Subalgebras of the rank two semisimple Lie algebras

Abstract

Abstract. In this expository article, we describe the classification of the subalgebras of the rank 2 semisimple Lie algebras. Their semisimple subalgebras are well-known, and in a recent series of papers, we completed the classification of the subalgebras of the classical rank 2 semisimple Lie algebras. Finally, Mayanskiy finished the classification of the subalgebras of the remaining rank 2 semisimple Lie algebra, the exceptional Lie algebra G 2 . We identify subalgebras of the classification in terms of a uniform classification scheme of Lie algebras of low dimension. The classification is up to inner automorphism, and the ground field is the complex numbers.

Decompositions of algebras and post-associative algebra structures

“…Comparing both sides we conclude that l jta klb,mpc = 0 when a = b or j = k and l jta jla,mpc = l uta ula,mpc for any 1 ≤ j, u ≤ n a . In the same way by (7) we obtain r stc ija,klb = 0 when b = c or l = t and r slb ija,klb = r svb ija,kvb for any 1 ≤ l, v ≤ n b . Then (9) yields the equality l ima ija,klb = r jna klb,mna for all 1 ≤ i, j, m, n ≤ n a , 1 ≤ k, l ≤ n b and 1 ≤ a, b ≤ q.…”

“…The above theorem has an easier proof for the exceptional Lie algebras G 2 and E 8 , not using the classification by Onishchik. For G 2 , the statement follows from the description of all its subalgebras [7,19]. For E 8 , it follows from the information on the centralizers of all its semisimple subalgebras [20, Table 14].…”

Geodesic orbit spaces of compact Lie groups of rank two

“…up to conjugation by inner automorphisms), where g is a semisimple Lie algebra of rank two. The main sources for this table are [15], [14] and [21].…”

“…For the description of the spaces G/H with G semisimple and h = su(2), we take into account the explicit description of the embeddings of sl 2 C in the complexified Lie algebra g C of G in [15] (see also Table I). The corresponding embedding of su(2) in g is shown in Table II.…”

Rota–Baxter operators and post-Lie algebra structures on semisimple Lie algebras

“…In this section we want to show an analogous result for n = sl 2 (C) × sl 2 (C). Here we will use RB-operators on n and an explicit classification by Douglas and Repka [18] of all subalgebras of n. This classification is up to inner automorphisms, but we will only need the subalgebras up to isomorphisms. Let us fix a basis (X 1 , Y 1 , H 1 , X 2 , Y 2 , H 2 ) of n consisting of the following 4 × 4 matrices.…”