Quantum Groupoid

A (finite) quantum groupoid over k is a finite-dimensional k-vector space H with the structures of an associative algebra (H, m, 1) with multiplication m : H ⊗_k H → H and unit 1 ∈ H and a coassociative coalgebra (H, ∆, ε) with comultiplication ∆ : H → H ⊗_k H and counit ε : H → k such that:

1. The comultiplication ∆ is a (not necessarily unit-preserving) homomorphism of algebras such that

(∆ ⊗ id)∆(1) = (∆(1) ⊗ 1) (1 ⊗ ∆(1)) = (1 ⊗ ∆(1)) (∆(1) ⊗ 1) —– (1)

2.  The counit is a k-linear map satisfying the identity:

ε(fgh) = ε(fg₍₁₎)ε(g₍₂₎h) = ε(f_g(2))ε(_g(1)h), (2) ∀ f, g, h ∈ H —– (2)

3.   There is an algebra and coalgebra anti-homomorphism S : H → H, called an antipode, such that, ∀ h ∈ H ,

m(id ⊗ S) ∆(h) = (ε ⊗ id) ∆(1)(h ⊗ 1) —– (3)

m(S ⊗ id) ∆(h) = (id ⊗ ε)(1 ⊗ h) ∆(1) —– (4)

A quantum groupoid is a Hopf algebra iff one of the following equivalent conditions holds: (i) the comultiplication is unit preserving or (ii) the counit is a homomorphism of algebras.

A morphism of quantum groupoids is a map between them which is both an algebra and a coalgebra morphism preserving unit and counit and commuting with the antipode. The image of such a morphism is clearly a quantum groupoid. The tensor product of two quantum groupoids is defined in an obvious way.

The set of axioms is self-dual. This allows to define a natural quantum groupoid  structure on the dual vector space H’ = Hom_k (H, k) by “reversing the arrows”:

⟨h,φ ψ⟩ = ∆(h), φ ⊗ ψ —– (5)

⟨g ⊗ h, ∆'(φ)⟩ = ⟨gh, φ⟩ —– (6)

⟨h, S'(φ)⟩ = ⟨S(h), φ⟩ —– (7)

∀ φ, ψ ∈ H’, g, h ∈ H. The unit 1ˆ ∈ H’ is ε and counit ε’ is φ → ⟨φ,1⟩. The linear endomorphisms of H defined by

h → m(id ⊗ S) ∆(h), h → m(S ⊗ id) ∆(h) —– (8)

are called the target and source counital maps and denoted ε_t and ε_s, respectively.

From axioms (3) and (4),

ε_t(h) = (ε ⊗ id) ∆(1)(h ⊗ 1), ε_s(h) = (id ⊗ ε) (1 ⊗ h)∆(1) . (9)

In the Hopf algebra case ε_t(h) = ε_s(h) = ε(h)1.

We have S ◦ ε_s = ε_t ◦ S and ε_s ◦ S = S ◦ ε_t. The images of these maps ε_t and ε_s

H_t = ε_t (H) = {h ∈ H | ∆(h) =∆(1)(h ⊗ 1)} —– (10)

H_s = ε_s (H) = {h ∈ H | ∆(h) = (1⊗h) ∆(1)} —– (11)

are subalgebras of H, called the target (respectively source) counital subalgebras. They play the role of ground algebras for H. They commute with each other and

H_t = {(φ ⊗ id) ∆(1)|φ ∈ H’,

H_s = (id ⊗ φ) ∆(1)| φ ∈ H’,

i.e., H_t (respectively H_s) is generated by the right (respectively left) tensorands of ∆(1). The restriction of S defines an algebra anti-isomorphism between H_t and H_s. Any morphism H → K of quantum groupoids preserves counital subalgebras, i.e., H_t ≅ K_t and H_s ≅ K_s.

In what follows we will use the Sweedler arrows, writing ∀ h ∈ H , φ ∈ H’:

h ⇀ φ = φ₍₁₎⟨h, φ₍₂₎⟩,

φ ↼ h = ⟨h, φ₍₁₎⟩φ₍₂₎ —– (12)

∀ h ∈ H, φ ∈ H’. Then the map z → (z ⇀ ε) is an algebra isomorphism between H_t and H. Similarly, the map y → (ε ↼ y) is an algebra isomorphism between H and H’_t. Thus, the counital subalgebras of H’ are canonically anti-isomorphic to those of H. A quantum groupoid H is called connected if H_s ∩ Z(H) = k, or, equivalently, H_t ∩ Z(H ) = k, where Z(H) denotes the center of H. A k-algebra A is separable if the multiplication epimorphism m : A ⊗_k A → A has a right inverse as an A − A bimodule homomorphism. When the characteristic of k is 0, this is equivalent to the existence of a separability element e ∈ A ⊗_k A such that m(e) = 1 and (a ⊗ 1)e = e(1 ⊗ a), (1 ⊗ a)e = e(a ⊗ 1) ∀ a ∈ A. The counital subalgebras H_t and H_s are separable, with separability elements e_t = (S ⊗ id)∆(1) and e_s = (id ⊗S)∆(1), respectively. Observe that the adjoint actions of 1 ∈ H give rise to non-trivial maps

H → H : h → 1₍₁₎h_S(1₍₂₎) = Ad^l₁(h), h → S(1₍₁₎)h1₍₂₎ = Ad^r1(h), h ∈ H —– (13) …….

 

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Sustainability of Debt

For economies with fractional reserve-generated fiat money, balancing the budget is characterized by an exponential growth D(t) ≈ D₀(1 + r)^t of any initial debt D₀ subjected to interest r as a function of time t due to the compound interest; a fact known since antiquity. At the same time, besides default, this increasing debt can only be reduced by the following five mostly linear, measures:

(i) more income or revenue I (in the case of sovereign debt: higher taxation or higher tax base);

(ii) less spending S;

(iii) increase of borrowing L;

(iv) acquisition of external resources, and

(v) inflation; that is, devaluation of money.

Whereas (i), (ii) and (iv) without inflation are essentially measures contributing linearly (or polynomially) to the acquisition or compensation of debt, inflation also grows exponentially with time t at some (supposedly constant) rate f ≥ 1; that is, the value of an initial debt D₀, without interest (r = 0), in terms of the initial values, gets reduced to F(t) = D₀/f^t. Conversely, the capacity of an economy to compensate debt will increase with compound inflation: for instance, the initial income or revenue I will, through adaptions, usually increase exponentially with time in an inflationary regime by If^t.

Because these are the only possibilities, we can consider such economies as closed systems (with respect to money flows), characterized by the (continuity) equation

If^t + S + L ≈ D₀(1+r)^t, or

L ≈ D₀(1 + r)^t − If^t − S.

Let us concentrate on sovereign debt and briefly discuss the fiscal, social and political options. With regards to the five ways to compensate debt the following assumptions will be made: First, in non-despotic forms of governments (e.g., representative democracies and constitutional monarchies), increases of taxation, related to (i), as well as spending cuts, related to (ii), are very unpopular, and can thus be enforced only in very limited, that is polynomial, forms.

Second, the acquisition of external resources, related to (iv), are often blocked for various obvious reasons; including military strategy limitations, and lack of opportunities. We shall therefore disregard the acquisition of external resources entirely and set A = 0.

As a consequence, without inflation (i.e., for f = 1), the increase of debt

L ≈ D₀(1 + r)^t − I − S

grows exponentially. This is only “felt” after trespassing a quasi-linear region for which, due to a Taylor expansion around t = 0, D(t) = D₀(1 + r)^t ≈ D₀ + D₀rt.

So, under the political and social assumptions made, compound debt without inflation is unsustainable. Furthermore, inflation, with all its inconvenient consequences and re-appropriation, seems inevitable for the continuous existence of economies based on fractional reserve generated fiat money; at least in the long run.

Activism and Militancy: Empire of the Sands. Note Quote.

Negri writes:

In the post-modern era, as the figure of the people dissolves, the militant is the one who best expresses the life of the multitude: the agent of biopolitical production and resistance against Empire […] When we speak of the militant, we are not thinking of anything like the sad, ascetic agent of the Third International […] We are thinking of nothing like that and of no one who acts on the basis of duty and discipline, who pretends his or her actions are deduced from an ideal plan […] Today the militant cannot even pretend to be a representative, even of the fundamental human needs of the exploited. Revolutionary political militancy today, on the contrary, must rediscover what has always been its proper form: not representational but constituent activity.[…] Militants resist imperial command in a creative way. In other words, resistance is linked immediately with a constitutive investment in the biopolitical realm and to the formation of co-operative apparatuses of production and community.[…] There is an ancient legend that might serve to illuminate the future life of communist militancy: that of Saint Francis of Assisi. Consider his work. To denounce the poverty of the multitude he adopted that common condition and discovered there the ontological power of a new society. The communist militant does the same, identifying in the common condition of the multitude its enormous wealth. Francis in opposition to nascent capitalism refused every instrumental discipline, and in opposition to the mortification of the flesh (in poverty and in the constituted order) he posed a joyous life, including all of being and nature […] Once again in postmodernity we find ourselves in Francis’s situation, posing against the misery of power the joy of being. This is a revolution that no power will control – because biopower and communism, co-operation and revolution remain together, in love, simplicity, and also innocence. This is the irrepressible lightness and joy of being communist.

Once again it is particularly difficult to find any ideas that bear any relation to classical Marxism in the extract above. For Negri, the militant [activist] becomes an individualist who confronts the capitalist system in a “creative” way and who draws his own revolutionary strength from his or her own very uniqueness and his or her capacity to identify with the conditions of the masses. On top of this, the hero of this type of militancy is St. Francis of Assisi! In reality, genuine Marxist activists are able to place themselves at the vanguard of the working class, not only because they have won the trust and respect of workers through their ideas but also because they are able to connect with the political consciousness of the working class at a particular given moment and raise it towards the accomplishment of the socialist transformation of society. These types of activists never act on the basis of their own individuality, but know how to use it by linking it up with the individualities of other activists and put it at the service of the revolution. The political activist is in no way some sort of dour killjoy, but is the driving force of a whole class, the proletariat.

For the activist, being part of the proletariat also means not being afraid to represent it. On the contrary, each day of the activist’s life is dedicated to advancing the working class in its quest for the final victory. The Marxist activist’s revolutionary duty is to organise and lead, without ever becoming separated from his or her own class. Lenin, in a critique of Rosa Luxemburg’s conception of party organisation – which he saw as a vanguard based on revolutionary discipline – says the following in “Left-wing communism, an infantile disorder” about how the discipline of the proletariat’s revolutionary party can be maintained.

First, by the class-consciousness of the proletarian vanguard and by its devotion to the revolution, by its tenacity, self-sacrifice and heroism. Second, by its ability to link up, maintain the closest contact, and – if you wish – merge, in certain measure, with the broadest masses of the working people – primarily with the proletariat, but also with the non-proletarian masses of working people. Third, by the correctness of the political leadership exercised by this vanguard, by the correctness of its political strategy and tactics, provided the broad masses have seen, from their own experience, that they are correct.“

All this has little to do with the ideal kind of activist described in the pages of Empire. In conclusion, we have a good suggestion for bringing Negri’s theory face to face with stark reality. What would happen if Negri’s “activist” went to a factory gate, or any other workplace at the beginning of the day’s shift, and invited the workers to “have fun” and “disobey”, in order to subvert the established order? We do not claim to know the conditions of every single workplace or factory, but we are certain that in those places that we know and where we often go to give out leaflets and organise campaigns, the level of alienation and fatigue caused by waged labour under the control of the capitalists is very high. Activists going to workers and proposing to them the type of activity that Negri suggests would be lucky to get away with less than a scratch! Again, once petit-bourgeois theories are confronted with the reality of the situation, they show their completely bankrupt nature.

Prometheus and Hinduism. Note Quote.

Prometheus: Yes, I caused mortals to cease foreseeing their doom.
Chorus: Of what sort was the cure that you found for this affliction?
Prometheus: I caused blind hopes to dwell within their breasts.
Chorus: A great benefit was this you gave to mortals.
Prometheus: In addition, I gave them fire.
Chorus: What! Do creatures of a day now have flame-eyed fire?
Prometheus: Yes, and from it they shall learn many arts.
— Aeschylus, Prometheus Bound

After the coming into being of the world, the stories tell us, everything had found its place. But one creature, capable of lofty thought, was still missing. When Kronos ruled Olympus, the deathless gods decided to fashion a golden race of mortal men. Knowing of the divine seed that slumbered in the Earth, only recently separated from the heavenly aether, Prometheus mixed clay and running water. He shaped this into likenesses of the all-controlling gods, including also qualities taken from all the animals. This is how the first humans came into existence. In the course of ages, after Zeus had banished the ancient gods to Tartarus, humans populated the Earth, but lumbered witless as if in a dream. They did not know how to see, hear, or understand, or how to create things with their hands. Prometheus’s empathy led him to steal for them the forbidden divine fire hidden by Zeus, which allowed him to teach them skills and sciences that used all their potentials.

Angered by the theft, Zeus plotted revenge on both humanity and its benefactor. He had Hephaestus, smith of the gods, create a beautiful virgin, Pandora, who was furnished with disastrous gifts by Athena, Aphrodite, and other gods. When a box she was carrying was opened on Earth, all evils and diseases escaped from it and spread among mankind. One single good thing, Hope, had not escaped when she clapped the lid closed. Meanwhile Prometheus was dragged to earth’s remotest wilderness and bound to a rock over a terrifying abyss with chains that could not be undone. Every day an eagle came and ate from his liver, which would regenerate each night. He endured this torment for centuries until the hero Hercules set him free.

This myth calls to mind stories of divine fire brought to mankind in many other traditions.

The allegory of the fire of Prometheus is another version of the rebellion of the proud Lucifer [“light-bringer”], who was hurled down to the bottomless pit, or simply unto our Earth, to live as man. The Hindu Lucifer, the Mahasura, is also said to have become envious of the Creator’s resplendent light, and, at the head of inferior Asuras (not gods, but spirits), to have rebelled against Brahma; for which Siva hurled him down to Patala. But, as philosophy goes hand in hand with allegorical fiction in Hindu myths, the devil is made to repent, and is afforded the oppor­tunity to progress.

Also in Hinduism are the Manasaputras or “sons of mind,” who brought mankind the fire of thought. In the Nordic Edda the name of the god Loki – a blood brother of Odin – comes from the old word liuhan, “to illuminate.”

What then is the inner significance behind these particular stories? Long ago the early human race had undergone a certain amount of evolution but “thinkers” had not yet been born: nature had succeeded in developing a suitable body but the soul-giving principle, the fire of self-conscious thought, had not yet been awakened. Adam and Eve, to a certain extent, existed in paradise without self-awareness. Like Lucifer, Prometheus is an allegorical representation of the incarnation of our higher self, the awakening of the active, self-reflective capacity for thought. This subject is therefore of the highest significance for human evolution.

It is owing to this rebellion of intellectual life against the morbid inactivity of pure spirit, that we are what we are — self-conscious, thinking men, with the capabilities and attributes of Gods in us, for good as much as for evil. Hence the rebels are our saviours. . . . It is only by the attractive force of the contrasts that the two opposites — Spirit and Matter — can be cemented on Earth, and, smelted in the fire of self-conscious experience and suffering, find themselves wedded in Eternity.

By the gods allying themselves with us for this period of development, it became possible for us to attain knowledge and wisdom. But why was Prometheus harshly punished? Other legends suggest a motive; for instance Genesis reports:

And the Lord God said, “The man has now become like one of us, knowing good and evil. He must not be allowed to reach out his hand and take also from the tree of life and eat, and live forever.” So the Lord God banished him from the Garden of Eden to work the ground from which he had been taken. After he drove the man out, he placed on the east side of the Garden of Eden cherubim and a flaming sword flashing back and forth to guard the way to the tree of life. – 3:22-4

As with Pandora’s box, all evil in this world arose from the joining of the spiritual with the human world. Many commentators see in this a question of guilt, but this misses the crux of the matter: Adam and Eve are driven out of paradise because, with the power of thought, there is no longer any paradise for them. The human being, equipped with the divine capacity for self-reflective thought, can use this newly-won strength to create or destroy, to accomplish wonderful things or great crimes. One day we will ascend again and establish a human race worthy of the gods, but there is still a long way to go in overcoming “I am I” (egoic awareness) to reach “I am” (universal consciousness).

This evolutionary process is also clearly reflected in symbolism. Spirit, represented by a vertical line, is linked with the material world, represented by a horizontal line. Together these give rise to the cross, the son or third logos. If this logos becomes active, as with the awakening capacity for thought through the incarnation of the higher self, then this cross begins to turn. The turning of the cross produces the swastika, a symbol found in many religions. Quite a number of terracotta discs were found under the ruins of ancient Troy that contained this symbol in two forms:  and . Again, Pramantha, the Vedic divine carpenter, unites himself with Arani, nature or Maya (illusion). They produce the divine boy Agni, god of fire. In the Bible too, Joseph is a carpenter, a master builder, and Mary is very reminiscent of Maya. Their child is mankind, with the fire of self-aware thought bestowed by the Holy Spirit. The son of the creator nailed to the cross – is he not a symbol of this process that speaks the same clear language as the legend of Prometheus, spirit chained to the cross of matter? The Crucified Titan is the personified symbol of the collective Logos, the ‘Host’, and of the ‘Lords of Wisdom’ or the heavenly man, who incarnated in Humanity.

We are also told that the suffering of Prometheus – the “one who foresees” – will end. He who has sacrificed himself for mankind is redeemed out of pity by the demigod Hercules, a son of Zeus, despite the fact that he may not take off the indestructible chains (karma). Still, the vulture – our base nature – will no longer come to eat the Titan’s liver. Human mental development, accelerated by the incarnation of the higher self, became unbalanced, with physical and moral development unable to keep pace. Once we regain our inner equilibrium, we will recognize our true destiny and nature, release the god chained within us, and as mankind conquer the darkness of ignorance.