Extensions, automorphisms, and involutions of Leavitt path algebras of polynomial growth: completing the ideal-chain structure theory

For a finite directed graph $\Gamma$ whose cycles are pairwise vertex-disjoint, the Leavitt path algebra $L(\Gamma)$ has polynomial growth (finite Gelfand–Kirillov dimension) and admits a canonical finite chain of ideals $(0) \le I_0 < I_1 < \dots < I_s = L(\Gamma)$, invariant under $\mathrm{Aut}(L(\Gamma))$, whose successive factors are finite direct sums of matrix algebras and finitary matrix algebras (infinite matrices with only finitely many nonzero entries) over the field $F$ or over the Laurent polynomial ring $F[t^{-1},t]$. This record asks for the three complementary pieces of structure theory that the ideal chain leaves undetermined, as left open by Alahmedi, Alsulami, Jain, and Zelmanov (PNAS, 2013): (1) the nature and classification of the extensions $0\to I_i \to I_{i+1} \to I_{i+1}/I_i \to 0$ that glue the chain, for which only the non-splitting of the Toeplitz step and a split/unsplit criterion for extensions of $F[t^{-1},t]$ by $M_\infty(F)$ are known; (2) the description of the automorphism group and the classification of involutions of $L(\Gamma)$ for general polynomial-growth $\Gamma$, beyond the algebraic Toeplitz algebra $L(\Gamma_1)$, presented by $\langle x,y \mid xy=1 \rangle$, where $\mathrm{Aut} \cong F^\times \ltimes GL_\infty(F)$, the semidirect product of the multiplicative group of the field with the finitary infinite general linear group; and (3) whether isomorphic Leavitt path algebras of finite graphs are isomorphic as involutive algebras (i.e. by an isomorphism compatible with their canonical involutions) — settled over $\mathbb{C}$ for finite graphs via the complete classification of unital graph C*-algebras, but open over general fields, the setting in which the source question is posed.

Background

A Leavitt path algebra $L(\Gamma)$ is the associative $F$-algebra generated by the vertices and edges of a finite directed graph $\Gamma$ together with formal adjoints $e^*$ of the edges, subject to Cuntz–Krieger-type relations that make the vertices pairwise orthogonal idempotents, make $s(e)e = er(e) = e$ and $r(e)e^* = e^*s(e) = e^*$, make the adjoints behave like matrix units ($e^*f = \delta_{e,f}r(e)$), and let each non-sink vertex be the sum of the projections $ee^*$ of the edges it emits. Introduced by Abrams–Aranda Pino and by Ara–Moreno–Pardo around 2004 as purely algebraic analogues of graph C*-algebras — the norm-closed algebras of operators on Hilbert space generated by the same vertices and edges — these algebras are $\mathbb{Z}$-graded, carry a canonical involution fixing the vertices and exchanging $e$ with $e^*$, and are finitely presented, which makes them a laboratory for finitely generated algebras with extreme properties. The Gelfand–Kirillov (GK) dimension measures the polynomial rate of growth of a finitely generated algebra; Alahmedi, Alsulami, Jain and Zelmanov (J. Algebra Appl., 2012) proved that $L(\Gamma)$ has polynomially bounded growth if and only if no two distinct cycles of $\Gamma$ share a vertex, and computed $\mathrm{GKdim}\, L(\Gamma) = \max(2d_1-1, 2d_2)$, where $d_1$ and $d_2$ are the maximal lengths of chains of cycles (respectively, of chains of cycles with an exit) — a chain of cycles being a sequence of distinct cycles connected by directed paths, and an exit being an extra edge emitted by a vertex of a cycle but not part of it. In the PNAS paper, the same authors established the structure theorem for this class: $L(\Gamma)$ possesses a finite chain of ideals $(0) \le I_0 < I_1 < \dots < I_s = L(\Gamma)$, with each $I_i$ invariant under all automorphisms of $L(\Gamma)$, in which $I_0$ is the locally finite-dimensional radical (equivalently the socle, i.e. the ideal spanned by the minimal idempotents $pq^*$) and is a finite direct sum of matrix algebras $M_n(F)$ and finitary matrix algebras $M_\infty(F)$ — infinite $\mathbb{N}\times\mathbb{N}$ matrices with only finitely many nonzero entries — while every factor $I_{i+1}/I_i$ with $i \ge 1$ is a finite direct sum of matrix algebras and finitary matrix algebras over the Laurent polynomial algebra $F[t^{-1},t]$. The prototype is the algebraic Toeplitz algebra $L(\Gamma_1)$ of the graph with one vertex, one loop, and one exit edge, presented as $\langle x,y \mid xy=1\rangle$ (Jacobson's algebra): there the chain is $0 \to M_\infty(F) \to L(\Gamma_1) \to F[t^{-1},t] \to 0$, the extension does not split (PNAS Theorem 2), the automorphism group is the semidirect product $F^\times \ltimes GL_\infty(F)$ of the multiplicative group of the field with the finitary infinite general linear group (re-deriving a 2012 theorem of Bavula), and, when every element of $F$ is a square, the standard involution is the only involution up to isomorphism. The involution question connects to a broader classification problem of Abrams–Tomforde (Trans. AMS, 2011), the Isomorphism Conjecture for graph algebras, which asks whether an isomorphism of Leavitt path algebras as ordinary algebras forces an isomorphism as -algebras (involutive algebras), and whether it forces -isomorphism of the associated graph C-algebras; they verified it for acyclic (ultramatricial) and simple row-finite graphs. Substantial later progress on the classification side — the complete geometric and strong classification of unital graph C-algebras of finite graphs by Eilers, Restorff, Ruiz and Sørensen (Duke Math. J., 2021), which settled the Abrams–Tomforde conjectures for graphs with finitely many vertices over $\mathbb{C}$, and the Graded Classification Conjecture program of Hazrat and coworkers, which predicts that two Leavitt path algebras are graded isomorphic (isomorphic by a grading-preserving map) precisely when their graded K-theory groups are isomorphic — provides powerful neighbouring results, but each of the three components above survives as an open problem in the generality in which the PNAS paper leaves them.

Problem Statement

Let $\Gamma$ be a finite directed graph in which no two distinct cycles share a vertex (equivalently, $L(\Gamma)$ has polynomial growth, that is, finite Gelfand–Kirillov dimension), let $F$ be a field, and let $(0) \le I_0 < I_1 < \dots < I_s = L(\Gamma)$ be the canonical $\mathrm{Aut}(L(\Gamma))$-invariant ideal chain of Theorem 1 of Alahmedi–Alsulami–Jain–Zelmanov (2013), whose factors are finite direct sums of matrix algebras and finitary matrix algebras over $F$ and over $F[t^{-1},t]$. Determine the three pieces of structure that the identification of the factors leaves open: (1) the nature and classification of the extensions $0 \to I_i \to I_{i+1} \to I_{i+1}/I_i \to 0$ binding the chain — decide which extensions split, and classify the extension classes up to the natural equivalence (isomorphism of the middle algebra compatible with the ideal and quotient), of which the non-split Toeplitz extension $0 \to M_\infty(F) \to L(\Gamma_1) \to F[t^{-1},t] \to 0$ is the prototype: for this single step a split/unsplit criterion is known (Bossaller 2022), but a classification of the extension classes is open already there, and no comparable criterion exists for the remaining layers of a general chain; (2) the automorphism group $\mathrm{Aut}(L(\Gamma))$ and the involutions of $L(\Gamma)$ up to isomorphism of involutive algebras, for general polynomial-growth $\Gamma$, generalizing the Toeplitz case $\mathrm{Aut}(L(\Gamma_1)) \cong F^\times \ltimes GL_\infty(F)$ and the uniqueness of the standard involution under the hypothesis $F^2 = F$ (every element of $F$ is a square); and (3) whether two Leavitt path algebras of finite graphs that are isomorphic as $F$-algebras are necessarily isomorphic as involutive algebras (with their canonical involutions) — a question asked in general by Abrams–Tomforde (2011) and taken up by the source paper; it is established for finite graphs over $\mathbb{C}$ via the complete classification of unital graph C*-algebras, and it remains open over a general field $F$, the setting in which the source poses it (in particular for the polynomial-growth class of this record).

References

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