The Leavitt path algebra of a graph

By G. Abrams, G. Pino. 2005

444 citations recorded by the source corpus.

Abstract For any row-finite graph E and any field K we construct the Leavitt path algebra L ( E ) having coefficients in K . When K is the field of complex numbers, then L ( E ) is the algebraic analog of the Cuntz–Krieger algebra C ∗ ( E ) described in [I. Raeburn, Graph algebras, in: CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., 2005]. The matrix rings M n ( K ) and the Leavitt algebras L ( 1 , n ) appear as algebras of the form L ( E ) for various graphs E . In our main result, we give necessary and sufficient conditions on E which imply that L ( E ) is simple.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3a37253dd6993333ef7b5994dc16fdd4102885ec

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Citations in the corpus

The Theory of Prime Ideals of Leavitt Path Algebras over Arbitrary Graphs (2011, methodology): “11 of [1]) when LK(E) is a prime ring.”
Graph inverse semigroups: their characterization and completion (2011, background): “There are clear parallels between the Cuntz algebras On and the Leavitt algebras L(1, n) observed in [1] and [49].”
Exchange Leavitt path algebras and stable rank (2005, methodology): “If n = 1, then L(E) is a simple ring and then E is cofinal by Lemma 2.8 and [ 1, Theorem 3.11 ].”
Uniqueness theorems and ideal structure for Leavitt path algebras (2006, methodology): “In 2005, Abrams and Aranda-Pino defined the Leavitt path algebra LK(E) to be the universal K-algebra generated by elements satisfying relations similar to those of the generators for the graph C∗-algebra of E [1].”
Isomorphism and Morita equivalence of graph algebras (2008, background): “Such K-algebras include many well-known classes of algebras and have been studied intensely in the algebraic community since their introduction in 2005 [6, 13].”
Finitely presented simple modules over Leavitt path algebras (2013, unknown): “Leavitt path algebras were introduced and initially studied in [1], [11], as algebraic analogues of graph C-algebras and as natural generalizations of the Leavitt algebras of type (1, n) built in [19].”
Chain conditions for Leavitt path algebras (2009, background): “For example, conditions on the graph allow us to decide when the corresponding Leavitt path algebra is simple [1], purely infinite simple [2], exchange [9], finite-dimensional [4].”
Finite-dimensional Leavitt path algebras (2007, background): “In [1], [2] and [3], a somewhat different description of LK(E) is given (in terms of quotients of path algebras).”
The graded Grothendieck group and the classification of Leavitt path algebras (2011, background): “Modeled on this, Leavitt path algebras are introduced [1, 7], which attach to a directed graph a certain algebra.”
Graded Rings and Graded Grothendieck Groups (2014, methodology): “Motivated by this algebra, the Leavitt path algebras were introduced in [2, 6], which associate to a direct graph a certain algebra.”
Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions☆ (2012, background): “Their associated graph C*-algebras C(E,C) and Leavitt path algebras LK(E,C) (for an arbitrary field K), provide generalizations of the usual graph C*-algebras ([47]) and Leavitt path algebras ([2], [10]) associated to directed graphs, although these algebras behave quite differently from the usual g”
Socle theory for Leavitt path algebras of arbitrary graphs (2008, background): “This was first don e in the analytic setting (see [15, 27, 28] among others), while the seminal results on Leavitt path algebras of row finite graphs appeared in [1] and [9], so starting a flurry of ac tivity.”
Locally finite Leavitt path algebras (2008, methodology): “3 (1)], LK(E/H) ∼= LK(E)/I(H) with I(H) a nonzero graded ideal of LK(E), which is impossible by the hypothesis.”
Leavitt path algebras and direct limits (2007, background): “f C*-algebras [15]. Generalizations of these led to a large class of C*-algebras built from directed graphs, and the construction was carried to the algebraic category by G. Abrams and G. Aranda Pino [1]. For both historical and technical reasons, the graphs used in constructing graph C*-algebras an”
Irreducible representations of Leavitt path algebras (2011, background): “Q/ of Q with coefficients in k is relatively new, which is introduced in [1, 7, 8].”
LEAVITT PATH ALGEBRAS OF FINITE GELFAND–KIRILLOV DIMENSION (2012, methodology): “The chain is said to have an exit if the cycle Ck has an exit (see [1]), that is, if there exists an edge e such that s(e) ∈ V (Ck), but e does not belong to Ck.”
Leavitt path algebras: the first decade (2014, methodology): “So even though [7] appeared in print eighteen months prior to the appearance in print of [36], in fact most the mathematical work done to produce the latter preceded that of the former.”