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Automorphic Forms and Lie Superalgebras

Urmie Ray

Chapter 2

Borcherds-Kac-Moody Lie Superalgebras - all with Video Answers

Educators

Section 1

Definitions and Elementary Properties

Problem 1

Show that a finite dimensional Lie superalgebra $L$ is semisimple if and only if it has no non-trivial abelian ideal.

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Problem 2

Let $L$ be a finite dimensional Lie superalgebra.
(i) Prove that the Killing form $K$ on $L$ is an invariant bilinear form, i.e. for all $x, y, z \in L, K([x, y], z)=K(x,[y, z])$ and $K$ is bilinear.
(ii) Deduce that the kernel of the Killing form is an ideal of $L$.
(iii) Suppose that $L$ is a Lie algebra. Deduce that the Killing form is nondegenerate if the Lie algebra $L$ is semisimple.
(iv) If the Killing form is non-degenerate on $L$, prove the Lie superalgebra $L$ is semisimple.

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Problem 3

Let $B$ be a symmetrizable matrix satisfying conditions $\left(i^{\prime}\right)-(i v)$ of Remark 2.1.11. We keep the same notation as in Remark 2.1.11.
(i) Show that for $i, j \in I,\left(h_i^{\prime}, h_j^{\prime}\right)=d_i^{-1} d_j^{-1} a_{i j}$, where $d_i$ is the (i,i)-th entry of the diagonal matrix $D$.
(ii) Deduce that the Lie superalgebras $G(A, H, S)$ and $G(B, H, S)$ are isomorphic.

We remind the reader that the generalized Cartan matrix is assumed to be indecomposable.

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Problem 4

Show that the centre of the BKM superalgebra $G$ is the subspace
$$
\left\{h \in H:\left(h, h_i\right)=0 \quad \forall i \in I\right\} .
$$

Deduce that if $H=\left\langle h_i: i \in I\right\rangle$ and the form is non-degenerate on $H$, then the BKM superalgebra $G$ is simple.

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Problem 5

Consider the Lie superalgebra $\tilde{G}$.
(i) Show that $\tilde{G}=\tilde{N}_{-} \oplus H \oplus \tilde{N}_{+}$as vector spaces, where $\tilde{N}_{+}$(resp. $\tilde{N}_{-}$) is the Lie sub-superalgebra generated by the elements $e_i$ (resp. $f_i$ ), $i \in I$, satisfying properties (1) - (3) of Definition 2.1.7.

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Problem 6

Let $G=\oplus_{i \in \mathbf{Z}} G_i$ be a $\mathbf{Z}$-graded Lie superalgebra. Show the existence of a minimal Lie superalgebra generated by $G_{-1}+G_0+G_1$.
For a solution, see [Kac6, Proposition 1.2.2] and [Kac1, Proposition 4]

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Problem 7

Let $\hat{G}$ be the Lie superalgebra generated by the subspace $\hat{H}$ containing vectors $h_{i j}, i, j \in I$ with $h_{i j} \neq 0$ if and only if $a_{k i}=a_{k j}$ for all $k \in I$, satisfying $\left(h_{i i}, h_{j j}\right)=a_{i j}$, and elements $e_i, f_i, i \in I$ satisfying relations (1) - (6) of Definition 2.1.7. and the following extra ones

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