and Gow does not change the Morita-equivalence class of the associated Leavitt path R-algebra, and therefore a number of graphical constructions which yield Morita equivalent C⁎-algebras also yield Morita equivalent Leavitt path algebras.

Steinberg–Leibniz algebras and superalgebras

As a universal central extension of the special linear Lie algebra sl ( n , A ) over a unital associative algebra A, the Steinberg algebras st ( n , A ) and stl ( n , A ) were studied in several papers. In this paper, we mainly study the Steinberg–Leibniz algebra stl ( n , D ) defined over a dialgebra D. We prove that it is the universal central extension of the special linear Leibniz algebra sl ( n , D ) with kernel HHS 1 ( D ) , the quotient of the first Hochschild homology group HH 1 ( D ) of the dialgebra D by the ideal generated by a ⊗ ( b ⊣ c ) − a ⊗ ( b ⊢ c ) for all a , b , c ∈ D . We also obtain a similar theorem for the Steinberg–Leibniz superalgebra stl ( m , n , D ) . This research plays a key role in studying the Leibniz algebras (superalgebras) graded by finite root systems and is also connected with ‘Leibniz K-theory.’