Restricted Lie algebra: Difference between revisions

Content deleted Content added

Inline

Revision as of 20:56, 14 December 2023

In mathematics, a restricted Lie algebra (or p-Lie algebra) is a Lie algebra over a field of characteristic $p>0$ together with an additional "pth power" operation. Most naturally occurring Lie algebras in characteristic p come with this structure, because the Lie algebra of a group scheme over a field of characteristic p is restricted.

Definition

Let L be a Lie algebra over a field k of characteristic p>0. The adjoint representation of L is defined by $({\text{ad }}X)(Y)=[X,Y]$ for $X,Y\in L$ . A p-mapping on L is a function $X\mapsto X^{[p]}$ satisfying^[1]

$\mathrm {ad} (X^{[p]})=(\mathrm {ad} \;X)^{p}$ for all $X\in L$ ,
$(tX)^{[p]}=t^{p}X^{[p]}$ for all $t\in k$ and $X\in L$ ,
$(X+Y)^{[p]}=X^{[p]}+Y^{[p]}+\sum _{i=1}^{p-1}s_{i}(X,Y)$ , for all $X,Y\in L$ , where $s_{i}(X,Y)$ is $1/i$ times the coefficient of $t^{i-1}$ in the formal expression $(\mathrm {ad} \;tX+Y)^{p-1}(X)$ .

A restricted Lie algebra over k is a Lie algebra over k together with a p-mapping. A Lie algebra is restrictable if it has at least one p-mapping.

For example:

For p = 2, the last condition says that $(X+Y)^{[2]}=X^{[2]}+[Y,X]+Y^{[2]}$ .
For p = 3, the last condition says that $(X+Y)^{[3]}=X^{[3]}+{\frac {1}{2}}[X,[Y,X]]+[Y,[Y,X]]+Y^{[3]}$ . Since ${\frac {1}{2}}=-1$ in a field of characteristic 3, this can be rewritten as $(X+Y)^{[3]}=X^{[3]}-[X,[Y,X]]+[Y,[Y,X]]+Y^{[3]}$ .

Examples

For an associative algebra A over a field k of characteristic p, the bracket operation $[X,Y]:=XY-YX$ and the p-mapping $X^{[p]}:=X^{p}$ make A into a restricted Lie algebra $\mathrm {Lie} (A)$ .^[1] In particular, taking A to be the ring of n x n matrices shows that the Lie algebra ${\mathfrak {gl}}(n)$ of n x n matrices over k is a restricted Lie algebra, with the p-mapping being the pth power of a matrix. This "explains" the definition of a restricted Lie algebra: the complicated formula for $(X+Y)^{[p]}$ is needed to express the pth power of the sum of two matrices over k, $(X+Y)^{p}$ , given that X and Y typically do not commute.

Let A be an algebra over a field k. (Here A is a possibly non-associative algebra.) Then the derivations of A over k form a Lie algebra ${\text{Der}}_{k}(A)$ , with the Lie algebra being the commutator, $[D_{1},D_{2}]=D_{1}D_{2}-D_{2}D_{1}$ . When k has characteristic p>0, then the pth power of a derivation is also a derivation, and this makes ${\text{Der}}_{k}(A)$ into a restricted Lie algebra.^[1]

Let G be a group scheme over a field k of characteristic p>0, and let $\mathrm {Lie} (G)$ be the Zariski tangent space at the identity element of G. Then $\mathrm {Lie} (G)$ is a restricted Lie algebra over k. This is essentially a special case of the previous example. Indeed, each element of $\mathrm {Lie} (G)$ determines a left-invariant vector field on G, and hence a derivation on the ring of regular functions on G. The pth power of this derivation is again a derivation, in fact the derivation associated to an element $X^{[p]}$ of $\mathrm {Lie} (G)$ .

Restricted enveloping algebra

The functor $A\mapsto \mathrm {Lie} (A)$ has a left adjoint $L\mapsto u(L)$ called the restricted enveloping algebra. To construct this, let $U(L)$ be the universal enveloping algebra of L (ignoring the p-mapping of L). Let I be the two-sided ideal generated by elements of the form $x^{p}-x^{[p]}$ ; then the restricted enveloping algebra is $u(L)=U(L)/I$ . It satisfies a form of the Poincaré–Birkhoff–Witt theorem: if $e_{1},\ldots ,e_{n}$ is a basis for L as a k-vector space, then a basis for $u(L)$ is given by all ordered products $e_{1}^{i_{1}}\cdots e_{n}^{i_{n}}$ with $0\leq i_{j}\leq p-1$ for each j. In particular, the map $L\to u(L)$ is injective, and if L has dimension n as a vector space, then $u(L)$ has dimension $p^{n}$ as a vector space.^[2]

Notes

^ ^a ^b ^c Strade & Farnsteiner (1988), section 2.1.
^ Strade & Farnsteiner (1988), section 2.5.

References

Borel, Armand (1991) [1969], Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), New York: Springer Nature, doi:10.1007/978-1-4612-0941-6, ISBN 0-387-97370-2, MR 1102012
Block, Richard E.; Wilson, Robert Lee (1988), "Classification of the restricted simple Lie algebras", Journal of Algebra, 114 (1): 115–259, doi:10.1016/0021-8693(88)90216-5, ISSN 0021-8693, MR 0931904.
Montgomery, Susan (1993), Hopf algebras and their actions on rings. Expanded version of ten lectures given at the CBMS Conference on Hopf algebras and their actions on rings, which took place at DePaul University in Chicago, USA, August 10-14, 1992, Regional Conference Series in Mathematics, vol. 82, Providence, RI: American Mathematical Society, p. 23, ISBN 978-0-8218-0738-5, Zbl 0793.16029.
Strade, Helmut; Farnsteiner, Rolf (1988), Modular Lie algebras and their representations, Marcel Dekker, ISBN 0-8247-7594-5, MR 0929682

[definition-1] Strade & Farnsteiner (1988), section 2.1.

[2] Strade & Farnsteiner (1988), section 2.5.

[1]

[2]

@@ Line 1: / Line 1: @@
-In [[mathematics]], a '''restricted Lie algebra''' is a [[Lie algebra]] together with an additional "''p'' operation."
+In [[mathematics]], a '''restricted Lie algebra''' (or ''' ''p''-Lie algebra''') is a [[Lie algebra]] over a [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] <math>p>0</math> together with an additional "''p''th power" operation. Most naturally occurring Lie algebras in characteristic ''p'' come with this structure, because the Lie algebra of a [[group scheme]] over a field of characteristic ''p'' is restricted.
 ==Definition==
-Let ''L'' be a Lie algebra over a field ''k'' of characteristic ''p>0''. A ''p'' '''operation''' on ''L'' is a map <math>X \mapsto X^{[p]}</math> satisfying
+Let ''L'' be a Lie algebra over a field ''k'' of characteristic ''p>0''. The [[adjoint representation#Adjoint representation of a Lie algebra|adjoint representation]] of ''L'' is defined by <math>(\text{ad }X)(Y)=[X,Y]</math> for <math>X,Y\in L</math>. A ''p''-'''mapping''' on ''L'' is a function <math>X \mapsto X^{[p]}</math> satisfying<ref name="definition">Strade & Farnsteiner (1988), section 2.1.</ref>
-* <math>\mathrm{ad}(X^{[p]}) = \mathrm{ad}(X)^p</math> for all <math>X \in L</math>,
+* <math>\mathrm{ad}(X^{[p]}) = (\mathrm{ad}\; X)^p</math> for all <math>X \in L</math>,
-* <math>(tX)^{[p]} = t^pX^{[p]}</math> for all <math>t \in k, X \in L</math>,
+* <math>(tX)^{[p]} = t^pX^{[p]}</math> for all <math>t \in k</math> and <math>X \in L</math>,
-* <math>(X+Y)^{[p]} = X^{[p]} + Y^{[p]} + \sum_{i=1}^{p-1} \frac{s_i(X,Y)}{i}</math>, for all <math>X,Y \in L</math>, where <math>s_i(X,Y)</math> is the coefficient of <math>t^{i-1}</math> in the formal expression <math>\mathrm{ad}(tX+Y)^{p-1}(X)</math>.
+* <math>(X+Y)^{[p]} = X^{[p]} + Y^{[p]} + \sum_{i=1}^{p-1} s_i(X,Y)</math>, for all <math>X,Y \in L</math>, where <math>s_i(X,Y)</math> is <math>1/i</math> times the coefficient of <math>t^{i-1}</math> in the formal expression <math>(\mathrm{ad}\; tX+Y)^{p-1}(X)</math>.
+A '''restricted Lie algebra''' over ''k'' is a Lie algebra over ''k'' together with a ''p''-mapping. A Lie algebra is '''restrictable''' if it has at least one ''p''-mapping.
+For example:
-If the characteristic of ''k'' is 0, then ''L'' is a restricted Lie algebra where the ''p'' operation is the identity map.
+* For ''p'' = 2, the last condition says that <math>(X+Y)^{[2]}=X^{[2]}+[Y,X]+Y^{[2]}</math>.
+* For ''p'' = 3, the last condition says that <math>(X+Y)^{[3]}=X^{[3]}+\frac{1}{2}[X,[Y,X]]+[Y,[Y,X]]+Y^{[3]}</math>. Since <math>\frac{1}{2}=-1</math> in a field of characteristic 3, this can be rewritten as <math>(X+Y)^{[3]}=X^{[3]}-[X,[Y,X]]+[Y,[Y,X]]+Y^{[3]}</math>.
 ==Examples==
-For any associative algebra ''A'' defined over a field of characteristic ''p'', the bracket operation <math>[X,Y] := XY-YX</math> and ''p'' operation <math>X^{[p]} := X^p</math> make ''A'' into a restricted Lie algebra <math>\mathrm{Lie}(A)</math>.
+For an [[associative algebra]] ''A'' over a field ''k'' of characteristic ''p'', the bracket operation <math>[X,Y] := XY-YX</math> and the ''p''-mapping <math>X^{[p]} := X^p</math> make ''A'' into a restricted Lie algebra <math>\mathrm{Lie}(A)</math>.<ref name="definition" /> In particular, taking ''A'' to be the [[matrix ring|ring of ''n'' x ''n'' matrices]] shows that the Lie algebra <math>\mathfrak{gl}(n)</math> of ''n'' x ''n'' matrices over ''k'' is a restricted Lie algebra, with the ''p''-mapping being the ''p''th power of a matrix. This "explains" the definition of a restricted Lie algebra: the complicated formula for <math>(X+Y)^{[p]}</math> is needed to express the ''p''th power of the sum of two matrices over ''k'', <math>(X+Y)^p</math>, given that ''X'' and ''Y'' typically do not commute.
-Let ''G'' be an algebraic group over a field k of characteristic ''p'', and <math>\mathrm{Lie}(G)</math> be the [[Zariski tangent space]] at the identity element of ''G''. Each element of <math>\mathrm{Lie}(G)</math> uniquely defines a left-invariant vector field on ''G'', and the commutator of vector fields defines a Lie algebra structure on <math>\mathrm{Lie}(G)</math> just as in the [[Lie group]] case. If ''p>0'', the [[Frobenius map]] <math>x \mapsto x^p</math> defines a ''p'' operation on <math>\mathrm{Lie}(G)</math>.
+Let ''A'' be an algebra over a field ''k''. (Here ''A'' is a possibly [[non-associative algebra]].) Then the [[Lie algebra#Derivations|derivations]] of ''A'' over ''k'' form a Lie algebra <math>\text{Der}_k(A)</math>, with the Lie algebra being the commutator, <math>[D_1,D_2]=D_1D_2-D_2D_1</math>. When ''k'' has characteristic ''p>0'', then the ''p''th power of a derivation is also a derivation, and this makes <math>\text{Der}_k(A)</math> into a restricted Lie algebra.<ref name="definition" />
+Let ''G'' be a group scheme over a field ''k'' of characteristic ''p>0'', and let <math>\mathrm{Lie}(G)</math> be the [[Zariski tangent space]] at the identity element of ''G''. Then <math>\mathrm{Lie}(G)</math> is a restricted Lie algebra over ''k''. This is essentially a special case of the previous example. Indeed, each element of <math>\mathrm{Lie}(G)</math> determines a left-invariant [[vector field]] on ''G'', and hence a derivation on the ring of [[regular function]]s on ''G''. The ''p''th power of this derivation is again a derivation, in fact the derivation associated to an element <math>X^{[p]}</math> of <math>\mathrm{Lie}(G)</math>.
-==Restricted universal enveloping algebra==
-The functor <math>A \mapsto \mathrm{Lie}(A)</math> has a [[left adjoint]] <math>L \mapsto U^{[p]}(L)</math> called the '''restricted universal enveloping algebra'''. To construct this, let <math>U(L)</math> be the [[universal enveloping algebra]] of ''L'' forgetting the ''p'' operation. Letting ''I'' be the two-sided ideal generated by elements of the form <math>x^p - x^{[p]}</math>, we set <math>U^{[p]}(L) = U(L) / I</math>. It satisfies a form of the [[PBW theorem]].
+==Restricted enveloping algebra==
+The functor <math>A \mapsto \mathrm{Lie}(A)</math> has a [[left adjoint]] <math>L \mapsto u(L)</math> called the '''restricted enveloping algebra'''. To construct this, let <math>U(L)</math> be the [[universal enveloping algebra]] of ''L'' (ignoring the ''p''-mapping of ''L''). Let ''I'' be the two-sided ideal generated by elements of the form <math>x^p - x^{[p]}</math>; then the restricted enveloping algebra is <math>u(L) = U(L) / I</math>. It satisfies a form of the [[Poincaré–Birkhoff–Witt theorem]]: if <math>e_1,\ldots,e_n</math> is a [[basis (vector space)|basis]] for ''L'' as a ''k''-vector space, then a basis for <math>u(L)</math> is given by all ordered products <math>e_1^{i_1}\cdots e_n^{i_n}</math> with <math>0\leq i_j\leq p-1</math> for each ''j''. In particular, the map <math>L\to u(L)</math> is injective, and if ''L'' has dimension ''n'' as a vector space, then <math>u(L)</math> has dimension <math>p^n</math> as a vector space.<ref>Strade & Farnsteiner (1988), section 2.5.</ref>
 ==See also==
-Restricted Lie algebras are used in [[Nathan Jacobson|Jacobson]]'s Galois correspondence for [[purely inseparable extension]]s of fields of exponent 1.
+Restricted Lie algebras are used in [[Nathan Jacobson]]'s Galois correspondence for [[purely inseparable extension]]s of fields of exponent 1.
+==Notes==
+{{reflist|30em}}
 ==References==
-* {{citation | first=Armand | last=Borel | authorlink=Armand Borel | title=Linear Algebraic Groups | edition=2nd | series=[[Graduate Texts in Mathematics]] | volume=126 | publisher=[[Springer-Verlag]] | zbl=0726.20030 | year=1991 }}.
+* {{citation|last=Borel|first=Armand|author-link=Armand Borel|title=Linear Algebraic Groups|volume=126|edition=2nd|location=New York| publisher=[[Springer Nature]] |isbn=0-387-97370-2 | year=1991 | orig-year=1969 | mr=1102012 | doi=10.1007/978-1-4612-0941-6|series=Graduate Texts in Mathematics}}
-* {{citation | last1=Block | first1=Richard E. | last2=Wilson | first2=Robert Lee | title=Classification of the restricted simple Lie algebras | doi=10.1016/0021-8693(88)90216-5 | mr=931904 | year=1988 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=114 | issue=1 | pages=115–259| doi-access=free }}.
+* {{citation | author1-last=Block | author1-first=Richard E. | author1-link=Richard Earl Block | author2-last=Wilson | author2-first=Robert Lee | title=Classification of the restricted simple Lie algebras | doi=10.1016/0021-8693(88)90216-5 | mr=931904 | year=1988 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=114 | issue=1 | pages=115–259| doi-access=free }}.
 * {{citation | last=Montgomery | first=Susan | authorlink=Susan Montgomery | title=Hopf algebras and their actions on rings. Expanded version of ten lectures given at the CBMS Conference on Hopf algebras and their actions on rings, which took place at DePaul University in Chicago, USA, August 10-14, 1992 | zbl=0793.16029 | series=Regional Conference Series in Mathematics | volume=82 | location=Providence, RI | publisher=American Mathematical Society | year=1993 | isbn=978-0-8218-0738-5 | page=23 }}.
+* {{citation | author1-last=Strade| author1-first=Helmut | author2-last=Farnsteiner | author2-first=Rolf | title=Modular Lie algebras and their representations | publisher=[[Marcel Dekker]] | year=1988 | mr=0929682 |isbn=0-8247-7594-5 }}
 [[Category:Algebraic groups]]