# What is the image of the diagonal map on the cohomology of Lie groups

Consider a simple Lie group $$G$$ and its mod $$p$$ cohomology $$H^*(G, \mathbb{Z}_p)$$.

A good reference is the book

Mimura, Mamoru; Toda, Hirosi, Topology of Lie groups, I and II. Transl. from the Jap. by Mamoru Mimura and Hirosi Toda, Translations of Mathematical Monographs. 91. Providence, RI: American Mathematical Society (AMS). iv, 451 p. (1991). ZBL0757.57001.

Consider the diagonal map $$\Delta: H^*(G, \mathbb{Z}_p) \rightarrow H^*(G, \mathbb{Z}_p) \otimes H^*(G, \mathbb{Z}_p)$$. Let $$x_i$$ for $$i=1,...,n$$ denote the generators of $$H^*(G, \mathbb{Z}_p)$$. Assume there is an element $$x^a_i\otimes x^b_j$$ contained in $$\Delta(x_m)$$, with $$a,b \neq 0$$.

Question 1: Is it true that at least one of the numbers $$a, b$$ is equal to $$1$$?

Question 2: Is it true that an element $$x_ix_j \otimes x_l$$ or $$x_i \otimes x_jx_l$$ can never be contained in $$\Delta(x_m)$$?

I assume that both answers are "yes", after checking many examples calculated in the mentioned book and also a paper on this topic released by Mimura and Kono.

The second question basically asks, wether the diagonal map can "mix" generators.

In the book the diagonal map is named $$\phi$$ and $$\phi'(x):= \phi(x) - 1\otimes x - x\otimes 1$$. The results are presented in terms of $$\phi'$$, which shouldnt change anything.

• Is it "tmage" or "image"?
– efs
Oct 9, 2020 at 0:38
• What do you call the "diagonal map"? The coproduct induced by the multiplication $G \times G \to G$? Oct 9, 2020 at 8:30
• Could you be more precise about the meaning of the word "contain" here? Oct 9, 2020 at 13:19
• By "generators", do you mean an additive basis (as $\mathbb{F} _p$ vector space) or something else? Oct 9, 2020 at 15:46
• Yes $\Delta$ is also known the coproduct map. Usually $\Delta(x)$ is a sum of elements $a \otimes b$. We say that $\Delta(x)$ contains a certain $a \otimes b$ if it ocurrs in this sum. The generators generate $H(G, \mathbb{Z}_p)$ additively and multiplicatively. We consider the ring structure of $H$.
– nxir
Oct 9, 2020 at 16:22