# Lie algebras of the unit group of a finite dimensional algebra

Given a finite dimmensional associative algebra $$A$$, it can be proven that the unit group $$A^{\ast}:=\{a\in A \mid \exists\, b \in A \mbox{ such that } ab=ba=1_A\}$$ is always a Lie group (The unit group of a finite dimensional associative algebra is a Lie group?), but I have problems trying to identify its Lie algebra. I think that its Lie algebra $$(\mathfrak{a},[.,.])$$ should be isomorphic (as Lie algebras) to $$(A,[.,.]_c)$$, where $$[a,b]_c=ab-ba$$ (as the case $$A=M_n(\mathbb{K})$$). It is true?? I hope somebody can help me

• You should add the assumption of the linked question and answer that $\Bbb K$ is a complete valued field (as otherwise the concept of "Lie group" makes little sense to begin with). – Torsten Schoeneberg Feb 11 at 1:31
• Well, of course $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. – GaSa Feb 11 at 1:36
• Of course ... (Many people are investigating $p$-adic Lie groups, but of course that need not be your business.) Your conjecture sounds plausible, but what have you tried? And what construction of the Lie algebra of a Lie group are you using in this generality -- invariant vector fields, tangent space, invariant derivations ...? – Torsten Schoeneberg Feb 11 at 1:39
• I have tried considering $\mathfrak{a}$ as tangent space but the problem is choose a basis on $A$ (based in the case $A=M_n(\mathbb{K}$)). – GaSa Feb 11 at 1:43