Regular element of a Lie algebra#Definition

In the specific case of $\backslash mathfrak\{gl\}\_n(\backslash mathbb\{k\})$, the Lie algebra of $n\; \backslash times\; n$ matrices over an algebraically closed field $\backslash mathbb\{k\}$ (such as the complex numbers), a regular element $M$ is an element whose Jordan normal form contains a single Jordan block for each eigenvalue (in other words, the geometric multiplicity of each eigenvalue is 1).

The centralizer of a regular element is the set of polynomials of degree less than $n$ evaluated at the matrix $M$, and therefore the centralizer has dimension $n$ (which equals the rank of $\backslash mathfrak\{gl\}\_n$, but is not necessarily an algebraic torus).

If the matrix $M$ is diagonalisable, then it is regular if and only if there are $n$ different eigenvalues. To see this, notice that $M$ will commute with any matrix $P$ that stabilises each of its eigenspaces. If there are $n$ different eigenvalues, then this happens only if $P$ is diagonalisable on the same basis as $M$; in fact $P$ is a linear combination of the first $n$ powers of $M$, and the centralizer is an algebraic torus of complex dimension $n$ (real dimension $2n$); since this is the smallest possible dimension of a centralizer, the matrix $M$ is regular. However if there are equal eigenvalues, then the centralizer is the product of the general linear groups of the eigenspaces of $M$, and has strictly larger dimension, so that $M$ is not regular.

For a connected compact Lie group $G$, the regular elements form an open dense subset, made up of $G$-conjugacy classes of the elements in a maximal torus $T$ which are regular in $G$. The regular elements of $T$ are themselves explicitly given as the complement of a set in $T$, a set of codimension-one subtori corresponding to the root system of $G$. Similarly, in the Lie algebra $\backslash mathfrak\{g\}$ of $G$, the regular elements form an open dense subset which can be described explicitly as adjoint $G$-orbits of regular elements of the Lie algebra of $T$, the elements outside the hyperplanes corresponding to the root system.{{cite book |first=Mark R. |last=Sepanski |title=Compact Lie Groups |url=https://books.google.com/books?id=F3NgD_25OOsC&pg=PA156 |date=2006 |publisher=Springer |isbn=978-0-387-30263-8 |pages=156}}

Let $\backslash mathfrak\{g\}$ be a finite-dimensional Lie algebra over an infinite field.Editorial note: the definition of a regular element over a finite field is unclear. For each $x\; \backslash in\; \backslash mathfrak\{g\}$, let

:$p\_x(t)\; =\; \backslash det(t\; -\; \backslash operatorname\{ad\}(x))\; =\; \backslash sum\_\{i=0\}^\{\backslash dim\; \backslash mathfrak\{g\}\}\; a\_i(x)\; t^i$

be the characteristic polynomial of the adjoint endomorphism $\backslash operatorname\{ad\}(x)\; :\; y\; \backslash mapsto\; [x,\; y]$ of $\backslash mathfrak\; g$. Then, by definition, the rank of $\backslash mathfrak\{g\}$ is the least integer $r$ such that $a\_r(x)\; \backslash ne\; 0$ for some $x\; \backslash in\; \backslash mathfrak\; g$ and is denoted by $\backslash operatorname\{rk\}(\backslash mathfrak\{g\})$.{{harvnb|Bourbaki|1981|loc=Ch. VII, § 2.2. Definition 2.}} For example, since $a\_\{\backslash dim\; \backslash mathfrak\; g\}(x)\; =\; 1$ for every x, $\backslash mathfrak\; g$ is nilpotent (i.e., each $\backslash operatorname\{ad\}(x)$ is nilpotent by Engel's theorem) if and only if $\backslash operatorname\{rk\}(\backslash mathfrak\{g\})\; =\; \backslash dim\; \backslash mathfrak\; g$.

Let $\backslash mathfrak\{g\}\_\{\backslash text\{reg\}\}\; =\; \backslash \{\; x\; \backslash in\; \backslash mathfrak\{g\}\; |\; a\_\{\backslash operatorname\{rk\}(\backslash mathfrak\{g\})\}\; (x)\; \backslash ne\; 0\; \backslash \}$. By definition, a regular element of $\backslash mathfrak\{g\}$ is an element of the set $\backslash mathfrak\{g\}\_\{\backslash text\{reg\}\}$. Since $a\_\{\backslash operatorname\{rk\}(\backslash mathfrak\{g\})\}$ is a polynomial function on $\backslash mathfrak\{g\}$, with respect to the Zariski topology, the set $\backslash mathfrak\{g\}\_\{\backslash text\{reg\}\}$ is an open subset of $\backslash mathfrak\{g\}$.

Over $\backslash mathbb\{C\}$, $\backslash mathfrak\{g\}\_\{\backslash text\{reg\}\}$ is a connected set (with respect to the usual topology),{{harvnb|Serre|2001|loc=Ch. III, § 1. Proposition 1.}} but over $\backslash mathbb\{R\}$, it is only a finite union of connected open sets.{{harvnb|Serre|2001|loc=Ch. III, § 6.}}

Over an infinite field, a regular element can be used to construct a Cartan subalgebra, a self-normalizing nilpotent subalgebra. Over a field of characteristic zero, this approach constructs all the Cartan subalgebras.

Given an element $x\; \backslash in\; \backslash mathfrak\{g\}$, let

:$\backslash mathfrak\{g\}^0(x)\; =\; \backslash sum\_\{n\; \backslash ge\; 0\}\; \backslash ker(\backslash operatorname\{ad\}(x)^n\; :\; \backslash mathfrak\{g\}\; \backslash to\; \backslash mathfrak\{g\})$

be the generalized eigenspace of $\backslash operatorname\{ad\}(x)$ for eigenvalue zero. It is a subalgebra of $\backslash mathfrak\; g$.This is a consequence of the binomial-ish formula for ad. Note that $\backslash dim\; \backslash mathfrak\{g\}^0(x)$ is the same as the (algebraic) multiplicityRecall that the geometric multiplicity of an eigenvalue of an endomorphism is the dimension of the eigenspace while the algebraic multiplicity of it is the dimension of the generalized eigenspace. of zero as an eigenvalue of $\backslash operatorname\{ad\}(x)$; i.e., the least integer m such that $a\_m(x)\; \backslash ne\; 0$ in the notation in {{slink||Definition}}. Thus, $\backslash operatorname\{rk\}(\backslash mathfrak\; g)\; \backslash le\; \backslash dim\; \backslash mathfrak\{g\}^0(x)$ and the equality holds if and only if $x$ is a regular element.

The statement is then that if $x$ is a regular element, then $\backslash mathfrak\{g\}^0(x)$ is a Cartan subalgebra.{{harvnb|Bourbaki|1981|loc=Ch. VII, § 2.3. Theorem 1.}} Thus, $\backslash operatorname\{rk\}(\backslash mathfrak\; g)$ is the dimension of at least some Cartan subalgebra; in fact, $\backslash operatorname\{rk\}(\backslash mathfrak\; g)$ is the minimum dimension of a Cartan subalgebra. More strongly, over a field of characteristic zero (e.g., $\backslash mathbb\{R\}$ or $\backslash mathbb\{C\}$),{{harvnb|Bourbaki|1981|loc=Ch. VII, § 3.3. Theorem 2.}}

• every Cartan subalgebra of $\backslash mathfrak\{g\}$ has the same dimension; thus, $\backslash operatorname\{rk\}(\backslash mathfrak\; g)$ is the dimension of an arbitrary Cartan subalgebra,
• an element x of $\backslash mathfrak\; g$ is regular if and only if $\backslash mathfrak\{g\}^0(x)$ is a Cartan subalgebra, and
• every Cartan subalgebra is of the form $\backslash mathfrak\{g\}^0(x)$ for some regular element $x\; \backslash in\; \backslash mathfrak\; g$.

For a Cartan subalgebra $\backslash mathfrak\; h$ of a complex semisimple Lie algebra $\backslash mathfrak\; g$ with the root system $\backslash Phi$, an element of $\backslash mathfrak\; h$ is regular if and only if it is not in the union of hyperplanes $\backslash bigcup\_\{\backslash alpha\; \backslash in\; \backslash Phi\}\; \backslash \{\; h\; \backslash in\; \backslash mathfrak\{h\}\; \backslash mid\; \backslash alpha(h)\; =\; 0\; \backslash \}$.{{harvnb|Procesi|2007|loc=Ch. 10, § 3.2.}} This is because: for $r\; =\; \backslash dim\; \backslash mathfrak\; h$,

• For each $h\; \backslash in\; \backslash mathfrak\{h\}$, the characteristic polynomial of $\backslash operatorname\{ad\}(h)$ is {{nowrap|$t^r\; \backslash left(t^\{\backslash dim\; \backslash mathfrak\; g\; -\; r\}\; -\; \backslash sum\_\{\backslash alpha\; \backslash in\; \backslash Phi\}\; \backslash alpha(h)\; t^\{\backslash dim\; \backslash mathfrak\; g\; -\; r\; -\; 1\}\; +\; \backslash cdots\; \backslash pm\; \backslash prod\_\{\backslash alpha\; \backslash in\; \backslash Phi\}\; \backslash alpha(h)\backslash right)$.}}

This characterization is sometimes taken as the definition of a regular element (especially when only regular elements in Cartan subalgebras are of interest).

{{reflist}}

• {{citation|last=Bourbaki|first=N.|series=Éléments de Mathématique|title=Groupes et Algèbres de Lie|publisher=Hermann|year=1981}}
• {{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | last2=Harris | first2=Joe | author2-link=Joe Harris (mathematician) | title=Representation Theory, A First Course | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-97495-8 |mr=1153249 | year=1991 | volume=129}}
• {{citation |author-link=Claudio Procesi |first=Claudio |last=Procesi |title=Lie Groups: an approach through invariants and representation |publisher=Springer |year=2007 |isbn=9780387260402 }}
• {{citation |first=Jean-Pierre |last=Serre |title=Complex Semisimple Lie Algebras |publisher=Springer |year=2001 |isbn=3-5406-7827-1 }}

Category:Lie groups

Category:Lie algebras