3

I'm looking for an example of an isomorph Lie Algebra. 2 algebras are isomorph, if there exists an bijective linear function $g_1 \rightarrow g_2$ which maps all $X,Y \in g_1$ like $\phi([X,Y]) = [\phi(X),\phi(Y)]$.

So 2 Lie algebras I could think of would be the cross product in ${\rm I\!R}^3$ and the Commutator algebra of a left invariant Vectorfield but I can't think of a function that maps them like I stated before.

glS
  • 7,963
gamma
  • 255

1 Answers1

10

Examples, roughly ordered from easy to hard:

  1. Let $\mathfrak g$ be any Lie algebra. The identity map $x \mapsto x$ is an isomorphism from $\mathfrak g$ to itself.

  2. Let $V$, $W$ be vector spaces over a field $k$, and define Lie brackets on them as $[v_1, v_2] = 0$ and $[w_1,w_2]=0$ for all $v_1,v_2 \in V$, $w_1,w_2 \in W$. Show that the Lie algebras $V$ and $W$ (with these brackets) are isomorphic if and only if $V$ and $W$ have the same dimension. (This should be just a check you understand isomorphisms of vector spaces, the absolute basis of linear algebra.)

  3. Let $k$ be any field and $\mathfrak{gl}_n(k)$ the Lie algebra given by all $n \times n$-matrices over $k$, with the Lie bracket given by the matrix commutator $[A,B]:= A\cdot B-B\cdot A$ (where $\cdot$ is the usual matrix multiplication). Let $g$ be any invertible $n\times n$-matrix over $k$, i.e. an element of $\mathrm{GL}_n(k)$. Show that the map $$ A \mapsto g\cdot A \cdot g^{-1}$$ is an isomorphism from $\mathfrak{gl}_n(k)$ to itself, i.e. an automorphism of $\mathfrak{gl}_n(k)$.

  4. Let $\mathfrak{gl}_n(k)$ be as in the previous example. The map which sends each matrix to its negative transpose, $$ A \mapsto -A^T$$ is an isomorphism from $\mathfrak{gl}_n(k)$ to itself, i.e. an automorphism of $\mathfrak{gl}_n(k)$.

  5. Let $k$ be any field, $c \in k^\times$, $\mathfrak g_1$ a two-dimensional $k$-vector space with basis $v_1, v_2$ and Lie bracket $[v_1, v_2] = v_2$. Let $\mathfrak g_2$ be another two-dimensional $k$-vector space with basis $w_1,w_2$ and $[w_1,w_2]= c\cdot w_2$. Find an isomorphism of the Lie algebras $\mathfrak g_1$ and $\mathfrak g_2$.

  6. Let $\mathfrak g_1$ and $\mathfrak g_2$ be as in the previous example, except that now the Lie bracket on $\mathfrak g_2$ is given by $[w_1,w_2] = a w_1 + c w_2$ where $c \in k^\times$ and $a \in k$. Again find an isomorphism $\mathfrak g_1 \simeq \mathfrak g_2$. (For this and the previous example, cf. Classsifying $1$- and $2$-Dimensional Lie Algebras, up to Isomorphism, How to get an explicit isomorphism (explicitly defined) between any two nonabelian Lie algebras of dimension $2$, What are the possible two-dimensional Lie algebras?, What can we say about generic two-dimensional Lie algebras? .)

  7. Let $k$ be any field of characteristic $\neq 2$, $\mathfrak{sl}_2(k) := \{ A \in \mathfrak{gl}_2(k): Tr(A)=0\}$ the Lie algebra of traceless $2 \times 2$-matrices (with Lie bracket given as in example 3). Let $\mathfrak{so}_3(k) := \{ \pmatrix{a&0&-f\\0&-a&-e\\e&f&0} : a,e,f \in k \}$ (the "split form of $\mathfrak{so}_3$") also with Lie bracket given by matrix commutator. Find an isomorphism between these two Lie algebras. (Compare Is there an intelligent way to find the Lie algebra isomorphism $\mathfrak{sl}_2(\mathbb{C})\simeq\mathfrak{o}_3(\mathbb{C})$?, What is a direct proof of the isomorphism $\mathfrak{so}(3)_{\mathbb C}\simeq\mathfrak{sl}(2,\mathbb C)$?, An Explicit Isomorphism Between the Three Dimensional Orthogonal Lie Algebra and the Special Linear Lie Algebra of Dimension $3$ and links therein.)

  8. Let $\mathfrak{su}_2 := \{\pmatrix{ai&b+ci\\-b+ci&-ai} : a,b,c \in \mathbb R \}$ (a three-dimensional real subspace of the $2 \times 2$ complex matrices); convince yourself that again with the Lie bracket given by the matrix commutator (as in example 3), this is a Lie algebra. Show it is isomorphic to $(\mathbb R^3, \times)$ i.e. the three-dimensional real Lie algebra with Lie bracket given by the cross product. (Compare Why is there a factor of $2$ in the isomorphism $\operatorname{Lie}(S^3)\cong\mathbb{R}^3$?. This seems to be what you allude to in the question.)

  9. Find an isomorphism between $\mathfrak{sl}_2(\mathbb C) \oplus \mathfrak{sl}_2(\mathbb C)$ and the skew-symmetric $4\times 4$ matrices over $\mathbb C$. (Cf. Explicit isomorphism between the four dimensional orthogonal Lie algebra and the direct sum of special linear Lie algebras of dimension 3. )

  10. Find an isomorphism between: a) the direct sum of skew-symmetric $3 \times 3$ real matrices with itself; and b) the $4 \times 4$ real skew-symmetric matrices. (Cf. Isomorphism between $\mathfrak o(4,\mathbb R)$ and $\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $ )

10.5. Find an isomorphism between: a) the skew-symmetric complex $6 \times 6$ matrices b) the traceless complex $4 \times$ matrices. (Cf. How should I show that the Lie algebra $\mathfrak{sl}(4,\mathbb{C})$ is isomorphic to the Lie algebra $\mathfrak{so}(6,\mathbb{C})$? and after going through the next one, come back and compare this to How should I show that the Lie algebra so(6) of SO(6) is isomorphic to the Lie algebra su(4) of SU(4)? .)

  1. For $\mathfrak g$ a real Lie algebra, the scalar extension / complexification $\mathbb C \otimes \mathfrak g$ is a complex Lie algebra with Lie bracket given by bilinear extension of $[a \otimes x, b \otimes y]:=ab\otimes [x,y]$. Easy: Show that the complexification of $\mathfrak{sl}_2(\mathbb R)$ is isomorphic to $\mathfrak{sl}_2(\mathbb C)$. Harder: For $\mathfrak{su}_2$ as defined in example 8, show that the complexification $\mathbb C \otimes \mathfrak{su}_2$ is also isomorphic to $\mathfrak{sl}_2(\mathbb C)$. Bonus: Show that in spite of that, the real Lie algebras $\mathfrak{sl}_2(\mathbb R)$ and $\mathfrak{su}_2$ are not isomorphic to each other. (Compare Precise connection between complexification of $\mathfrak{su}(2)$, $\mathfrak{so}(1,3)$ and $\mathfrak{sl}(2, \mathbb{C})$, Are Lie algebra complexifications $\mathfrak g_{\mathbb C}$ equivalent to Lie algebra structures on $\mathfrak g\oplus \mathfrak g$?, and probably many more.)

  2. Show that $\mathfrak{sl}_2(\mathbb C)$, but viewed as a (six-dimensional) real Lie algebra, is isomorphic to the "Lorentz Lie algebra" called $\mathfrak{so}(1,3)$. (Cf. Relationship between $\mathfrak{sl}(2,\mathbb{C})$ and $\mathfrak{so}(1,3)$ and understand that this is a variant of nos. 9 and 10 above mixed with techniques from no. 11.)

Also, try Finding Lie algebra isomorphisms, and Question regarding isomorphisms in low rank Lie algebras .

Torsten Schoeneberg
  • 29,325