Free skew fields have many ∗-orderings

Cimpric, Jaka

doi:10.1016/j.jalgebra.2004.01.033

Cited by 7 publications

(4 citation statements)

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“…For each Lie k-algebra L, he constructed a division k-algebra D(L) that contains U (L) and it is generated by it. In the study of division rings, D(L) plays a similar role to k(G) for the group algebra k[G] of an ordered group G. In [3], Cimprič proved that the principal involution on U (L) can be extended to D(L). In [16], Lichtman also proved that if k is a field of characteristic zero and L is not commutative, any division ring that contains U (L) must contain a free algebra of rank 2.…”

Section: The Map K[g] → K[g]mentioning

confidence: 99%

“…The contents of Section 1 are known. We sketch the construction of D(L) following Lichtman [15], and show how the princial involution is extended to D(L) as shown in [3]. We also present some results on the completion of skew polynomial rings that will be needed in Section 3.…”

Section: The Map K[g] → K[g]mentioning

confidence: 99%

“…For further details, the interested reader is referred to [4, Section 2.6] or the original paper [15]. Most of our exposition is taken from [3,Section 2].…”

Section: Preliminariesmentioning

confidence: 99%

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Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras

Ferreira

Gonçalves

Sánchez

2015

Int. J. Algebra Comput.

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For any Lie algebra L over a field, its universal enveloping algebra U (L) can be embedded in a division ring D(L) constructed by Lichtman. If U (L) is an Ore domain, D(L) coincides with its ring of fractions.It is well known that the principal involution of L, x → −x, can be extended to an involution of U (L), and Cimpric has proved that this involution can be extended to one on D(L).For a large class of noncommutative Lie algebras L over a field of characteristic zero, we show that D(L) contains noncommutative free algebras generated by symmetric elements with respect to (the extension of) the principal involution. This class contains all noncommutative Lie algebras such that U (L) is an Ore domain.

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Section: The Map K[g] → K[g]mentioning

confidence: 99%

Section: The Map K[g] → K[g]mentioning

confidence: 99%

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Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras

Ferreira

Gonçalves

Sánchez

2015

Int. J. Algebra Comput.

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“…Let a be a fixed nonzero element of D. We shall say that a is residually central (with respect to V ) if it satisfies any (and, hence, both) of the following equivalent conditions. Let us mention in passing that Conditions C + & C S + were studied by various authors (See [6,13,17] & [2,12,[14][15][16] respectively). For notational convenience, let us make the following Definition B' .…”

Section: Introductionmentioning

confidence: 99%

An Involutorial Version of a Theorem of Gräter

Chacron

2015

Communications in Algebra

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Let D be a division ring with centre Z and with involution ( * ). Let V be a valuation of D with value group , a linearly ordered additive group (non necessarily commutative) together with a symbol (positive infinity). We assume that for each nonzero symmetric element s = s * of D, which is algebraic over Z, we have for all nonzero elements x of D, V xa − ax > V ax . We define the residue characteristic exponent p of V to be the characteristic of the associated residue division ring written as D V , if = 0, and p = 1, if = 0. We show here that if F is a finite dimensional commutative subalgebra of D over Z, which is * -closed (i.e., F * = F ), and if ( * ) is of the first kind (i.e., each central element of D must be symmetric), then F Z = 2 r p m where m is a nonnegative integer and r = 0 or 1 according as the restricted involution in F is trivial or not. The case of an involution ( * ) of the second kind (i.e., some central element of D is not symmetric) requires (for this author) a stronger type of valuation, namely, V is a * -valuation, that is to say, for all elements x of D, we have V x * = V x , a condition which readily implies must be Abelian. Here, we can show that for F as in the preceding, F Z = p m , where m is again a nonnegative integer. The preceding results generalize a theorem of Gräter and improve in parts recent theorems of this author in [2]. In the special case p = 2 the results provide a modicum of answers to the questions opened informally in [2] (see concluding paragraph in [2] or here Question 3.2.1). More is to be said in the third and final section of this work.

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*-orderable semigroups

Klep

Moravec

2009

Semigroup Forum

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Free skew fields have many ∗-orderings

Cited by 7 publications

References 16 publications

Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras

Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras

An Involutorial Version of a Theorem of Gräter

*-orderable semigroups

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