Celebratio Mathematica

G. Benkart and E. Zelmanov: “Lie algebras graded by root systems,” pp. 31–38 in Non-associative algebra and its applications (12–17 July 1993, Oviedo, Spain). Edited by S. González. Mathematics and its Applications 303. Kluwer Academic (Dordrecht), 1994. MR 1338154 Zbl 0826.17030 incollection

Let \( \Delta \) be a finite irreducible reduced root system. Assume \( \Gamma \) is the integer lattice generated by \( \Delta \). In [1992] S. Berman and R. Moody initiated the study of Lie algebras graded by the root system \( \Delta \). Following them we say that a Lie algebra is graded by \( \Delta \) or is \( \Delta \)-graded if

\( L \) has a \( \Gamma \)-gradation \( L = \sum_{\gamma\in\Gamma}L_{\gamma} \) in which \( L_{\gamma}\neq (0) \) if and only if \( \gamma\in \Delta\cup\{0\} \);
the split simple Lie algebra \( S \) whose root system is \( \Delta \) is a subalgebra of \( L \), and relative to some split Cartan subalgebra \( \mathcal{H} \) of \( \mathcal{G} \) we have \( L_{\gamma} \supseteq \mathcal{G}_{\gamma} \) for all \( \gamma \in \Delta\cup\{0\} \);
for all \( h\in\mathcal{H} \) the operator \( ad(h) \) acts diagonally on \( L_{\gamma} \) with eigenvalue \( \gamma(h) \); and
\( L \) is generated by its nonzero root spaces \( L_{\gamma} \) where \( \gamma\in\Delta \).

G. Benkart and E. Zelmanov: “Lie algebras graded by finite root systems and intersection matrix algebras,” Invent. Math. 126 : 1 (1996), pp. 1–45. MR 1408554 Zbl 0871.17024 article

G. Benkart: “Down-up algebras and Witten’s deformations of the universal enveloping algebra of \( \mathfrak{sl}_2 \),” pp. 29–45 in Recent progress in algebra (11–15 August 1997, Taejon, South Korea). Edited by S. G. Hahn, H. C. Myung, and E. Zelmanov. Contemporary Mathematics 224. American Mathematical Society (Providence, RI), 1999. MR 1653061 Zbl 0922.17007 incollection

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G. Benkart and A. Elduque: Cross products, invariants, and centralizers. Preprint, June 2016. Dedicated to Efim Zelmanov on the occasion of his 60th birthday. ArXiv 1606.07588 techreport

An algebra \( \mathrm{V} \) with a cross product \( \times \) has dimension 3 or 7. In this work, we use 3-tangles to describe, and provide a basis for, the space of homomorphisms from \( \mathrm{V}^{\otimes n} \) to \( \mathrm{V}^{\otimes m} \) that are invariant under the action of the automorphism group \( \operatorname{Aut}(\mathrm{V},\times) \) of \( V \), which is a special orthogonal group when \( \dim \mathrm{V} = 3 \), and a simple algebraic group of type \( \mathrm{G}_2 \) when \( \dim \mathrm{V} = 7 \). When \( m = n \), this gives a graphical description of the centralizer algebra \[ \operatorname{End}_{\operatorname{Aut}(\mathrm{V},\times)}(\mathrm{V}^{\otimes n}) ,\] and therefore, also a graphical realization of the \( \operatorname{Aut}(\mathrm{V},\times) \)-invariants in \( \mathrm{V}^{\otimes n}) \) equivalent to the First Fundamental Theorem of Invariant Theory. We show how the 3-dimensional simple Kaplansky Jordan superalgebra can be interpreted as a cross product (super)algebra and use 3-tangles to obtain a graphical description of the centralizers and invariants of the Kaplansky superalgebra relative to the action of the special orthosymplectic group.

Efim Isaakovich Zelmanov	Related
Alberto Elduque	Related

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