Matrix representations of iterated wreath products of one-dimensional Lie algebras
DOI:
https://doi.org/10.17721/1812-5409.2025/2.2Keywords:
Lie algebra representation, wreath product, infinitely iterated wreath product, matrix schemeAbstract
The wreath product is a classical construction in group theory, known for its ability to generate complex groups from simpler ones and its deep connections to permutation and automorphism groups. Its analogue in Lie algebra theory, however, remains less developed, despite several notable contributions by A. Shmelkin, F. Sullivan, V. Sushchansky, N. Bondarenko, V. Petrogradsky, and others. In particular, the Lie algebras Lp,n, associated via the Lazard correspondence with Sylow p-subgroups of symmetric groups Sym(pn), admit a decomposition as iterated wreath products of one-dimensional Lie algebras over the field Fp, along with a tableau-based representation of elements. In this paper, we consider the Lie algebras Ln := L2,n and their inverse limit L∞, which is the infinitely iterated wreath product of one-dimensional Lie algebras over F2. We construct explicit embeddings of Ln into the Lie algebra UTm(F2) of strictly upper triangular matrices of minimal possible order m = 2n−1 +1, and extend this to a recursive embedding of the Lie algebra L∞ into the infinite-dimensional Lie algebra UT∞(F2). The construction employs matrix schemes inspired by earlier representations of iterated wreath products of cyclic groups. These results provide a concrete realization of L∞ and further deepen the connection between combinatorial Lie algebra structures and their matrix representations.
Pages of the article in the issue: 12 - 18
Language of the article: English
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