# Indecomposable Objects – Part 1

An object X in a category C with an initial object is called indecomposable if X is not the initial object and X is not isomorphic to a coproduct of two noninitial objects. A group G is called indecomposable if it cannot be expressed as the internal direct product of two proper normal subgroups of G. This is equivalent to saying that G is not isomorphic to the direct product of two nontrivial groups.

A quiver Q is a directed graph, specified by a set of vertices Q0, a set of arrows Q1, and head and tail maps

h, t : Q1 → Q0

We always assume that Q is finite, i.e., the sets Q0 and Q1 are finite. A (complex) representation of a quiver Q consists of complex vector spaces Vi for i ∈ Qand linear maps

φa : Vt(a) → Vh(a)

for a ∈ Q1. A morphism between such representations (V, φ) and (W, ψ) is a collection of linear maps fi : Vi → Wi for i ∈ Q0 such that the diagram commutes ∀ a ∈ Q1. A representation of Q is finite-dimensional if each vector space Vi is. The dimension vector of such a representation is just the tuple of non-negative integers (dim Vi)i∈Q0.

Rep(Q) is the category of finite-dimensional representations of Q. This category is additive; we can add morphisms by adding the corresponding linear maps fi, the trivial representation in which each Vi = 0 is a zero object, and the direct sum of two representations is obtained by taking the direct sums of the vector spaces associated to each vertex. If Q is the one-arrow quiver, • → •, then the classification of indecomposable objects of Rep(Q), yields the objects E ∈ Rep(Q) which do not have a non-trivial direct sum decomposition E = A ⊕ B. An object of Rep(Q) is just a linear map of finite-dimensional vector spaces f: V1 → V2. If W = im(f) is a nonzero proper subspace of V2, then the splitting V2 = U ⊕ W, and the corresponding object of Rep(Q) splits as a direct sum of the two representations

V1 →ƒ W and 0 → W

Thus if an object f: V1 → V2 of Rep(Q) is indecomposable, the map f must be surjective. Similarly, if ƒ is nonzero, then it must also be injective. Continuing in this way, one sees that Rep(Q) has exactly three indecomposable objects up to isomorphism:

C → 0, 0 → C, C →id C

Every other object of Rep(Q) is a direct sum of copies of these basic representations.

# Conjuncted: Avarice

Greed followed by avarice….We consider the variation in which events occur at a rate equal to the difference in capital of the two traders. That is, an individual is more likely to take capital from a much poorer person rather than from someone of slightly less wealth. For this “avaricious” exchange, the corresponding rate equations are

dck/dt = ck-1j=1k-1(k – 1 – j)cj + ck+1j=k+1(j – k – 1)cj – ckj=1|k – j|cj —– (1)

while the total density obeys,

dN/dt = -c1(1 – N) —– (2)

under the assumption that the total wealth density is set equal to one, ∑kck = 1

These equations can be solved by again applying scaling. For this purpose, it is first expedient to rewrite the rate equation as,

dck/dt = (ck-1 – ck)∑j=1k-1(k – j)cj – ck-1j=1k-1cj + (ck+1 – ck)∑j=k+1(j – k)cj – ck+1j=k+1cj —– (3)

taking the continuum limits

∂c/∂t = ∂c/∂k – N∂/∂k(kc) —— (3)

We now substitute the scaling ansatz,

ck(t) ≅ N2C(x), with x = kN to yield

C(0)[2C + xC′] = (x − 1)C′ + C —– (4)

and

dN/dt = -C(0)N2 —– (5)

Solving the above equations gives N ≅ [C(0)t]−1 and

C(x) = (1 + μ)(1 + μx)−2−1/μ —– (6)

with μ = C(0) − 1. The scaling approach has thus found a family of solutions which are parameterized by μ, and additional information is needed to determine which of these solutions is appropriate for our system. For this purpose, note that equation (6) exhibits different behaviors depending on the sign of μ. When μ > 0, there is an extended non-universal power-law distribution, while for μ = 0 the solution is the pure exponential, C(x) = e−x. These solutions may be rejected because the wealth distribution cannot extend over an unbounded domain if the initial wealth extends over a finite range.

The accessible solutions therefore correspond to −1 < μ < 0, where the distribution is compact and finite, with C(x) ≡ 0 for x ≥ xf = −μ−1. To determine the true solution, let us re-examine the continuum form of the rate equation, equation (3). From naive power counting, the first two terms are asymptotically dominant and they give a propagating front with kf exactly equal to t. Consequently, the scaled location of the front is given by xf = Nkf. Now the result N ≃ [C(0)t]−1 gives xf = 1/C(0). Comparing this expression with the corresponding value from the scaling approach, xf = [1 − C(0)]−1, selects the value C(0) = 1/2. Remarkably, this scaling solution coincides with the Fermi distribution that found for the case of constant interaction rate. Finally, in terms of the unscaled variables k and t, the wealth distribution is

ck(t) = 4/t2, k < t

= 0, k ≥ 0 —– (7)

This discontinuity is smoothed out by diffusive spreading. Another interesting feature is that if the interaction rate is sufficiently greedy, “gelation” occurs, whereby a finite fraction of the total capital is possessed by a single individual. For interaction rates, or kernels K(j, k) between individuals of capital j and k which do not give rise to gelation, the total density typically varies as a power law in time, while for gelling kernels N(t) goes to zero at some finite time. At the border between these regimes N(t) typically decays exponentially in time. We seek a similar transition in behavior for the capital exchange model by considering the rate equation for the density

dN/dt = -c1k=1k(1, k)ck —– (8)

For the family of kernels with K(1, k) ∼ kν as k → ∞, substitution of the scaling ansatz gives N ̇ ∼ −N3−ν. Thus N(t) exhibits a power-law behavior N ∼ t1/(2−ν) for ν < 2 and an exponential behavior for ν = 2. Thus gelation should arise for ν > 2.

# Tantric Reality आत्मा त्वं गिरिजा मतिः सहचराः प्राणाः शरीरं गृहं पूजा ते विषयोपभोगरचना निद्रा समाधिस्थितिः।
सञ्चारः पदयोः प्रदक्षिणविधिः स्तोत्राणि सर्वा गिरो यद्यत्कर्म करोमि तत्तदखिलं शम्भो तवाराधनम्॥

Ātmā tvaṃ Girijā matiḥ sahacarāḥ prāṇāḥ śarīraṃ gṛham
Pūjā te viṣayopabhoga-racanā nidrā samādhi-sthitiḥ |

You (tvam) (are) the Self (ātmā) and Girijā –an epithet of Pārvatī, Śiva’s wife, meaning “mountain-born”– (girijā) (is) the intelligence (matiḥ). The vital energies (prāṇāḥ) (are Your)companions (sahacarāḥ). The body (śarīram) (is Your) house (gṛham). Worship (pūjā) of You (te) is prepared (racanā) with the objects (viṣaya) (known as sensual) enjoyments (upabhoga). Sleep (nidrā) (is Your) state (sthitiḥ) of Samādhi –i.e. perfect concentration or absorption– (samādhi). (My) wandering (sañcāraḥ) (is) the ceremony (vidhiḥ) of circumambulation from left to right (pradakṣiṇa) of (Your) feet (padayoḥ) –this act is generally done as a token of respect–. All (sarvāḥ) (my) words (giraḥ) (are) hymns of praise (of You) (stotrāṇi). Whatever (yad yad) action (karma) I do (karomi), all (akhilam) that (tad tad) is adoration (ārādhanam) of You (tava), oh Śambhu — an epithet of Śiva meaning “beneficent, benevolent”.

This Self is an embodiment of the Light of Consciousness; it is Śiva, free and autonomous. As an independent play of intense joy, the Divine conceals its own true nature [by manifesting plurality], and may also choose to reveal its fullness once again at any time. All that exists, throughout all time and beyond, is one infinite divine Consciousness, free and blissful, which projects within the field of its awareness a vast multiplicity of apparently differentiated subjects and objects: each object an actualization of a timeless potentiality inherent in the Light of Consciousness, and each subject the same plus a contracted locus of self-awareness. This creation, a divine play, is the result of the natural impulse within Consciousness to express the totality of its self-knowledge in action, an impulse arising from love. The unbounded Light of Consciousness contracts into finite embodied loci of awareness out of its own free will. When those finite subjects then identify with the limited and circumscribed cognitions and circumstances that make up this phase of their existence, instead of identifying with the transindividual overarching pulsation of pure Awareness that is their true nature, they experience what they call “suffering.” To rectify this, some feel an inner urge to take up the path of spiritual gnosis and yogic practice, the purpose of which is to undermine their misidentification and directly reveal within the immediacy of awareness the fact that the divine powers of Consciousness, Bliss, Willing, Knowing, and Acting comprise the totality of individual experience as well – thereby triggering a recognition that one’s real identity is that of the highest Divinity, the Whole in every part. This experiential gnosis is repeated and reinforced through various means until it becomes the nonconceptual ground of every moment of experience, and one’s contracted sense of self and separation from the Whole is finally annihilated in the incandescent radiance of the complete expansion into perfect wholeness. Then one’s perception fully encompasses the reality of a universe dancing ecstatically in the animation of its completely perfect divinity.”

# Organic and the Orgiastic. Cartography of Ground and Groundlessness in Deleuze and Heidegger. Thought of the Day 43.0 In his last hermeneutical Erörterung of Leibniz, The Principle of Ground, Heidegger traces back metaphysics to its epochal destiny in the twofold or duplicity (Zwiefalt) of Being and Thought and thus follows the ground in its self-ungrounding (zugrundegehen). Since the foundation of thought is also the foundation of Being, reason and ground are not equal but belong together (zusammenhören) in the Same as the ungrounded yet historical horizon of the metaphysical destiny of Being: On the one hand we say: Being and ground: the Same. On the other hand we say: Being: the abyss (Ab-Grund). What is important is to think the univocity (Einsinnigkeit) of both Sätze, those Sätze that are no longer Sätze. In Difference and Repetition, similarly, Deleuze tells us that sufficient reason is twisted into the groundless. He confirms that the Fold (Pli) is the differenciator of difference engulfed in groundlessness, always folding, unfolding, refolding: to ground is always to bend, to curve and recurve. He thus concludes:

Sufficient reason or ground is strangely bent: on the one hand, it leans towards what it grounds, towards the forms of representation; on the other hand, it turns and plunges into a groundless beyond the ground which resists all forms and cannot be represented.

Despite the fundamental similarity of their conclusions, however, our short overview of Deleuze’s transformation of the Principle of Sufficient Reason has already indicated that his argumentation is very different from Heideggerian hermeneutics. To ground, Deleuze agrees, is always to ground representation. But we should distinguish between two kinds of representation: organic or finite representation and orgiastic or infinite representation. What unites the classicisms of Kant, Descartes and Aristotle is that representation retains organic form as its principle and the finite as its element. Here the logical principle of identity always precedes ontology, such that the ground as element of difference remains undetermined and in itself. It is only with Hegel and Leibniz that representation discovers the ground as its principle and the infinite as its element. It is precisely the Principle of Sufficient Reason that enables thought to determine difference in itself. The ground is like a single and unique total moment, simultaneously the moment of the evanescence and production of difference, of disappearance and appearance. What the attempts at rendering representation infinite reveal, therefore, is that the ground has not only an Apollinian, orderly side, but also a hidden Dionysian, orgiastic side. Representation discovers within itself the limits of the organized; tumult, restlessness and passion underneath apparent calm. It rediscovers monstrosity.

The question then is how to evaluate this ambiguity that is essential to the ground. For Heidegger, the Zwiefalt is either naively interpreted from the perspective of its concave side, following the path of the history of Western thought as the belonging together of Being and thought in a common ground; or it is meditated from its convex side, excavating it from the history of the forgetting of Being the decline of the Fold (Wegfall der Zwiefalt, Vorenthalt der Zwiefalt) as the pivotal point of the Open in its unfolding and following the path that leads from the ground to the abyss. Instead of this all or nothing approach, Deleuze takes up the question in a Nietzschean, i.e. genealogical fashion. The attempt to represent difference in itself cannot be disconnected from its malediction, i.e. the moral representation of groundlessness as a completely undifferentiated abyss. As Bergson already observed, representational reason poses the problem of the ground in terms of the alternative between order and chaos. This goes in particular for the kind of representational reason that seeks to represent the irrepresentable: Representation, especially when it becomes infinite, is imbued with a presentiment of groundlessness. Because it has become infinite in order to include difference within itself, however, it represents groundlessness as a completely undifferentiated abyss, a universal lack of difference, an indifferent black nothingness. Indeed, if Deleuze is so hostile to Hegel, it is because the latter embodies like no other the ultimate illusion inseparable from the Principle of Sufficient Reason insofar as it grounds representation, namely that groundlessness should lack differences, when in fact it swarms with them.

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# Simultaneity Let us introduce the concept of space using the notion of reflexive action (or reflex action) between two things. Intuitively, a thing x acts on another thing y if the presence of x disturbs the history of y. Events in the real world seem to happen in such a way that it takes some time for the action of x to propagate up to y. This fact can be used to construct a relational theory of space à la Leibniz, that is, by taking space as a set of equitemporal things. It is necessary then to define the relation of simultaneity between states of things.

Let x and y be two things with histories h(xτ) and h(yτ), respectively, and let us suppose that the action of x on y starts at τx0. The history of y will be modified starting from τy0. The proper times are still not related but we can introduce the reflex action to define the notion of simultaneity. The action of y on x, started at τy0, will modify x from τx1 on. The relation “the action of x on y is reflected to x” is the reflex action. Historically, Galileo introduced the reflection of a light pulse on a mirror to measure the speed of light. With this relation we will define the concept of simultaneity of events that happen on different basic things. Besides we have a second important fact: observation and experiment suggest that gravitation, whose source is energy, is a universal interaction, carried by the gravitational field.

Let us now state the above hypothesis axiomatically.

Axiom 1 (Universal interaction): Any pair of basic things interact. This extremely strong axiom states not only that there exist no completely isolated things but that all things are interconnected.

This universal interconnection of things should not be confused with “universal interconnection” claimed by several mystical schools. The present interconnection is possible only through physical agents, with no mystical content. It is possible to model two noninteracting things in Minkowski space assuming they are accelerated during an infinite proper time. It is easy to see that an infinite energy is necessary to keep a constant acceleration, so the model does not represent real things, with limited energy supply.

Now consider the time interval (τx1 − τx0). Special Relativity suggests that it is nonzero, since any action propagates with a finite speed. We then state

Axiom 2 (Finite speed axiom): Given two different and separated basic things x and y, such as in the above figure, there exists a minimum positive bound for the interval (τx1 − τx0) defined by the reflex action.

Now we can define Simultaneity as τy0 is simultaneous with τx1/2 =Df (1/2)(τx1 + τx0)

The local times on x and y can be synchronized by the simultaneity relation. However, as we know from General Relativity, the simultaneity relation is transitive only in special reference frames called synchronous, thus prompting us to include the following axiom:

Axiom 3 (Synchronizability): Given a set of separated basic things {xi} there is an assignment of proper times τi such that the relation of simultaneity is transitive.

With this axiom, the simultaneity relation is an equivalence relation. Now we can define a first approximation to physical space, which is the ontic space as the equivalence class of states defined by the relation of simultaneity on the set of things is the ontic space EO.

The notion of simultaneity allows the analysis of the notion of clock. A thing y ∈ Θ is a clock for the thing x if there exists an injective function ψ : SL(y) → SL(x), such that τ < τ′ ⇒ ψ(τ) < ψ(τ′). i.e.: the proper time of the clock grows in the same way as the time of things. The name Universal time applies to the proper time of a reference thing that is also a clock. From this we see that “universal time” is frame dependent in agreement with the results of Special Relativity.

# Weil Conjectures. Note Quote. Solving Diophantine equations, that is giving integer solutions to polynomials, is often unapproachably difficult. Weil describes one strategy in a letter to his sister, the philosopher Simone Weil: Look for solutions in richer fields than the rationals, perhaps fields of rational functions over the complex numbers. But these are quite different from the integers:

We would be badly blocked if there were no bridge between the two. And voilà god carries the day against the devil: this bridge exists; it is the theory of algebraic function fields over a finite field of constants.

A solution modulo 5 to a polynomial P(X,Y,..Z) is a list of integers X,Y,..Z making the value P(X,Y,..Z) divisible by 5, or in other words equal to 0 modulo 5. For example, X2 + Y2 − 3 has no integer solutions. That is clear since X and Y would both have to be 0 or ±1, to keep their squares below 3, and no combination of those works. But it has solutions modulo 5 since, among others, 32 + 32 − 3 = 15 is divisible by 5. Solutions modulo a given prime p are easier to find than integer solutions and they amount to the same thing as solutions in the finite field of integers modulo p.

To see if a list of polynomial equations Pi(X, Y, ..Z) = 0 have a solution modulo p we need only check p different values for each variable. Even if p is impractically large, equations are more manageable modulo p. Going farther, we might look at equations modulo p, but allow some irrationals, and ask how the number of solutions grows as we allow irrationals of higher and higher degree—roots of quadratic polynomials, roots of cubic polynomials, and so on. This is looking for solutions in all finite fields, as in Weil’s letter.

The key technical points about finite fields are: For each prime number p, the field of integers modulo p form a field, written Fp. For each natural number r > 0 there is (up to isomorphism) just one field with pr elements, written as Fpr or as Fq with q = pr. This comes from Fp by adjoining the roots of a degree r polynomial. These are all the finite fields. Trivially, then, for any natural number s > 0 there is just one field with qs elements, namely Fp(r+s) which we may write Fqs. The union for all r of the Fpr is the algebraic closure Fp. By Galois theory, roots for polynomials in Fpr, are fixed points for the r-th iterate of the Frobenius morphism, that is for the map taking each x ∈ Fp to xpr.

Take any good n-dimensional algebraic space (any smooth projective variety of dimension n) defined by integer polynomials on a finite field Fq. For each s ∈ N, let Ns be the number of points defined on the extension field F(qs). Define the zeta function Z(t) as an exponential using a formal variable t:

Z(t) = exp(∑s=1Nsts/s)

The first Weil conjecture says Z(t) is a rational function:

Z(t) = P(t)/Q(t)

for integer polynomials P(t) and Q(t). This is a strong constraint on the numbers of solutions Ns. It means there are complex algebraic numbers a1 . . . ai and b1 . . . bj such that

Ns =(as1 +…+ asi) − (bs1 +…+ bsj)

And each algebraic conjugate of an a (resp. b) also an a (resp. b).

The second conjecture is a functional equation:

Z(1/qnt) = ± qnE/2tEZ(t)

This says the operation x → qn/x permutes the a’s (resp. the b’s).The third is a Riemann Hypothesis

Z(t) = (P1(t)P3(t) · · · P2n−1(t))/(P0(t)P2(t) · · · P2n(t))

where each Pk is an integer polynomial with all roots of absolute value q−k/2. That means each a has absolute value qk for some 0 ≤ k ≤ n. Each b has absolute value q(2k−1)/2 for some 0 ≤ k ≤ n.

Over it all is the conjectured link to topology. Let B0, B1, . . . B2n be the Betti numbers of the complex manifold defined by the same polynomials. That is, each Bk gives the number of k-dimensional holes or handles on the continuous space of complex number solutions to the equations. And recall an algebraically n-dimensional complex manifold is topologically 2n-dimensional. Then each Pk has degree Bk. And E is the Euler number of the manifold, the alternating sum

k=02n (−1)kBk

On its face the topology of a continuous manifold is worlds apart from arithmetic over finite fields. But the topology of this manifold tells how many a’s and b’s there are with each absolute value. This implies useful numerical approximations to the numbers of roots Ns. Special cases of these conjectures, with aspects of the topology, were proved before Weil, and he proved more. All dealt with curves (1-dimensional) or hypersurfaces (defined by a single polynomial).

Weil presented the topology as motivating the conjectures for higher dimensional varieties. He especially pointed out how the whole series of conjectures would follow quickly if we could treat the spaces of finite field solutions as topological manifolds. The topological strategy was powerfully seductive but seriously remote from existing tools. Weil’s arithmetic spaces were not even precisely defined. To all appearances they would be finite or (over the algebraic closures of the finite fields) countable and so everywhere discontinuous. Topological manifold methods could hardly apply.

# Smooth Manifolds: Frölicher space as Weil exponentiable

Thus spoke André Weil,

Nothing is more fruitful – all mathematicians know it – than those obscure analogies, those disturbing reflections of one theory in another; those furtive caresses, those inexplicable discords; nothing also gives more pleasure to the researcher. The day comes when the illusion dissolves; the yoked theories reveal their common source before disappearing. As the Gita teaches, one achieves knowledge and indifference at the same time. The notion of Weil algebra is ordinarily defined for a Lie algebra g. In mathematics, the Weil algebra of a Lie algebra g, introduced by Cartan based on unpublished work of André Weil, is a differential graded algebra given by the Koszul algebra Λ (g*) ⊗ S(g*) of its dual g*.

A Frölicher space is one flavour of a generalized smooth space. Frölicher smooth spaces are determined by a rule for
• how to map the real line smoothly into it,
• and how to map out of the space smoothly to the real line.

In the general context of space and quantity, Frölicher spaces take an intermediate symmetric position: they are both presheaves as well as copresheaves on their test domain (which here is the full subcategory of manifolds on the real line) and both of these structures determine each other.

After assigning, to each pair (X, W ) of a Frölicher space X and a Weil algebra W , another Frölicher space X ⊗ W , called the Weil prolongation of X with respect to W, which naturally extends to a bifunctor FS × W → FS, where FS is the category of Frölicher spaces and smooth mappings, and W is the category of Weil algebras. We also know

The functor · ⊗ W : FS → FS is product-preserving for any Weil algebra W.

Weil Exponentiability

A Frölicher space X is called Weil exponentiable if (X ⊗ (W1 W2))Y = (X ⊗ W1)Y ⊗ W2 —– (1)

holds naturally for any Frölicher space Y and any Weil algebras W1 and W2. If Y = 1, then (1) degenerates into

X ⊗ (W1 W2) = (X ⊗ W1) ⊗ W2 —– (2)

If W1 = R, then (1) degenerates into

(X ⊗ W2)Y = XY ⊗ W2 —– (3)

Proposition: Convenient vector spaces are Weil exponentiable.
Corollary: C-manifolds are Weil exponentiable.

Proposition: If X is a Weil exponentiable Frölicher space, then so is X ⊗ W for any Weil algebra W.

Proposition: If X and Y are Weil exponentiable Frölicher spaces, then so is X × Y.

Proposition: If X is a Weil exponentiable Frölicher space, then so is XY for any Frölicher space Y .

Theorem: Weil exponentiable Frölicher spaces, together with smooth mappings among them, form a Cartesian closed subcategory FSWE of the category FS.

Generally speaking, limits in the category FS are bamboozling. The notion of limit in FS should be elaborated geometrically.

A finite cone D in FS is called a transversal limit diagram providing that D ⊗ W is a limit diagram in FS for any Weil algebra W , where the diagram D ⊗ W is obtained from D by putting ⊗ W to the right of every object and every morphism in D. By taking W = R, we see that a transversal limit diagram is always a limit diagram. The limit of a finite diagram of Frölicher spaces is said to be transversal providing that its limit diagram is a transversal limit diagram.

Lemma: If D is a transversal limit diagram whose objects are all Weil exponentiable, then DX is also a transversal limit diagram for any Frölicher space X, where DX is obtained from D by putting X as the exponential over every object and every morphism in D.

Proof: Since the functor ·X : FS → FS preserves limits, we have DX ⊗ W = (D ⊗ W)X

for any Weil algebra W , so that we have the desired result.

Lemma: If D is a transversal limit diagram whose objects are all Weil exponentiable, then D ⊗ W is also a transversal limit diagram for any Weil algebra W.

Proof: Since the functor W ⊗ · : W → W preserves finite limits, we have (D ⊗ W) ⊗ W′ = D ⊗ (W ⊗ W′)

for any Weil algebra W′, so that we have the desired result.