A quiver approach to studying orbit spaces of ... - Semantic Scholar
Systems & Control Letters 73 (2014) 1–5
Contents lists available at ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
A quiver approach to studying orbit spaces of linear systems Mihai-Sorin Stupariu ∗ University of Bucharest, Faculty of Mathematics and Computer Science, Academiei Str. 14, 010014 Bucharest, Romania
article
info
Article history: Received 26 September 2012 Received in revised form 2 August 2014 Accepted 9 September 2014 Available online 29 September 2014 Keywords: Linear dynamical system Quiver representation Stability Compactification
abstract Orbit spaces associated to linear actions are of particular interest in control theory. Their geometrical properties can be naturally investigated by using the representations of quivers as an abstract framework. The aim of the paper is to bring into attention an application of this approach and to show how the use of quivers makes it easy handling concepts arising in control theory. Specifically, the natural duality between controllable and observable systems, as well as the construction of compactifications for the associated orbit spaces is interpreted in terms of opposite quivers. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The use of techniques from algebraic geometry has been proved to be a very fruitful approach in solving problems arising in control theory. In the late ’70s, several pioneering works (e.g. those by Hazewinkel and Kalman [1] or by Byrnes and Hurt [2]) highlighted some meaningful algebro-geometric tools in constructing moduli spaces for linear dynamical systems; an excellent synthesis of these techniques, as well as a source for relevant references is the monograph by Tannenbaum [3]. Recent developments (see [4,5]) brought into attention the opportunity of using (partial) representations of quivers as framework for constructing moduli spaces for certain classes of linear systems or for studying their topological properties. Moreover, it was proved that the compactification proposed by Helmke and Shayman in [6] for the orbit space of controllable state space systems and that one proposed by Helmke (see [7,8]) for the space of rational transfer functions can be interpreted in terms of orbit spaces associated to quiver representations. Actually, these orbit spaces are constructed by adapting ideas from Mumford’s Geometric Invariant Theory (see [9]) to the case of linear actions, particularly to the case of representations of quivers (see [10,11]). Furthermore, it was shown in [12] that, by appropriately adapting the concepts and the definitions, all these results can be obtained by completely remaining within the framework of linear algebra. The aim of this note is to bring into attention other applications of the quiver approach for problems arising in control theory. The
paper is organized as follows. Section 2 is dedicated to revising the main concepts referring to stability, respectively to the construction of orbit spaces associated to quiver factorization problems. Section 3 is divided in two parts. Section 3.1 deals with results that are abstract in nature, referring to the natural relationship between stability concepts (Proposition 1), respectively between orbit spaces (Theorem 1) associated to opposite quivers. Section 3.2 is oriented towards applications. The well-known duality between controllability and observability is naturally interpreted in Proposition 2 in terms of quiver representations. The interpretation is extended in Theorem 2 for the compactifications of the associated orbit spaces of controllable and observable linear systems. 2. Orbit spaces associated to quiver representations Quiver factorization problems. Let Q = (N , A, h, t ) be a quiver (here N and A are the sets of nodes, respectively of arrows, while h, t : A → N are the head, respectively the tail maps). A representation r = (V, ψ) of Q over the field K (K ∈ {R, C}) is a pair consisting of a family of K-vector spaces V = (Vν )ν∈N (assumed throughout the paper to be finite dimensional) and a family of linear maps (ψa )a∈A . The set of all representations of Q having the same family of vector spaces V as support is WV =
Hom(Vt (a) , Vh(a) ).
a∈A
The group
ν∈N
GL(Vν ) acts naturally on the space WV , by
(gν )ν · (ψa )a := (gh(a) ◦ ψa ◦ gt−(a1) )a . ∗
Tel.: +40 213143508. E-mail addresses: [email protected], [email protected].
http://dx.doi.org/10.1016/j.sysconle.2014.09.001 0167-6911/© 2014 Elsevier B.V. All rights reserved.
However, in some situations considering a smaller symmetry group may provide geometric meaningful constructions (see e.g.
2
M.-S. Stupariu / Systems & Control Letters 73 (2014) 1–5
Example 2 below or the examples in Section 3.2). Let therefore M be a non-empty subset of nodes ∅ ̸= M ⊆ N and denote by GM the product ν∈M GL(Vν ). According to [4], the pair (Q , M ) is a quiver with marked vertices and, using the terminology from [13], the problem of constructing orbit spaces for the GM -action on WV is called a quiver factorization problem. Elements of Hermitian type. An element s of the Lie algebra gM of GM is of Hermitian type if there exist a compact subgroup KM of GM and a Cartan-type decomposition gM = kM ⊕ pM (see e.g. [14] for the terminology) such that s ∈ pM . The case K = C was discussed in [11, Definition 3.1] and, in this case, the equality pM = ikM holds—as in the decomposition gl(n, C) = u(n) ⊕ iu(n) of the vector space gl(n, C) as a direct sum between the vector space of unitary matrices and that one of Hermitian matrices. In the case K = R, there is no longer such a relationship between kM and pM —as in the decomposition gl(n, R) = o(n) ⊕ sym(n) of the vector space gl(n, R) as a direct sum between the vector space of orthogonal matrices and that one of symmetric matrices. The main point is that, in both cases, if s is of Hermitian type, then s has only real eigenvalues and it is diagonalizable. Furthermore, according to [11, Definition 3.1], if ρ : GM → GL(W ) is a representation of GM , then ρ∗ (s) has only real eigenvalues (here ρ∗ : gM → gl(W ) is the derivative of ρ ). We will denote by H (GM ) the set of elements of Hermitian type corresponding to the group GM . The following remark is a key ingredient for handling them. It refers only to the case when the group GM has the form GL(V ), but it naturally extends to the general case. Remark 1. Let V be a K-vector space (K ∈ {R, C}) and let s ∈ gl(V ) be an element of Hermitian type. Then s gives rise to the following data.
• A sequence of real numbers λ1 < λ2 < · · · < λq (the eigenvalues of s), together with the corresponding multiplicities mλ1 , . . . , mλq . • A filtration Fs : {0} = V0 ⊂ V1 ⊂ · · · ⊂ Vq = V of V , provided by the eigenspaces of s. We denoted Vi = ⊕λ≤λi V (λ), where V (λ) is the eigenspace corresponding to the eigenvalue λ. It obviously holds the relation dimK V (λ) = mλ .
q
It is worth to notice that tr(s) = i=1 mλi λi , that is the trace of s depends only on the eigenvalues of s and on the dimensions of the vector spaces arising in the filtration Fs . In particular, let g ∈ GL(V ). Define a new element of Hermitian type g ∗ s as follows. Chose a basis B = {b1 , . . . , bn }, where n = dimK V , such that s has diagonal form with respect to B and define g ∗ s such that it has the same diagonal form with respect to the basis {g · b1 , . . . , g · bn }. The eigenvalues of g ∗ s and their multiplicities are equal to those of s, while the filtration of g ∗ s is given by the inclusions {0} ⊂ g (V1 ) ⊂ g (V2 ) ⊂ · · · ⊂ g (Vq ). Although the filtrations associated to s and to (g ∗ s) do not coincide, the dimensions of the corresponding subspaces are equal and, particularly, one has tr s = tr (g ∗ s). Weights. For defining the concept of weight we first consider the case when GM = GL(V ), that is M contains only one node. Let TGM be the one-dimensional K-vector subspace of g∨ M generated by the trace, that is by the functional ξ → tr(ξ ) (g∨ M denotes the dual of the Lie algebra gM ). For any Cartan-type decomposition gM = kM ⊕ pM , by restricting the elements of TGM to pM , one gets real multiples of trace (independently if K = R or K = C and independently on the chosen Cartan-type decomposition). These functionals will be called in the sequel weights. We notice that the convention used here is slightly different from that used in [11,12]: in these papers, where only the complex case is treated, weights do actually belong to k∨ M and the identification pM = ikM makes it possible to relate weights and elements of Hermitian type. The definition proposed above has the advantage to be applicable both in real and complex cases and, via natural identifications, it can be
related to the previous one. Summing up, in the sequel a weight is nothing else but a functional θ of the form θ = α tr(·), α ∈ R, acting on elements of Hermitian type. For an arbitrary set of nodes M, giving a weight θ ∈ TGM for the group GM is equivalent to giving a family of weights (θν )ν∈M . Stability. Throughout this paper we will use the concept of θ (semi)stability, where θ ∈ TGM . This definition extends that one from [11], where only the case K = C is considered. An element w ∈ WV is called θ -semistable if for any s ∈ H (GM ) such that w ∈ WV≤0 (s), it holds ⟨θ , s⟩ ≥ 0. If w is θ -semistable and, fur-
≤0 thermore, for any s ∈ H (GM ) \ h such that w ∈ WV (s) it holds ⟨θ , s⟩ > 0, the element w is called θ -stable. Here h is the Lie al≤0 gebra of the kernel of the representation, while WV (s) is the vec-
tor space spanned by eigenvectors corresponding to nonpositive eigenvalues of ρ∗ (s) (notice that, since s is an element of Hermitian type, as noticed in Remark 1, the eigenvalues of ρ∗ (s) are all real). Let us first notice that stability is invariant under multiplication by a positive constant, i.e. if w is θ -stable, then it is αθ -stable for any positive constant α . We deduce that for each node there exist essentially three different stability conditions, corresponding to the weights θ0 := 0, θ1 := tr, θ−1 := −tr. In conclusion, giving a weight θ ∈ TGM for the group GM is equivalent to giving a family of weights (θεν )ν∈M , where for each marked node ν ∈ M one has εν ∈ {−1, 0, 1}. Furthermore, forany element of Hermitian type s = (sν )ν∈M , one has ⟨θ , s⟩ = ν∈M ⟨θεν , sν ⟩. The sets of all θ (s)s,θ
(semi)stable points will be denoted by WV . We claim that these sets are unions of orbits. Indeed, let us fix an element g ∈ GM of the symmetry group GM and a vector w ∈ WV . By Remark 1, one ≤0 ≤0 has that w ∈ WV (s) if and only if g · w ∈ WV (g ∗ s). On the other hand, again by Remark 1, we deduce that ⟨θ , s⟩ = ⟨θ , g ∗ s⟩. In conclusion, w is θ -(semi)stable if and only if g · w is θ -(semi)stable and this yields the desired conclusion. About the meaning of ‘stability’. The approach used throughout this paper for dealing with (semi)stability is self-contained, uses only linear algebra concepts and addresses both the real and the complex case. It is however, worth to notice that its roots go back to Mumford’s GIT (see [9]). In that framework, as pointed out in [15, Definition 1.25], stability refers to a specific behavior under a group action: stable points have closed orbits and finite stabilizer. In the complex case, the definition proposed above is equivalent to the original GIT definition, as applied in [10] to linear actions. The result relies on the Kempf–Ness theorem in [16] relating stability to the minima of a certain norm; for a comprehensive presentation and for further references, we refer the interested reader to [11]. In the real case, recent results (e.g. [17,18]) indicate that the Kempf–Ness theorem still holds and one expects to have an analogous interpretation of the definition given above. Anyway, the proofs presented in the sequel are independent on these relationships. QFP-quotients. According to the general theory (see e.g. [10,11]), the restriction of the action on the space of θ -semistable orbits provides a quotient with ‘nice’ geometric properties for the GM action. Furthermore, it is easy to see that, up to homeomorphism, such a quotient (called QFP-quotient) depends only on the combinatorial data (Q , M ), the dimension vector d = (dν )ν∈N (where dν = dimK Vν ) of the quiver representation and the weight θ . Some topological properties of a QFP-quotient can be easily described using this data. For instance, a non-empty QFP-quotient is compact if and only if Q contains no marked cycles and no oriented paths with unmarked source and sink (see [4]). We will denote by M (Q , M , d, θ ) the quotient obtained for the ‘standard’ family of vector spaces associated to Q and having dimension vector d, i.e. for any node ν one puts Vν = Kdν . We finish this preparatory subsection by describing two examples. The first one aims to detail the use of the quiver approach in
M.-S. Stupariu / Systems & Control Letters 73 (2014) 1–5
studying the stability of a given linear system. In the second example, one presents a quiver factorization problem as well as the associated QFP-quotient. Example 1. Let (A, B) ∈ K4×4 × K4×2 be the pair given by the matrices
1 0
A= α
β δ
γ
∗ ∗ , ∗ ∗
∗ ∗ ∗ ∗
0 1
1 0 B= 0 0
0 1 . 0 0
In the quiver framework, this pair can be related to the following data: the quiver Q with two vertices, one loop and one arrow, namely
/•d
◦
Q :
,
(the vertex denoted by ‘•’ is marked, and the one denoted by ‘◦’ is not marked), the dimension vector d = (2, 4) (4 corresponds to the marked vertex), the family of vector spaces V = (K2 , K4 ) and the natural GL(K4 )-action on the vector space WV = EndK (K4 ) × HomK (K2 , K4 ). In the sequel we will briefly investigate the θ (semi)-stability of the pair (A, B) for the weight θ := θ−1 acting by ⟨θ, ·⟩ = −tr(·). Let us first notice that for an element of Hermitian type s ∈ ≤0 H (GL(K4 )) with associated filtration Fs one has (A, B) ∈ WV (s) if and only if the filtration Fs is A-invariant [12, Example 1] and im(B) ⊆ Fs≤0 [12, Example 2] (one denotes by Fs≤0 the subspace of the filtration Fs generated by eigenvectors corresponding to nonpositive eigenvalues). Before proceeding, let us fix some notation. Since an important role in analyzing the θ-stability of the pair (A, B) is played by
the 2 × 2 sub-matrix
α γ
β δ , we will denote it by M. Let also
{e1 , e2 , e3 , e4 } be the canonical basis of K4 . Several cases might occur. (i) If rank (M ) = 0, one takes the element of Hermitian type s having the filtration Fs given by V1 = ⟨e1 , e2 ⟩, V2 = K4 and the eigenvalues λ1 = 0, λ2 = 1 (both with multiplicity 2). Then the plane V1 (and consequently the whole filtration Fs ) is A-invariant ≤0 and im (B) ⊆ Fs≤0 , which means that (A, B) ∈ WV (s). On the other hand, ⟨θ , s⟩ = −tr(s) < 0. Altogether, if rank (M ) = 0, then the pair (A, B) is not θ -(semi)stable. (ii) Let us now consider the case when rank (M ) = 1. We first claim that, instead of A, we may consider a matrix A′ of the form 1 0
0 1
A′ = ′ α 0
β′ 0
∗ ∗ ∗ η′
∗ ∗ , ∗ ∗
with (α , β ) ̸= (0, 0). Indeed, the equality rank (M ) = 1 translates to the fact that the projection of the space A(⟨e1 , e2 ⟩) onto the plane ⟨e3 , e4 ⟩ is one-dimensional; let {b3 } be a basis of this line, such that A(⟨e1 , e2 ⟩) = ⟨e1 , e2 , b3 ⟩. Choose now b4 such that ⟨b3 , b4 ⟩ = ⟨e3 , e4 ⟩. Changing the canonical basis with the basis {e1 , e2 , b3 , b4 } is equivalent to finding an element g ∈ GL(K4 ) such that (A, B) = g ·(A′ , B)· g −1 , where A′ has the form indicated above. But stability is a property of the orbits, so we may replace A by A′ , as claimed. Two sub-cases need now to be considered. (ii,a) If η′ = 0, the reasoning is similar to case (i). One considers the element of Hermitian type s having the filtration Fs given by V1 = ⟨e1 , e2 , b3 ⟩, V2 = K4 and the eigenvalues λ1 = 0 (with multiplicity 3), λ2 = 1 (with multiplicity 1). Then the space V1 (and consequently the whole filtration Fs ) is A′ -invariant and im (B) ⊆ Fs≤0 , which means that (A′ , B) ∈ WV≤0 (s). On the other hand, ⟨θ, s⟩ = −tr(s) < 0. Thus, the pair (A′ , B) (and consequently the pair (A, B)) is not θ -(semi)stable. ′
′
3
(ii,b) If η′ ̸= 0, the situation completely changes. Assume that U ⊂ K4 is an arbitrary A′ -invariant subspace such that im (B) ⊂ U. Since im (B) = ⟨e1 , e2 ⟩ ⊂ U and since (α ′ , β ′ ) ̸= (0, 0) and η′ ̸= 0, by using the A′ -invariance, it follows that b3 , b4 ∈ U and, eventually, U = K4 . This means that for any element of Hermitian type s ≤0 such that (A′ , B) ∈ WV (s) one has Fs≤0 = K4 ; in particular all its eigenvalues are non-positive. Thus, ⟨θ , s⟩ ≥ 0 and equality holds if and only if s = 0. In conclusion, the pair (A, B) is θ -(semi)stable. (iii) Assume finally that rank (M ) = 2. By applying directly to the matrix A a reasoning analogous to the case (ii,b) one proves that the pair (A, B) is θ -(semi)stable. It is worth to notice that the cases detailed above can be easily related to the well-known description of stability that uses the controllability matrix C (A, B) = [B AB A2 B A3 B]. Thus, the case (i) corresponds to the relation rank(C (A, B)) = 2; the case (ii,a) holds if and only if rank(C (A, B)) = 3 and the cases (ii,b), (iii) are related to the fulfillment of the full-rank condition rank(C (A, B)) = 4. In conclusion, the quiver approach is consistent with already established methods and provides a geometric insight to well-known algebraic approaches, such as the rank-based one. Example 2. Consider the quiver Q with two vertices and one arrow Q :
•
/◦
(again the vertex denoted by ‘•’ is the marked one). Take the dimension vector d = (r , n) (r ≤ n and r corresponds to the marked vertex). Actually, this means that we consider the family of vector spaces V = (Kr , Kn ) and the natural GM = GL(Kr )-action on the vector space WV = HomK (Kr , Kn ). Fix the weight θ1 , acting by ⟨θ1 , s⟩ = tr(s). We claim that ψ ∈ WV is θ1 -semistable if and only if rank ψ = r (or, equivalently, ker ψ = {0}). Let us briefly sketch the proof of this statement. First of all, according to [12, Example 3], the following equivalence holds: ψ ∈ ≤0 WV (s) if and only if ker ψ ⊇ Fs
Contents lists available at ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
A quiver approach to studying orbit spaces of linear systems Mihai-Sorin Stupariu ∗ University of Bucharest, Faculty of Mathematics and Computer Science, Academiei Str. 14, 010014 Bucharest, Romania
article
info
Article history: Received 26 September 2012 Received in revised form 2 August 2014 Accepted 9 September 2014 Available online 29 September 2014 Keywords: Linear dynamical system Quiver representation Stability Compactification
abstract Orbit spaces associated to linear actions are of particular interest in control theory. Their geometrical properties can be naturally investigated by using the representations of quivers as an abstract framework. The aim of the paper is to bring into attention an application of this approach and to show how the use of quivers makes it easy handling concepts arising in control theory. Specifically, the natural duality between controllable and observable systems, as well as the construction of compactifications for the associated orbit spaces is interpreted in terms of opposite quivers. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The use of techniques from algebraic geometry has been proved to be a very fruitful approach in solving problems arising in control theory. In the late ’70s, several pioneering works (e.g. those by Hazewinkel and Kalman [1] or by Byrnes and Hurt [2]) highlighted some meaningful algebro-geometric tools in constructing moduli spaces for linear dynamical systems; an excellent synthesis of these techniques, as well as a source for relevant references is the monograph by Tannenbaum [3]. Recent developments (see [4,5]) brought into attention the opportunity of using (partial) representations of quivers as framework for constructing moduli spaces for certain classes of linear systems or for studying their topological properties. Moreover, it was proved that the compactification proposed by Helmke and Shayman in [6] for the orbit space of controllable state space systems and that one proposed by Helmke (see [7,8]) for the space of rational transfer functions can be interpreted in terms of orbit spaces associated to quiver representations. Actually, these orbit spaces are constructed by adapting ideas from Mumford’s Geometric Invariant Theory (see [9]) to the case of linear actions, particularly to the case of representations of quivers (see [10,11]). Furthermore, it was shown in [12] that, by appropriately adapting the concepts and the definitions, all these results can be obtained by completely remaining within the framework of linear algebra. The aim of this note is to bring into attention other applications of the quiver approach for problems arising in control theory. The
paper is organized as follows. Section 2 is dedicated to revising the main concepts referring to stability, respectively to the construction of orbit spaces associated to quiver factorization problems. Section 3 is divided in two parts. Section 3.1 deals with results that are abstract in nature, referring to the natural relationship between stability concepts (Proposition 1), respectively between orbit spaces (Theorem 1) associated to opposite quivers. Section 3.2 is oriented towards applications. The well-known duality between controllability and observability is naturally interpreted in Proposition 2 in terms of quiver representations. The interpretation is extended in Theorem 2 for the compactifications of the associated orbit spaces of controllable and observable linear systems. 2. Orbit spaces associated to quiver representations Quiver factorization problems. Let Q = (N , A, h, t ) be a quiver (here N and A are the sets of nodes, respectively of arrows, while h, t : A → N are the head, respectively the tail maps). A representation r = (V, ψ) of Q over the field K (K ∈ {R, C}) is a pair consisting of a family of K-vector spaces V = (Vν )ν∈N (assumed throughout the paper to be finite dimensional) and a family of linear maps (ψa )a∈A . The set of all representations of Q having the same family of vector spaces V as support is WV =
Hom(Vt (a) , Vh(a) ).
a∈A
The group
ν∈N
GL(Vν ) acts naturally on the space WV , by
(gν )ν · (ψa )a := (gh(a) ◦ ψa ◦ gt−(a1) )a . ∗
Tel.: +40 213143508. E-mail addresses: [email protected], [email protected].
http://dx.doi.org/10.1016/j.sysconle.2014.09.001 0167-6911/© 2014 Elsevier B.V. All rights reserved.
However, in some situations considering a smaller symmetry group may provide geometric meaningful constructions (see e.g.
2
M.-S. Stupariu / Systems & Control Letters 73 (2014) 1–5
Example 2 below or the examples in Section 3.2). Let therefore M be a non-empty subset of nodes ∅ ̸= M ⊆ N and denote by GM the product ν∈M GL(Vν ). According to [4], the pair (Q , M ) is a quiver with marked vertices and, using the terminology from [13], the problem of constructing orbit spaces for the GM -action on WV is called a quiver factorization problem. Elements of Hermitian type. An element s of the Lie algebra gM of GM is of Hermitian type if there exist a compact subgroup KM of GM and a Cartan-type decomposition gM = kM ⊕ pM (see e.g. [14] for the terminology) such that s ∈ pM . The case K = C was discussed in [11, Definition 3.1] and, in this case, the equality pM = ikM holds—as in the decomposition gl(n, C) = u(n) ⊕ iu(n) of the vector space gl(n, C) as a direct sum between the vector space of unitary matrices and that one of Hermitian matrices. In the case K = R, there is no longer such a relationship between kM and pM —as in the decomposition gl(n, R) = o(n) ⊕ sym(n) of the vector space gl(n, R) as a direct sum between the vector space of orthogonal matrices and that one of symmetric matrices. The main point is that, in both cases, if s is of Hermitian type, then s has only real eigenvalues and it is diagonalizable. Furthermore, according to [11, Definition 3.1], if ρ : GM → GL(W ) is a representation of GM , then ρ∗ (s) has only real eigenvalues (here ρ∗ : gM → gl(W ) is the derivative of ρ ). We will denote by H (GM ) the set of elements of Hermitian type corresponding to the group GM . The following remark is a key ingredient for handling them. It refers only to the case when the group GM has the form GL(V ), but it naturally extends to the general case. Remark 1. Let V be a K-vector space (K ∈ {R, C}) and let s ∈ gl(V ) be an element of Hermitian type. Then s gives rise to the following data.
• A sequence of real numbers λ1 < λ2 < · · · < λq (the eigenvalues of s), together with the corresponding multiplicities mλ1 , . . . , mλq . • A filtration Fs : {0} = V0 ⊂ V1 ⊂ · · · ⊂ Vq = V of V , provided by the eigenspaces of s. We denoted Vi = ⊕λ≤λi V (λ), where V (λ) is the eigenspace corresponding to the eigenvalue λ. It obviously holds the relation dimK V (λ) = mλ .
q
It is worth to notice that tr(s) = i=1 mλi λi , that is the trace of s depends only on the eigenvalues of s and on the dimensions of the vector spaces arising in the filtration Fs . In particular, let g ∈ GL(V ). Define a new element of Hermitian type g ∗ s as follows. Chose a basis B = {b1 , . . . , bn }, where n = dimK V , such that s has diagonal form with respect to B and define g ∗ s such that it has the same diagonal form with respect to the basis {g · b1 , . . . , g · bn }. The eigenvalues of g ∗ s and their multiplicities are equal to those of s, while the filtration of g ∗ s is given by the inclusions {0} ⊂ g (V1 ) ⊂ g (V2 ) ⊂ · · · ⊂ g (Vq ). Although the filtrations associated to s and to (g ∗ s) do not coincide, the dimensions of the corresponding subspaces are equal and, particularly, one has tr s = tr (g ∗ s). Weights. For defining the concept of weight we first consider the case when GM = GL(V ), that is M contains only one node. Let TGM be the one-dimensional K-vector subspace of g∨ M generated by the trace, that is by the functional ξ → tr(ξ ) (g∨ M denotes the dual of the Lie algebra gM ). For any Cartan-type decomposition gM = kM ⊕ pM , by restricting the elements of TGM to pM , one gets real multiples of trace (independently if K = R or K = C and independently on the chosen Cartan-type decomposition). These functionals will be called in the sequel weights. We notice that the convention used here is slightly different from that used in [11,12]: in these papers, where only the complex case is treated, weights do actually belong to k∨ M and the identification pM = ikM makes it possible to relate weights and elements of Hermitian type. The definition proposed above has the advantage to be applicable both in real and complex cases and, via natural identifications, it can be
related to the previous one. Summing up, in the sequel a weight is nothing else but a functional θ of the form θ = α tr(·), α ∈ R, acting on elements of Hermitian type. For an arbitrary set of nodes M, giving a weight θ ∈ TGM for the group GM is equivalent to giving a family of weights (θν )ν∈M . Stability. Throughout this paper we will use the concept of θ (semi)stability, where θ ∈ TGM . This definition extends that one from [11], where only the case K = C is considered. An element w ∈ WV is called θ -semistable if for any s ∈ H (GM ) such that w ∈ WV≤0 (s), it holds ⟨θ , s⟩ ≥ 0. If w is θ -semistable and, fur-
≤0 thermore, for any s ∈ H (GM ) \ h such that w ∈ WV (s) it holds ⟨θ , s⟩ > 0, the element w is called θ -stable. Here h is the Lie al≤0 gebra of the kernel of the representation, while WV (s) is the vec-
tor space spanned by eigenvectors corresponding to nonpositive eigenvalues of ρ∗ (s) (notice that, since s is an element of Hermitian type, as noticed in Remark 1, the eigenvalues of ρ∗ (s) are all real). Let us first notice that stability is invariant under multiplication by a positive constant, i.e. if w is θ -stable, then it is αθ -stable for any positive constant α . We deduce that for each node there exist essentially three different stability conditions, corresponding to the weights θ0 := 0, θ1 := tr, θ−1 := −tr. In conclusion, giving a weight θ ∈ TGM for the group GM is equivalent to giving a family of weights (θεν )ν∈M , where for each marked node ν ∈ M one has εν ∈ {−1, 0, 1}. Furthermore, forany element of Hermitian type s = (sν )ν∈M , one has ⟨θ , s⟩ = ν∈M ⟨θεν , sν ⟩. The sets of all θ (s)s,θ
(semi)stable points will be denoted by WV . We claim that these sets are unions of orbits. Indeed, let us fix an element g ∈ GM of the symmetry group GM and a vector w ∈ WV . By Remark 1, one ≤0 ≤0 has that w ∈ WV (s) if and only if g · w ∈ WV (g ∗ s). On the other hand, again by Remark 1, we deduce that ⟨θ , s⟩ = ⟨θ , g ∗ s⟩. In conclusion, w is θ -(semi)stable if and only if g · w is θ -(semi)stable and this yields the desired conclusion. About the meaning of ‘stability’. The approach used throughout this paper for dealing with (semi)stability is self-contained, uses only linear algebra concepts and addresses both the real and the complex case. It is however, worth to notice that its roots go back to Mumford’s GIT (see [9]). In that framework, as pointed out in [15, Definition 1.25], stability refers to a specific behavior under a group action: stable points have closed orbits and finite stabilizer. In the complex case, the definition proposed above is equivalent to the original GIT definition, as applied in [10] to linear actions. The result relies on the Kempf–Ness theorem in [16] relating stability to the minima of a certain norm; for a comprehensive presentation and for further references, we refer the interested reader to [11]. In the real case, recent results (e.g. [17,18]) indicate that the Kempf–Ness theorem still holds and one expects to have an analogous interpretation of the definition given above. Anyway, the proofs presented in the sequel are independent on these relationships. QFP-quotients. According to the general theory (see e.g. [10,11]), the restriction of the action on the space of θ -semistable orbits provides a quotient with ‘nice’ geometric properties for the GM action. Furthermore, it is easy to see that, up to homeomorphism, such a quotient (called QFP-quotient) depends only on the combinatorial data (Q , M ), the dimension vector d = (dν )ν∈N (where dν = dimK Vν ) of the quiver representation and the weight θ . Some topological properties of a QFP-quotient can be easily described using this data. For instance, a non-empty QFP-quotient is compact if and only if Q contains no marked cycles and no oriented paths with unmarked source and sink (see [4]). We will denote by M (Q , M , d, θ ) the quotient obtained for the ‘standard’ family of vector spaces associated to Q and having dimension vector d, i.e. for any node ν one puts Vν = Kdν . We finish this preparatory subsection by describing two examples. The first one aims to detail the use of the quiver approach in
M.-S. Stupariu / Systems & Control Letters 73 (2014) 1–5
studying the stability of a given linear system. In the second example, one presents a quiver factorization problem as well as the associated QFP-quotient. Example 1. Let (A, B) ∈ K4×4 × K4×2 be the pair given by the matrices
1 0
A= α
β δ
γ
∗ ∗ , ∗ ∗
∗ ∗ ∗ ∗
0 1
1 0 B= 0 0
0 1 . 0 0
In the quiver framework, this pair can be related to the following data: the quiver Q with two vertices, one loop and one arrow, namely
/•d
◦
Q :
,
(the vertex denoted by ‘•’ is marked, and the one denoted by ‘◦’ is not marked), the dimension vector d = (2, 4) (4 corresponds to the marked vertex), the family of vector spaces V = (K2 , K4 ) and the natural GL(K4 )-action on the vector space WV = EndK (K4 ) × HomK (K2 , K4 ). In the sequel we will briefly investigate the θ (semi)-stability of the pair (A, B) for the weight θ := θ−1 acting by ⟨θ, ·⟩ = −tr(·). Let us first notice that for an element of Hermitian type s ∈ ≤0 H (GL(K4 )) with associated filtration Fs one has (A, B) ∈ WV (s) if and only if the filtration Fs is A-invariant [12, Example 1] and im(B) ⊆ Fs≤0 [12, Example 2] (one denotes by Fs≤0 the subspace of the filtration Fs generated by eigenvectors corresponding to nonpositive eigenvalues). Before proceeding, let us fix some notation. Since an important role in analyzing the θ-stability of the pair (A, B) is played by
the 2 × 2 sub-matrix
α γ
β δ , we will denote it by M. Let also
{e1 , e2 , e3 , e4 } be the canonical basis of K4 . Several cases might occur. (i) If rank (M ) = 0, one takes the element of Hermitian type s having the filtration Fs given by V1 = ⟨e1 , e2 ⟩, V2 = K4 and the eigenvalues λ1 = 0, λ2 = 1 (both with multiplicity 2). Then the plane V1 (and consequently the whole filtration Fs ) is A-invariant ≤0 and im (B) ⊆ Fs≤0 , which means that (A, B) ∈ WV (s). On the other hand, ⟨θ , s⟩ = −tr(s) < 0. Altogether, if rank (M ) = 0, then the pair (A, B) is not θ -(semi)stable. (ii) Let us now consider the case when rank (M ) = 1. We first claim that, instead of A, we may consider a matrix A′ of the form 1 0
0 1
A′ = ′ α 0
β′ 0
∗ ∗ ∗ η′
∗ ∗ , ∗ ∗
with (α , β ) ̸= (0, 0). Indeed, the equality rank (M ) = 1 translates to the fact that the projection of the space A(⟨e1 , e2 ⟩) onto the plane ⟨e3 , e4 ⟩ is one-dimensional; let {b3 } be a basis of this line, such that A(⟨e1 , e2 ⟩) = ⟨e1 , e2 , b3 ⟩. Choose now b4 such that ⟨b3 , b4 ⟩ = ⟨e3 , e4 ⟩. Changing the canonical basis with the basis {e1 , e2 , b3 , b4 } is equivalent to finding an element g ∈ GL(K4 ) such that (A, B) = g ·(A′ , B)· g −1 , where A′ has the form indicated above. But stability is a property of the orbits, so we may replace A by A′ , as claimed. Two sub-cases need now to be considered. (ii,a) If η′ = 0, the reasoning is similar to case (i). One considers the element of Hermitian type s having the filtration Fs given by V1 = ⟨e1 , e2 , b3 ⟩, V2 = K4 and the eigenvalues λ1 = 0 (with multiplicity 3), λ2 = 1 (with multiplicity 1). Then the space V1 (and consequently the whole filtration Fs ) is A′ -invariant and im (B) ⊆ Fs≤0 , which means that (A′ , B) ∈ WV≤0 (s). On the other hand, ⟨θ, s⟩ = −tr(s) < 0. Thus, the pair (A′ , B) (and consequently the pair (A, B)) is not θ -(semi)stable. ′
′
3
(ii,b) If η′ ̸= 0, the situation completely changes. Assume that U ⊂ K4 is an arbitrary A′ -invariant subspace such that im (B) ⊂ U. Since im (B) = ⟨e1 , e2 ⟩ ⊂ U and since (α ′ , β ′ ) ̸= (0, 0) and η′ ̸= 0, by using the A′ -invariance, it follows that b3 , b4 ∈ U and, eventually, U = K4 . This means that for any element of Hermitian type s ≤0 such that (A′ , B) ∈ WV (s) one has Fs≤0 = K4 ; in particular all its eigenvalues are non-positive. Thus, ⟨θ , s⟩ ≥ 0 and equality holds if and only if s = 0. In conclusion, the pair (A, B) is θ -(semi)stable. (iii) Assume finally that rank (M ) = 2. By applying directly to the matrix A a reasoning analogous to the case (ii,b) one proves that the pair (A, B) is θ -(semi)stable. It is worth to notice that the cases detailed above can be easily related to the well-known description of stability that uses the controllability matrix C (A, B) = [B AB A2 B A3 B]. Thus, the case (i) corresponds to the relation rank(C (A, B)) = 2; the case (ii,a) holds if and only if rank(C (A, B)) = 3 and the cases (ii,b), (iii) are related to the fulfillment of the full-rank condition rank(C (A, B)) = 4. In conclusion, the quiver approach is consistent with already established methods and provides a geometric insight to well-known algebraic approaches, such as the rank-based one. Example 2. Consider the quiver Q with two vertices and one arrow Q :
•
/◦
(again the vertex denoted by ‘•’ is the marked one). Take the dimension vector d = (r , n) (r ≤ n and r corresponds to the marked vertex). Actually, this means that we consider the family of vector spaces V = (Kr , Kn ) and the natural GM = GL(Kr )-action on the vector space WV = HomK (Kr , Kn ). Fix the weight θ1 , acting by ⟨θ1 , s⟩ = tr(s). We claim that ψ ∈ WV is θ1 -semistable if and only if rank ψ = r (or, equivalently, ker ψ = {0}). Let us briefly sketch the proof of this statement. First of all, according to [12, Example 3], the following equivalence holds: ψ ∈ ≤0 WV (s) if and only if ker ψ ⊇ Fs
