Celebratio Mathematica

G. Benkart: “A Kac–Moody bibliography and some related references,” pp. 111–135 in Lie algebras and related topics (26 June–6 July 1984, Windsor, ON). Edited by D. J. Britten, F. W. Lemire, and R. V. Moody. CMS Conference Proceedings 5. American Mathematical Society (Providence, RI), 1986. MR 832196 Zbl 0578.17013 incollection

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G. Benkart: “Cartan subalgebras in Lie algebras of Cartan type,” pp. 157–187 in Lie algebras and related topics (26 June–6 July 1984, Windsor, ON). Edited by D. J. Britten, F. W. Lemire, and R. V. Moody. CMS Conference Proceedings 5. American Mathematical Society (Providence, RI), 1986. MR 832198 Zbl 0581.17006 incollection

The known finite dimensional simple Lie algebras over an algebraically closed field of characteristic \( p\geq 7 \) are either classical (analogues of complex Lie algebras) or of Cartan type. Each Lie algebra \( L \) of Cartan type possesses a certain maximal subalgebra \( L_0 \) which is invariant under the automorphisms of \( L \), and every Cartan subalgebra \( L_0 \) is a Cartan subalgebra of \( L \). The main result of this work is to show that the Cartan subalgebras of \( L_0 \) are conjugate under the group generated by the generalized exponential mappings defined by \[ E^z = 1 + \sum_{j=1}^{p-1} \left(\prod_{\nu = 1}^j (\nu \vert -\psi)^{-1}\right)(adz)^j \] where \( z\in L_0 \) has \( (adz)^{p^k} = 0 \), and where \[ \psi = (adz)^p + \cdots + (adz)^{p^{k-1}} .\] As a consequence, all the Cartan subalgebras of \( L \) in \( L_0 \) have the same dimension, and this dimension is computed for each type of algebra.

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G. M. Benkart, D. J. Britten, and F. W. Lemire: Stability in modules for classical Lie algebras — a constructive approach. Memoirs of the American Mathematical Society 430. American Mathematical Society (Providence, RI), 1990. MR 1010997 Zbl 0706.17003 book

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G. Benkart, D. Britten, and F. Lemire: “Projection maps for tensor products of \( \mathfrak{gl}(r,\mathbb{C}) \)-representations,” Publ. Res. Inst. Math. Sci. 28 : 6 (1992), pp. 983–1010. MR 1203757 Zbl 0830.17004 article

We investigate the tensor product \[ \mathcal{T} = V(\lambda^1)\otimes \cdots \otimes V(\lambda^m) \] of the finite dimensional irreducible \( \mathcal{G} = \mathfrak{gl}(r,\mathbb{C}) \) modules labelled by partitions \( \lambda^1,\dots \), \( \lambda^m \) of \( m \) not necessarily distinct numbers \( n_1,\dots \), \( n_m \) respectively. We determine the centralizer algebra \( \operatorname{End}_{\mathcal{G}}(\mathcal{T}) \) and the projection maps of \( \mathcal{T} \) onto its irreducible \( \mathcal{G} \)-summands and give an explicit construction of the corresponding maximal vectors. In the special case that \( n_i = 1 \) for \( i = 1,\dots \), \( m \), the results reduce to the well-known results of Schur and Weyl.

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G. Benkart, D. Britten, and F. Lemire: “Modules with bounded weight multiplicities for simple Lie algebras,” Math. Z. 225 : 2 (June 1997), pp. 333–353. MR 1464935 Zbl 0884.17004 article

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