The Case of Morphisms of Representation Corresponding to A-Module Holomorphisms. Part 2

1-s2.0-S0001870811003495-fx003

Representations of a quiver can be interpreted as modules over a non-commutative algebra A(Q) whose elements are linear combinations of paths in Q.

Let Q be a quiver. A non-trivial path in Q is a sequence of arrows am…a0 such that h(ai−1) = t(ai) for i = 1,…, m:

Untitled

The path is p = am…a0. Writing t(p) = t(a0) and saying that p starts at t(a0) and, similarly, writing h(p) = h(am) and saying that p finishes at h(am). For each vertex i ∈ Q0, we denote by ei the trivial path which starts and finishes at i. Two paths p and q are compatible if t(p) = h(q) and, in this case, the composition pq can defined by juxtaposition of p and q. The length l(p) of a path is the number of arrows it contains; in particular, a trivial path has length zero.

The path algebra A(Q) of a quiver Q is the complex vector space with basis consisting of all paths in Q, equipped with the multiplication in which the product pq of paths p and q is defined to be the composition pq if t(p) = h(q), and 0 otherwise. Composition of paths is non-commutative; in most cases, if p and q can be composed one way, then they cannot be composed the other way, and even if they can, usually pq ≠ qp. Hence the path algebra is indeed non-commutative.

Let us define Al ⊂ A to be the subspace spanned by paths of length l. Then A = ⊕l≥0Al is a graded C-algebra. The subring A0 ⊂ A spanned by the trivial paths ei is a semisimple ring in which the elements ei are orthogonal idempotents, in other words eiej = ei when i = j, and 0 otherwise. The algebra A is finite-dimensional precisely if Q has no directed cycles.

The category of finite-dimensional representations of a quiver Q is isomorphic to the category of finitely generated left A(Q)-modules. Let (V, φ) be a representation of Q. We can then define a left module V over the algebra A = A(Q) as follows: as a vector space it is

V = ⊕i∈Q0 Vi

and the A-module structure is extended linearly from

eiv = v, v ∈ Mi

= 0, v ∈ Mj for j ≠ i

for i ∈ Qand

av = φa(vt(a)), v ∈ Vt(a)

= 0, v ∈ Vj for j ≠ t(a)

for a ∈ Q1. This construction can be inverted as follows: given a left A-module V, we set Vi = eiV for i ∈ Q0 and define the map φa: Vt(a) → Vh(a) by v ↦ a(v). Morphisms of representations of (Q, V) correspond to A-module homomorphisms.

Phenomenological Model for Stock Portfolios. Note Quote.

hamiltonian-cycles

The data analysis and modeling of financial markets have been hot research subjects for physicists as well as economists and mathematicians in recent years. The non-Gaussian property of the probability distributions of price changes, in stock markets and foreign exchange markets, has been one of main problems in this field. From the analysis of the high-frequency time series of market indices, a universal property was found in the probability distributions. The central part of the distribution agrees well with Levy stable distribution, while the tail deviate from it and shows another power law asymptotic behavior. In probability theory, a distribution or a random variable is said to be stable if a linear combination of two independent copies of a random sample has the same distributionup to location and scale parameters. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. The scaling property on the sampling time interval of data is also well described by the crossover of the two distributions. Several stochastic models of the fluctuation dynamics of stock prices are proposed, which reproduce power law behavior of the probability density. The auto-correlation of financial time series is also an important problem for markets. There is no time correlation of price changes in daily scale, while from more detailed data analysis an exponential decay with a characteristic time τ = 4 minutes was found. The fact that there is no auto-correlation in daily scale is not equal to the independence of the time series in the scale. In fact there is auto-correlation of volatility (absolute value of price change) with a power law tail.

Portfolio is a set of stock issues. The Hamiltonian of the system is introduced and is expressed by spin-spin interactions as in spin glass models of disordered magnetic systems. The interaction coefficients between two stocks are phenomenologically determined by empirical data. They are derived from the covariance of sequences of up and down spins using fluctuation-response theorem. We start with the Hamiltonian expression of our system that contain N stock issues. It is a function of the configuration S consisting of N coded price changes Si (i = 1, 2, …, N ) at equal trading time. The interaction coefficients are also dynamical variables, because the interactions between stocks are thought to change from time to time. We divide a coefficient into two parts, the constant part Jij, which will be phenomenologically determined later, and the dynamical part δJij. The Hamiltonian including the interaction with external fields hi (i = 1,2,…,N) is defined as

H [S, δ, J, h] = ∑<i,j>[δJij2/2Δij – (Jij + δJij)SiSj] – ∑ihiSi —– (1)

The summation is taken over all pairs of stock issues. This form of Hamiltonian is that of annealed spin glass. The fluctuations δJij are assumed to distribute according to Gaussian function. The main part of statistical physics is the evaluation of partition function that is given by the following functional in this case

Z[h] = ∑{si} ∫∏<i,j> dδJij/√(2πΔij) e-H [S, δ, J, h] —– (2)

The integration over the variables δJij is easily performed and gives

Z[h] = A {si} e-Heff[S, h] —– (3)

Here the effective Hamiltonian Heff[S,h] is defined as

Heff[S, h] = – <i,j>JijSiSj – ∑ihiSi —– (4)

and A = e(1/2 ∆ij) is just a normalization factor which is irrelevant to the following step. This form of Hamiltonian with constant Jij is that of quenched spin glass.

The constant interaction coefficients Jij are still undetermined. We use fluctuation-response theorem which relates the susceptibility χij with the covariance Cij between dynamical variables in order to determine those constants, which is given by the equation,

χij = ∂mi/∂hj |h=0 = Cij —– (5)

Thouless-Anderson-Palmer (TAP) equation for quenched spin glass is

mi =tanh(∑jJijmj + hi – ∑jJij2(1 – mj2)mi —– (6)

Equation (5) and the linear approximation of the equation (6) yield the equation

kik − Jik)Ckj = δij —– (7)

Interpreting Cij as the time average of empirical data over a observation time rather than ensemble average, the constant interaction coefficients Jij is phenomenologically determined by the equation (7).

The energy spectra of the system, simply the portfolio energy, is defined as the eigenvalues of the Hamiltonian Heff[S,0]. The probability density of the portfolio energy can be obtained in two ways. We can calculate the probability density from data by the equation

p(E) ΔE = p(E – ΔE/2 ≤ E ≤ E + ΔE/2) —– (8)

This is a fully consistent phenomenological model for stock portfolios, which is expressed by the effective Hamiltonian (4). This model will be also applicable to other financial markets that show collective time evolutions, e.g., foreign exchange market, options markets, inter-market interactions.