Finite generation of Lie algebras associated with associative algebras. (English) Zbl 1365.17002
A Lie algebra \(L\) over a field \(\mathbb F\) is an \(\mathbb F\)-vector space together with a bilinear map \([\;,\;] : L \times L \to L\), denoted by \((x, y) \mapsto [x, y]\) and called the bracket of \(x\) and \(y\) such that the following axioms are satisfied:
Any associative \(\mathbb F\)-algebra \(R\) gives rise to:
- (i)
- \([x, x] = 0\),
- (ii)
- \( [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0\) (Jacobi identity),
Any associative \(\mathbb F\)-algebra \(R\) gives rise to:
- \(\bullet\)
- A Lie algebra \(R^{-}\) with Lie bracket \([x,y]:=xy-yx\), for all \(x, y \in R\).
- \(\bullet\)
- If the algebra \(R\) is equipped with an involution \(* :R\to R\) then the space of the skew-symmetric elements \(K=Skew(R,*)\) can be seen as a subalgebra of the Lie algebra \(R^-\).
Reviewer: Miguel Angel Gómez Lozano (Malaga)
MSC:
17B05 | Structure theory for Lie algebras and superalgebras |
16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |
References:
[1] | Amitsur, S. A., Identities in rings with involutions, Israel J. Math., 7, 63-68 (1969) · Zbl 0179.33701 |
[2] | Baxter, W. E., Lie simplicity of a special case of associative rings II, Trans. Amer. Math. Soc., 87, 63-75 (1958) · Zbl 0082.25202 |
[3] | Herstein, I. N., Non-commutative Ring, Carus Mathematical Monographs, vol. 15 (1968), Math. Assoc. Amer.: Math. Assoc. Amer. Washington, DC · Zbl 0177.05801 |
[4] | Herstein, I. N., Rings with Involution, Mathematics Lecture Notes (1976), University of Chicago · Zbl 0343.16011 |
[5] | Jacobson, N., Structure and Representations of Jordan Algebras, AMS Coll. Publ., vol. 39 (1968), Providence · Zbl 0218.17010 |
[6] | Loos, O., Jordan Pairs, Lecture Notes in Math., vol. 460 (1975), Springer-Verlag: Springer-Verlag New York · Zbl 0301.17003 |
[7] | McCrimmon, K., A Taste of Jordan Algebras (2004), Springer-Verlag: Springer-Verlag New York · Zbl 1044.17001 |
[8] | Montgomery, S.; Small, L. W., Some remarks on affine rings, Proc. Amer. Math. Soc., 98, 537-544 (1986) · Zbl 0606.16007 |
[9] | Zhevlakov, K. A.; Slinko, A. M.; Shestakov, J. P.; Shirshov, A. I., Rings That are Nearly Associative (1982), Academic Press: Academic Press New York · Zbl 0487.17001 |
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