Category of a Quantum Groupoid

For a quantum groupoid H let Rep(H) be the category of representations of H, whose objects are finite-dimensional left H -modules and whose morphisms are H -linear homomorphisms. We shall show that Rep(H) has a natural structure of a monoidal category with duality.

For objects V, W of Rep(H) set

V ⊗ W = x ∈ V ⊗_k W|x = ∆(1) · x ⊂ V ⊗_k W —– (1)

with the obvious action of H via the comultiplication ∆ (here ⊗_k denotes the usual tensor product of vector spaces). Note that ∆(1) is an idempotent and therefore V ⊗ W = ∆(1) × (V ⊗_k W). The tensor product of morphisms is the restriction of usual tensor product of homomorphisms. The standard associativity isomorphisms (U ⊗ V ) ⊗ W → U ⊗ (V ⊗ W ) are functorial and satisfy the pentagon condition, since ∆ is coassociative. We will suppress these isomorphisms and write simply U ⊗ V ⊗ W.

The target counital subalgebra H_t ⊂ H has an H-module structure given by h · z = ε_t(hz),where h ∈ H, z ∈ H_t.

H_t is the unit object of Rep(H).

Define a k-linear homomorphism l_V : H_t ⊗ V → V by l_V(1₍₁₎ · z ⊗ 1₍₂₎ · v) = z · v, z ∈ H_t, v ∈ V.

This map is H-linear, since

l_V h · (1₍₁₎ · z ⊗ 1₍₂₎ · v) = l_V(h(1) · z ⊗ h₍₂₎ · v) = ε_t(h₍₁₎z)h₍₂₎ · v = hz · v = h · l_V (1₍₁₎ · z ⊗ 1₍₂₎ · v),

∀ h ∈ H. The inverse map l⁻¹_V: → H_t ⊗ V is given by V

l⁻¹_V(v) = S(1₍₁₎) ⊗ (1₍₂₎ · v) = (1₍₁₎ · 1) ⊗ (1₍₂₎ · v)

The collection {l_V}_V gives a natural equivalence between the functor H_t ⊗ (·) and the identity functor. Indeed, for any H -linear homomorphism f : V → U we have:

l_U ◦ (id ⊗ f)(1₍₁₎ · z ⊗ 1₍₂₎ · v) = l_U 1₍₁₎ · z ⊗ 1₍₂₎ · f(v) = z · f(v) = f(z·v) = f ◦ l_V(1₍₁₎ · z ⊗ 1₍₂₎ · v)

Similarly, the k-linear homomorphism r_V : V ⊗ H_t → V defined by r_V(1₍₁₎ · v ⊗ 1₍₂₎ · z) = S(z) · v, z ∈ H_t, v ∈ V, has the inverse r⁻¹_V(v) = 1₍₁₎ · v ⊗ 1₍₂₎ and satisfies the necessary properties.

Finally, we can check the triangle axiom idV ⊗ l_W = r_V ⊗ id_W : V ⊗ H_t ⊗ W → V ⊗ W ∀ objects V, W of Rep(H). For v ∈ V, w ∈ W we have

(id_V ⊗ l_W)(1₍₁₎ · v ⊗ 1′₍₁₎1₍₂₎ · z ⊗ 1′₍₂₎ · w)

= 1₍₁₎ · v ⊗ 1′₍₂₎z · w) = 1₍₁₎S(z) · v ⊗ 1₍₂₎ · w

=(r_V ⊗ id_W) (1′₍₁₎ · v ⊗ 1′₍₂₎ 1₍₁₎ · z ⊗ 1₍₂₎ · w),

therefore, id_V ⊗ l_W = r_V ⊗ id_W

Using the antipode S of H, we can provide Rep(H) with a duality. For any object V of Rep(H), define the action of H on V^∗ = Hom_k(V, k) by

(h · φ)(v) = φ S(h) · v —– (2)

where h ∈ H , v ∈ V , φ ∈ V^∗. For any morphism f : V → W , let f^∗: W^∗ → V^∗ be the morphism dual to f. For any V in Rep(H), we define the duality morphisms d_V : V^∗ ⊗ V → H_t, b_V : H_t → V ⊗ V^∗ as follows. For ∑_j φ^j ⊗ v_j ∈ V^* ⊗ V, set

d_V(∑_j φ^j ⊗ v_j)  = ∑_j φ^j (1₍₁₎ · v_j) 1₍₂₎ —– (3)

Let {f_i}_i and {ξⁱ}_i be bases of V and V^∗, respectively, dual to each other. The element ∑_i f_i ⊗ ξⁱ does not depend on choice of these bases; moreover, ∀ v ∈ V, φ ∈ V^∗ one has φ = ∑_i φ(f_i) ξⁱ and v = ∑_i f_i ξⁱ (v). Set

b_V(z) = z · (∑_i f_i ⊗ ξⁱ) —– (4)

The category Rep(H) is a monoidal category with duality. We know already that Rep(H) is monoidal, it remains to prove that d_V and b_V are H-linear and satisfy the identities

(id_V ⊗ d_V)(b_V ⊗ id_V) = id_V, (d_V ⊗ id_V^∗)(id_V^∗ ⊗ b_V) = id_V^∗.

Take ∑_j φ^j ⊗ v_j ∈ V^∗ ⊗ V, z ∈ H_t, h ∈ H. Using the axioms of a quantum groupoid, we have

h · d_V(∑_j φ^j ⊗ v_j) = ((∑_j φ^j (1₍₁₎ · v_j) ε_t(h1₍₂₎)

= (∑_j φ^j ⊗ ε_s(1₍₁₎h) · v_j 1₍₂₎ ∑_j φ^j S(h₍₁₎)1₍₁₎h₍₂₎ · v_j 1₍₂₎

= (∑_j h₍₁₎ · φ^j )(1₍₁₎ · (h₍₂₎ · v_j))1₍₂₎

= (∑_j d_V(h₍₁₎ · φ^j  ⊗ h₍₂₎ · v_j) = d_V(h · ∑_j φ^j ⊗ v_j)

therefore, d_V is H-linear. To check the H-linearity of b_V we have to show that h · b_V(z) =

b_V (h · z), i.e., that

∑_i h₍₁₎z · f_i ⊗ h₍₂₎ · ξⁱ = ∑_i 1₍₁₎ ε_t(h_z) · f_i ⊗ 1₍₂₎ · ξⁱ

Since both sides of the above equality are elements of V ⊗_k V^∗, evaluating the second factor on v ∈ V, we get the equivalent condition

h₍₁₎zS(h₍₂₎) · v = 1₍₁₎ε^t (hz)S(1₍₂₎) · v, which is easy to check. Thus, b_V is H-linear.

Using the isomorphisms l_V and r_V identifying H_t ⊗ V, V ⊗ H_t, and V, ∀ v ∈ V and φ ∈ V^∗ we have:

(id_V ⊗ d_V)(b_V ⊗ id_V)(v)

=(id_V ⊗ d_V)b_V(1₍₁₎ · 1) ⊗ 1₍₂₎ · v

= (id_V ⊗ d_V)b_V(1₍₂₎) ⊗ S⁻¹(1₍₁₎) · v

= ∑_i (id_V ⊗ d_V) 1₍₂₎ · f_i ⊗ 1₍₃₎ · ξⁱ ⊗ S⁻¹ (1₍₁₎) · v

= ∑_i 1₍₂₎ · f_i ⊗ 1₍₃₎ · ξⁱ (1′₍₁₎S^-1 (1₍₁₎) · v) 1′₍₂₎

= 1₍₂₎ S(1₍₃₎) 1′_{(1) S^-1} (1₍₁₎) · v ⊗ 1′₍₂₎ = v

(d_V ⊗ id_V^∗)(id_V^∗ ⊗ b_V)(φ)

= (d_V ⊗ id_V^∗) 1₍₁₎ · φ ⊗ b_V(1₍₂₎)

= ∑_i (d_V ⊗ id_V^∗)(1₍₁₎ · φ ⊗ 1₍₂₎ · 1₍₂₎ · 1₍₃₎ · ξⁱ )

= ∑_i (1₍₁₎ · φ (1′₍₁₎1₍₂₎ · f_i)1′₍₂₎ ⊗ 1₍₃₎ · ξⁱ

= 1′₍₂₎ ⊗ 1₍₃₎1₍₁₎ S(1′₍₁₎1₍₂₎) · φ = φ,

QED.

 

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Malthusian Catastrophe.

As long as wealth is growing exponentially, it does not matter that some of the surplus labor is skimmed. If the production of the laborers is growing x% and their wealth grows y% – even if y% < x%, and the wealth of the capital grows faster, z%, with z% > x% – everybody is happy. The workers minimally increased their wealth, even if their productivity has increased tremendously. Nearly all increased labor production has been confiscated by the capital, exorbitant bonuses of bank managers are an example. (Managers, by the way, by definition, do not ’produce’ anything, but only help skim the production of others; it is ‘work’, but not ‘production’. As long as the skimming [money in] is larger than the cost of their work [money out], they will be hired by the capital. For instance, if they can move the workers into producing more for equal pay. If not, out they go).

If the economy is growing at a steady pace (x%), resulting in an exponential growth (1+x/100)ⁿ, effectively today’s life can be paid with (promises of) tomorrow’s earnings, ‘borrowing from the future’. (At a shrinking economy, the opposite occurs, paying tomorrow’s life with today’s earnings; having nothing to live on today).

Let’s put that in an equation. The economy of today E_i is defined in terms of growth of economy itself, the difference between today’s economy and tomorrow’s economy, E_i+1 − E_i,

E_i = α(E_i+1 − E_i) —– (1)

with α related to the growth rate, GR ≡ (E_i+1 − E_i)/E_i = 1/α. In a time-differential equation:

E(t) = αdE(t)/dt —– (2)

which has as solution

E(t) = E₀e^1/α —– (3)

exponential growth.

The problem is that eternal growth of x% is not possible. Our entire society depends on a

continuous growth; it is the fiber of our system. When it stops, everything collapses, if the derivative dE(t)/dt becomes negative, economy itself becomes negative and we start destroying things (E < 0) instead of producing things. If the growth gets relatively smaller, E itself gets smaller, assuming steady borrowing-from-tomorrow factor α (second equation above). But that is a contradiction; if E gets smaller, the derivative must be negative. The only consistent observation is that if E shrinks, E becomes immediately negative! This is what is called a Malthusian Catastrophe.

Now we seem to saturate with our production, we no longer have x% growth, but it is closer to 0. The capital, however, has inertia (viz. The continuing culture in the financial world of huge bonuses, often justified as “well, that is the market. What can we do?!”). The capital continues to increase their skimming of the surplus labor with the same z%. The laborers, therefore, now have a decrease of wealth close to z%. (Note that the capital cannot have a decline, a negative z%, because it would refuse to do something if that something does not make profit).

Many things that we took for granted before, free health care for all, early pension, free education, cheap or free transport (no road tolls, etc.) are more and more under discussion, with an argument that they are “becoming unaffordable”. This label is utter nonsense, when you think of it, since

1) Before, apparently, they were affordable.

2) We have increased productivity of our workers.

1 + 2 = 3) Things are becoming more and more affordable. Unless, they are becoming unaffordable for some (the workers) and not for others (the capitalists).

It might well be that soon we discover that living is unaffordable. The new money M’ in Marx’s equation is used as a starting point in new cycle M → M’. The eternal cycle causes condensation of wealth to the capital, away from the labor power. M keeps growing and growing. Anything that does not accumulate capital, M’ – M < 0, goes bankrupt. Anything that does not grow fast enough, M’ – M ≈ 0, is bought by something that does, reconfigured to have M’ – M large again. Note that these reconfigurations – optimizations of skimming (the laborers never profit form the reconfigurations, they are rather being sacked as a result of them) – are presented by the media as something good, where words as ‘increased synergy’ are used to defend mergers, etc. It alludes to the sponsors of the messages coming to us. Next time you read the word ‘synergy’ in these communications, just replace it with ‘fleecing’.

The capital actually ‘refuses’ to do something if it does not make profit. If M’ is not bigger than M in a step, the step would simply not be done, implying also no Labour Power used and no payment for Labour Power. Ignoring for the moment philanthropists, in capitalistic Utopia capital cannot but grow. If economy is not growing it is therefore always at the cost of labor! Humans, namely, do not have this option of not doing things, because “better to get 99 paise while living costs 1 rupee, i.e., ‘loss’, than get no paisa at all [while living still costs one rupee (haha, excuse me the folly of quixotic living!]”. Death by slow starvation is chosen before rapid death.

In an exponential growing system, everything is OK; Capital grows and reward on labor as well. When the economy stagnates only the labor power (humans) pays the price. It reaches a point of revolution, when the skimming of Labour Power is so big, that this Labour Power (humans) cannot keep itself alive. Famous is the situation of Marie-Antoinette (representing the capital), wife of King Louis XVI of France, who responded to the outcry of the public (Labour Power) who demanded bread (sic!) by saying “They do not have bread? Let them eat cake!” A revolution of the labor power is unavoidable in a capitalist system when it reaches saturation, because the unavoidable increment of the capital is paid by the reduction of wealth of the labor power. That is a mathematical certainty.

Gnostic Semiotics. Thought of the Day 63.0

The question here is what is being composed? For the deferment and difference that is always already of the Sign, suggests that perhaps the composition is one that lies not within but without, a creation that lies on the outside but which then determines – perhaps through the reader more than anything else, for after all the meaning of a particular sign, be it a word or anything else, requires a form of impregnation by the receiver – a particular meaning.

Is there any choice but to assume a meaning in a sign? Only through the simulation, or ‘belief’ if you prefer (there is really no difference in the two concepts), of an inherent meaning in the sign can any transference continue. For even if we acknowledge that all communication is merely the circulation of empty signifiers, the impregnation of the signified (no matter how unconnected it may be to the other person’s signified) still ensures that the sign carries with it a meaning. Only through this simulation of a meaning is circulation possible – even if one posits that the sign circulates itself, this would not be possible if it were completely empty.

Since it is from without (even if meaning is from the reader, (s)he is external to the signification), this suggests that the meaning is a result, a consequence of forces – its signification is a result of the significance of various forces (convention, context, etc) which then means that inherently, the sign remains empty; a pure signifier leading to yet another signifier.

The interesting element though lies in the fact that the empty signifier then sucks the Other (in the form of the signified, which takes the form of the Absolute Other here) into it, in order to define an existence, but essentially remains an empty signifier, awaiting impregnation with meaning from the reader. A void: always full and empty or perhaps (n)either full (n)or empty. For true potentiality must always already contain the possibility of non-potentiality. Otherwise there would be absolutely no difference between potentiality and actualization – they would merely be different ends of the same spectrum.