Let me say that I'm pretty sure that all the things I'm going to ask are obviously wrong for an expert because I don't have much faith in this stuff. So take this question as a counterexample to uncomplicated things đź™‚

Accept $ G $ is a semisimple split affine algebraic group about $ k $,

For an algebraic group we refer to $ ad (G) $ the associated lie algebra.

We know that $ G $ is determined by the associated master date, which is structured as follows:

- To let $ T $ Be a maximum torus (divided by PS). Any representation of $ T $ Divisions in one-dimensional repetitions, determined by a "weight" $ alpha in display (T) ^ * $,
- Thus, $ res_T (ad G) $ gives us a (finite) list of weights in $ ad (T) ^ * $ that determines the root date.

Note that the weights displayed can also be calculated $ K_0 (res_T ((ad G))) $ at the level of the Grothendieck groups.

(1) Can you understand who is $ (ad G) $ in the $ K_0 (Rep G) $? In other words, when two groups are the same $ K_0 $are the element $ (Display G), (display G & # 39;) $ equal?

In the case of finite groups over complex numbers, this is easy to see $ H to G $ is injective iff for each representation $ V neq 0 $ from $ H $. $ ind_H ^ G V neq 0 $,

This can also be checked on Grothendieck groups. Record the dot product $ K_0 (Rep G) $ given by $ langle (V), (W) rangle = dim_C Hom (V, W) $, Then $ langle (ind) V, W rangle = langle V, (res) W rangle $what determines $ (ind) $ as an adjoint in the linear algebra of res because the dot product is determined positively.

(2) Does this criterion apply in the general context? Can you only check this criterion for Grothendieck groups?

If so, let me name a map with neutral tannacic categories $ C to D $, Pseudo-injective if the adjoint $ D to C $ Sends non-zero objects to non-zero objects (somehow the adjunct's "kernel" is zero), logically for cards between their Grothendieck rings.

Now I'm saying that $ T $ is a maximum torus is like saying that $ Rep G to Rep T $ is maximally pseudo-injective of $ Rep G $ to a category of the form $ Rep (k ^ *) ^ { otimes n} $, Note that $ K_0 (Rep (k ^ *)) simeq Z (x, x ^ {- 1}) $ in char zero and $ mathbb {Z} (x) / (x ^ {p-1} -1) $ in char p (maybe? Not sure about char p)

Is it true that $ K_0 (Rep G) to K_0 (Rep T) $ is a maximum pseudo-objective map $ K_0 (Rep G) $ into a ring of shape $ Z (x_1 ^ { pm 1}, ldots, x_n ^ { pm 1}) $ (and analog for char p)?

If 1 is wrong, there is a possibility that all of these cards are from maximum tori. The question in this case is:

Do you think there is a possibility that in the general case $ K_0 (Rep G) $ and $ (ad G) in K_0 (Rep G) $ Determine group?

In fact, there is currently no indication of a maximum torus.

Of course, you also have to remember the K_0 of the fiber optic radio $ Rep G to Vec_k $what gives a card $ K_0 (Rep G) to mathbb {Z} $ remember the dimension of things.

If 1 is true, there should be some maximum pseudo-objective maps of $ K_0 (Rep G) $ to a ring of this shape that does not originate from maximally tori, otherwise one could reconstruct it $ G $ from $ K_0 (Rep G) $, and this does not even apply to finite groups (knowing K_0 in this case is like knowing the character table).

In this case the question is:

Can you find additional dates to remember? $ Rep G $ – that can be formulated abstractly for a general group – if a map can be understood $ K_0 (Rep G) to mathbb {Z} (x_1 ^ { pm 1}, ldots, x_n ^ { pm 1}) $ comes from a maximum torus?