Involutions of a Path Algebra

Introduction

A path algebra is generated by the trivial paths at the vertices and by the arrows, subject only to the concatenation rule, so an involution of it is a rule that reverses products and permutes the generators. Two constructions supply such a rule. The first is a self-involution of the quiver: a relabelling of the vertices and the arrows that reverses each arrow and has order two, which extends to an anti-automorphism of the path algebra; the second is the reversal of a path, which reads the arrows in the opposite order and is an anti-isomorphism from the path algebra of a quiver to the path algebra of the reversed quiver, hence an involution of the path algebra exactly when the quiver is isomorphic to its reversal. When the quiver carries relations, an involution descends to the bound quiver algebra only if it preserves the relations.

This article develops the quiver involutions and the anti-automorphisms they induce, the reversal of paths and the identification of the path algebra of the reversed quiver with the opposite algebra, the descent to the algebras bound by an admissible ideal, and the examples of the polynomial algebra, the Kronecker quiver, the preprojective algebra and the lower triangular matrices. The quivers, the paths and the path algebra are the subject of Quiver Representations and Representation Type; the opposite algebra and the anti-isomorphisms are Opposite Algebras and Anti-Isomorphisms; the reversal of words in a free algebra is Involutions of a Free Algebra; the involutive graded algebras are Involutive Graded Algebras; and the general theory of the involution on the elements is Involutive Linear Algebras.

Throughout, $k$ is a field, $Q = (Q_0, Q_1, s, t)$ is a finite quiver, and $kQ$ is its path algebra with the concatenation product, the trivial paths $e_i$ and the length grading. The composition of paths is read from right to left as in Quiver Representations and Representation Type, so that a path $a_1a_2\cdots a_n$ of arrows with $t(a_i) = s(a_{i+1})$ composes with the first arrow acting last; the reversal of a path is the word $a_n\cdots a_1$, and the reversed quiver $Q^{\mathrm{rev}}$ is the quiver with the same vertices and with an arrow for each arrow of $Q$ having the two ends exchanged. The bound quiver algebra $kQ/I$ is the quotient by an admissible ideal $I$, and the relations are the elements of $I$.

The Reversal of the Quiver

The Reversed Quiver and the Opposite Algebra

Definition. The reversed quiver $Q^{\mathrm{rev}}$ has the vertex set $Q_0$, and for each arrow $a \in Q_1$ from $i$ to $j$ an arrow $a^{rev}$ from $j$ to $i$. A path of $Q^{\mathrm{rev}}$ is the reversal of a path of $Q$, and the trivial paths are the same.

Theorem. The path-reversal map

$$ \rho : kQ \longrightarrow kQ^{\mathrm{rev}}, \qquad \rho(a_1a_2\cdots a_n) = a_n^{\mathrm{rev}}\cdots a_1^{\mathrm{rev}}, \qquad \rho(e_i) = e_i, $$

is an isomorphism of $k$-algebras from $kQ$ onto the opposite algebra of $kQ^{\mathrm{rev}}$,

$$ kQ \cong (kQ^{\mathrm{rev}})^{\mathrm{op}} , $$

and it is the identification by which the reversal of the arrows computes the opposite algebra of a path algebra. In particular $kQ^{\mathrm{op}} \cong kQ^{\mathrm{rev}}$, and $kQ$ carries an involution induced by the reversal exactly when the quiver is isomorphic to its reversal.

Proof. A reversal of a path is a path of the reversed quiver: if $t(a_i) = s(a_{i+1})$, then $t(a_{i+1}^{\mathrm{rev}}) = s(a_{i+1}) = t(a_i) = s(a_i^{\mathrm{rev}})$, so the reversed arrows concatenate. The map reverses the order in a product, $\rho(pq) = \rho(q)\rho(p)$, and is bijective with inverse the reversal back; hence it is an anti-isomorphism, which is an isomorphism onto the opposite algebra. The final statement follows by composing with the identification $\iota_{kQ^{\mathrm{rev}}}$ of Opposite Algebras and Anti-Isomorphisms.

Corollary. The reversal of paths is the path-algebra analogue of the reversal of words of Involutions of a Free Algebra: on the free algebra on the arrows it is the canonical anti-involution, and on a path algebra it lands in the opposite algebra of the reversed quiver, so it becomes an involution of $kQ$ exactly when $Q \cong Q^{\mathrm{rev}}$.

The Quiver Involution

Definition. A quiver involution is a pair $\tau = (\tau_0, \tau_1)$ of bijections $\tau_0 : Q_0 \to Q_0$ and $\tau_1 : Q_1 \to Q_1$ with $\tau_0^2 = \mathrm{id}$, $\tau_1^2 = \mathrm{id}$, and

$$ s(\tau_1(a)) = \tau_0(t(a)), \qquad t(\tau_1(a)) = \tau_0(s(a)) $$

for every arrow $a$; the second condition says that $\tau_1$ reverses each arrow before relabelling its ends. A quiver with a quiver involution is a self-involutive quiver.

Theorem. A quiver involution $\tau$ induces an anti-automorphism of $kQ$ of order two, the relabelling involution $\sigma_\tau$, by

$$ \sigma_\tau(e_i) = e_{\tau_0(i)}, \qquad \sigma_\tau(a) = \tau_1(a), \qquad \sigma_\tau(a_1\cdots a_n) = \tau_1(a_n)\cdots\tau_1(a_1), $$

extended linearly. It is a graded involution for the length grading, it permutes the trivial paths, and every length-preserving involution of $kQ$ that carries arrows to arrows and trivial paths to trivial paths arises from a quiver involution.

Proof. For consecutive arrows the condition $t(\tau_1(a_i)) = \tau_0(s(a_i)) = \tau_0(t(a_{i-1})) = s(\tau_1(a_{i-1}))$ shows that the word $\tau_1(a_n)\cdots\tau_1(a_1)$ is a path of $Q$, so $\sigma_\tau$ is defined on the basis and extends linearly; it reverses the order in a product by construction and has order two because $\tau_1^2 = \mathrm{id}$ and $\tau_0^2 = \mathrm{id}$. A length-preserving involution carries $kQ_0$, the span of the trivial paths, to itself, so it permutes the $e_i$; it carries $kQ_1$, the span of the arrows, to itself, and this is the data of the two bijections satisfying the reversed-arrow condition.

Corollary. A self-involutive quiver has $|Q_0|$ fixed vertices and the arrows paired by $\tau_1$; the relabelling involution restricts to an involution of the semisimple algebra $kQ_0 = \prod_i k$ and to a linear involution of the arrow space $kQ_1$, and the pair of these two restrictions determines it.

The Relation with the Reversal

Proposition. A quiver involution $\tau$ and the path reversal are the same construction when $Q$ is identified with $Q^{\mathrm{rev}}$ by $\tau$: the composite of the relabelling involution with the identification $\rho$ of the path algebra with the opposite of the reversed path algebra is the reversal of paths, and conversely the reversal of paths composed with the isomorphism $kQ \to kQ^{\mathrm{rev}}$ induced by a quiver isomorphism of order two is a relabelling involution.

Proof. Both maps reverse the order of the factors of a path and act by a bijection on the generators, so they agree on the basis once the generators are matched by the identification; the identification sends the arrow $a$ of $Q$ to the arrow $\tau_1(a)$ of $Q^{\mathrm{rev}}$ reversed back into $Q$, which is exactly the action of $\sigma_\tau$ on $a$.

The Descent to a Bound Quiver

The Stability Condition

Definition. Let $I \subseteq kQ$ be an admissible ideal of relations, and let $\sigma$ be an anti-automorphism of $kQ$ of order two. Then $\sigma$ descends to $kQ/I$ if $\sigma(I) \subseteq I$, and the descended map is defined by $\bar\sigma(x+I) = \sigma(x)+I$.

Theorem. Let $Q$ carry a quiver involution $\tau$ and let $I$ be a two-sided ideal of $kQ$ stable under $\sigma_\tau$. Then the bound quiver algebra $kQ/I$ is an involutive algebra with the descended anti-automorphism $\bar\sigma_\tau$ of order two, the projection $kQ \to kQ/I$ is equivariant, and the induced involution on the semisimple algebra $kQ_0/(I \cap kQ_0)$ is the one induced by $\tau_0$ on the fixed vertices.

Proof. Stability of the ideal under the involution makes the map on the quotient well defined; the quotient inherits the order-two and the anti-multiplicativity from the path algebra, and the equivariance of the projection is the definition of the descended map. The statement about the semisimple part is the functoriality of the quotient by the intersection.

Corollary. The involutions of a bound quiver algebra are the involutions of the path algebra that preserve the ideal of relations, and increasing the ideal restricts the involution. The length-preserving involutions of the path algebra that permute the trivial paths and the arrows are exactly the relabelling involutions of the quiver involutions; the path reversal supplies the involutions that exchange the two length-graded orientations, and it is an involution of $kQ$ exactly when $Q \cong Q^{\mathrm{rev}}$.

The Examples

The polynomial algebra. For the quiver with one vertex and one loop $a$, the path algebra is $k[a] = k[x]$ by the identification $a = x$, and the admissible ideals are the powers $(x^m)$. A quiver involution fixes the vertex and carries $a$ to $a$, so the relabelling involution is the identity; the involution $x \mapsto -x$ is a length-preserving automorphism, not an anti-automorphism, and it is an involutive automorphism of $k[x]$. The reversal of the single-letter path is the path itself, so the path reversal is the identity here, in agreement with the general fact that a one-letter word is its own reversal.

The Kronecker quiver and the preprojective algebra. Let $Q$ have two vertices $1, 2$ and arrows $a, b : 1 \to 2$. First, the map $\tau$ with $\tau_0$ exchanging $1$ and $2$ and $\tau_1$ exchanging the two arrows is a quiver involution: $s(\tau_1(a)) = s(b) = 1 = \tau_0(t(a)) = \tau_0(2)$ and $t(\tau_1(a)) = t(b) = 2 = \tau_0(s(a)) = \tau_0(1)$. The doubled quiver $Q^{*}$ has, beside $a$ and $b$, the reverse arrows $a^{*}, b^{*} : 2 \to 1$; the map with $\tau_0 = \mathrm{id}$ and $\tau_1$ exchanging each arrow with its reverse, $\tau_1(a) = a^{*}$ and $\tau_1(a^{*}) = a$, is a quiver involution of $Q^{*}$: $s(\tau_1(a)) = s(a^{*}) = 2 = t(a) = \tau_0(t(a))$ and $t(\tau_1(a)) = t(a^{*}) = 1 = s(a) = \tau_0(s(a))$. The preprojective algebra of $Q$ is the quotient of $kQ^{*}$ by the ideal generated by the elements $a a^{*} - a^{*} a$ and $b b^{*} - b^{*} b$, and each generator is fixed by the induced relabelling involution: $\sigma(a a^{*} - a^{*} a) = \sigma(a^{*})\sigma(a) - \sigma(a)\sigma(a^{*}) = a a^{*} - a^{*} a$. Hence the relation is preserved and the preprojective algebra carries the descended involution, which pairs the two orientations; this is the central example of the theory.

The lower triangular matrices. Let $Q$ be the quiver $1 \to 2$ with one arrow $a$. Its path algebra is the algebra of the lower triangular $2 \times 2$ matrices, with basis $e_1, e_2, a$ and $a^2 = 0$. The map $\tau_0$ exchanging $1$ and $2$ and $\tau_1$ fixing $a$ is a quiver involution, since $s(\tau_1(a)) = s(a) = 1 = \tau_0(t(a)) = \tau_0(2)$ and $t(\tau_1(a)) = t(a) = 2 = \tau_0(s(a)) = \tau_0(1)$; the induced involution is $\sigma(e_1) = e_2$, $\sigma(e_2) = e_1$, $\sigma(a) = a$, which is the conjugation by $J = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ composed with the transpose, $\sigma(X) = JX^{\mathsf{T}}J$, the reflection of the matrix across its antidiagonal. The example shows that a quiver involution need not reverse an arrow: the arrow is moved by the permutation of its two ends, and the reflection of the underlying graph is the inverse of the reversal of the arrows.

The one-vertex group quiver. Let $Q$ be the quiver with one vertex and one loop $x_g$ for each element of a finite group $G$, and let $I$ be generated by the relations $x_gx_h - x_{gh}$ and $x_1 - 1$. The quotient $kQ/I$ is the group algebra $k[G]$ of Involutions of a Free Algebra. The quiver involution with $\tau_0 = \mathrm{id}$ and $\tau_1(x_g) = x_{g^{-1}}$ is a quiver involution, because a loop is carried to a loop at the same vertex, and it preserves the relations: $\sigma(x_gx_h - x_{gh}) = x_{h^{-1}}x_{g^{-1}} - x_{(gh)^{-1}} = x_{(gh)^{-1}} - x_{(gh)^{-1}} = 0$. Hence it descends, and the descended involution of $k[G]$ is the inversion $g \mapsto g^{-1}$, which is of order two exactly when every element of $G$ has order dividing two.

Summary

A path algebra $kQ$ carries two families of order-two anti-maps. The path reversal $\rho(a_1\cdots a_n) = a_n^{\mathrm{rev}}\cdots a_1^{\mathrm{rev}}$ is an isomorphism $kQ \cong (kQ^{\mathrm{rev}})^{\mathrm{op}}$, so $kQ^{\mathrm{op}} \cong kQ^{\mathrm{rev}}$ and the reversal is an involution of $kQ$ exactly when $Q$ is isomorphic to its reversal. A quiver involution $\tau = (\tau_0,\tau_1)$ is a pair of bijections with $\tau_0^2 = \tau_1^2 = \mathrm{id}$ and $s(\tau_1(a)) = \tau_0(t(a))$, $t(\tau_1(a)) = \tau_0(s(a))$; it induces the relabelling involution $\sigma_\tau$ with $\sigma_\tau(e_i) = e_{\tau_0(i)}$ and $\sigma_\tau(a) = \tau_1(a)$ on the generators, extended anti-multiplicatively to the paths, and every length-preserving involution of $kQ$ that permutes the trivial paths and the arrows comes from a quiver involution. An admissible ideal $I$ stable under $\sigma_\tau$ admits the descent, and the bound quiver algebra $kQ/I$ then carries the descended involution of order two; the involutions of the bound quiver algebras are exactly the involutions of the path algebras that preserve the relations. The examples are the identity involution and the automorphism $x \mapsto -x$ on the polynomial algebra, the involution of the preprojective algebra of the Kronecker quiver pairing the two orientations, the reflection $X \mapsto JX^{\mathsf{T}}J$ of the lower triangular matrices, which is the relabelling involution that moves the arrow by the permutation of its ends, and the inversion of the group algebra. The quivers and the path algebras are Quiver Representations and Representation Type; the opposite algebra is Opposite Algebras and Anti-Isomorphisms; the reversal of words is Involutions of a Free Algebra; and the general involution on the elements is Involutive Linear Algebras.

Summary of Notation

Symbol	Meaning
$Q = (Q_0,Q_1,s,t)$	the quiver, with vertices, arrows, source and target
$kQ$	the path algebra, the concatenation product
$Q^{\mathrm{rev}}$	the reversed quiver, every arrow reversed
$\rho$	the path reversal, an isomorphism $kQ \cong (kQ^{\mathrm{rev}})^{\mathrm{op}}$
$\tau = (\tau_0,\tau_1)$	a quiver involution
$\sigma_\tau$	the relabelling involution of $kQ$
$I$	an admissible ideal of relations
$kQ/I$	the bound quiver algebra
$J$	the reflection matrix in the lower triangular example