Algebra Symbols $\mathfrak h$ and $\mathfrak{so}$

Question

What do these symbols mean in algebra? I found them as follows: $$\mathfrak h_3(\Bbb O(\Bbb Z_p))$$ $$\mathfrak{so}(\Bbb O)\oplus\Bbb O^3$$

Answer 1

These symbols are written with the german fraktur script, traditionally used in several areas of mathematics following the founding texts of 19th and early 20th german-speaking mathematicians (Lie, Killing, F. Engel, Frobenius, ...) and their followers, notably Group Theory, Lie algebras, Jordan Algebras. You will find related notations and objects in latin alphabet capitals, like this for example

$${H_{3}}(\Bbb R)$$

$${SO_{n}}(\Bbb C) \quad {\rm or}\quad {SO}(n,\Bbb C)$$

${H_{n}}$ usually designates the Heisenberg Group and ${\frak h_{3}}$ or ${\frak H_{3}}$ is an abbreviation for the Heisenberg Algebra over $3$x$3$ matrices with elements built from the structure given as argument. See for a first reference the Heisenberg Group Wikipedia article. For the particular one you give as an example, I do not know about ${\frak h_{3}}(\Bbb O(\Bbb Z_p))$ but I know about ${\frak h_{3}}(\Bbb O)$, constructed over the octonions, it is usually called the (exceptional) Albert algebra, a Jordan algebra.

The ${\frak so}$ is an abbreviation for Special Orthogonal Lie Algebra : you might want to look at the Wikipedia article on the Orthogonal Group for an introduction to the Special Orthogonal Group of which ${\frak so}$ is an extension.