
	Where results make sense

Topic: Subalgebra

	PlanetMath: Borel subalgebra
		We have a root decomposition into the Cartan subalgebra and the root spaces
		be the direct sum of the Cartan subalgebra and the positive root spaces.
		This is version 3 of Borel subalgebra, born on 2002-12-05, modified 2004-04-09.
	planetmath.org /encyclopedia/BorelSubalgebra.html (76 words)

	Octonion - Open Encyclopedia (Site not responding. Last check: 2007-11-06)
		Note that each of the seven lines generates a subalgebra of O isomorphic to the quaternions H.
		This means that the subalgebra generated by any two elements is associative.
		Actually, one can show that the subalgebra generated by any two elements of O is isomorphic to R, C, or H, all of which are associative.
	open-encyclopedia.com /Octonion (799 words)

	Operations on Associative Algebras and their Elements (Site not responding. Last check: 2007-11-06)
		Given an associative algebra A and a subalgebra B of A, compute the idealizer of B in A, that is, the largest subalgebra of A in which B is an ideal.
		Let A and B be subalgebras of an associative algebra with underlying module M. This function returns the submodule of M which is spanned by the elements [a, b] = a * b - b * a, a in A, b in B. CommutatorIdeal(A, B) : AlgAss, AlgAss -> AlgAss
		For two subalgebras A and B of an associative algebra, return the left annihilator of B in A; that is, the subalgebra of A consisting of all elements a such that a * b = 0 for all b in B. RightAnnihilator(A, B) : AlgAss, AlgAss -> AlgAss, AlgAss
	www.math.niu.edu /help/math/magmahelp/text849.html (718 words)

	[No title]
		A principal three-dimensional subalgebra of $G_l$ is a subalgebra $A$ of type ${\rm sl}(2)$, with basis $\{E,F,H\}$ satisfying \beq [H,E]=2E,\qquad [H,F]=-2F,\qquad [E,F]=H, \eeq such that the number of irreducible components occurring in the complete reduction of the adjoint representation of $G_l$ with respect to $A$ is equal to $l$ (Kostand 1959).
		In this Letter the investigation of principal subalgebras of quantum enveloping algebras of type ${\rm gl}_q(l+1)$ or ${\rm sl}_q(l+1)$ is initiated.
		A principal subalgebra of ${\rm gl}_q(l+1)$ is defined as follows~: it is a subalgebra of ${\rm gl}_q(l+1)$ of type ${\rm sl}_q(2)$, i.e.~its generators $\{E,F,H\}$ satisfy \beq [H,E]=2E,\qquad [H,F]=-2F,\qquad [E,F]=[H], \label{5} \eeq and in the limit $q\rightarrow q$, this ${\rm sl}_q(2)$ subalgebra reduces to the principal subalgebra of ${\rm gl}(l+1)$.
	allserv.rug.ac.be /~jvdjeugt/files/tex/q-princ.tex (1831 words)

	6. Relating the Submonoid and Subalgebra Membership Problems in Monoids and Monoid Rings
		While the subgroup problem is thoroughly studied in the literature, the submonoid problem is less investigated except for some special cases like the free monoid case.
		Hence the submonoid cannot be described adequately in the monoid ring using the right ideal congruence as in the subgroup case studied before.
		The next theorem states that the submonoid problem for a monoid is equivalent to a special instance of the subalgebra membership problem in the corresponding monoid ring.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/14/paper_html/node6.html (348 words)

	PlanetMath: Cartan subalgebra
		Then a Cartan subalgebra is a maximal subalgebra of
		All Cartan subalgebras of a Lie algebra are conjugate by the adjoint action of any Lie group with algebra
		This is version 4 of Cartan subalgebra, born on 2002-12-27, modified 2005-09-26.
	planetmath.org /encyclopedia/CartanSubalgebra2.html (78 words)

	The Cohomology Algebra Of A Subalgebra Of The Steenrod Algebra (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		The Cohomology Algebra Of A Subalgebra Of The Steenrod Algebra (ResearchIndex)
		The Cohomology Algebra Of A Subalgebra Of The Steenrod Algebra
		We compute the cohomology algebra of P (1), the subalgebra of the Steenrod algebra generated by P 1 and P p.
	citeseer.ist.psu.edu /349552.html (312 words)

	Blocks of Finite Groups : The Hyperfocal Subalgebra of a Block. Engl.-Chin. (Springer Monographs in: ... (Site not responding. Last check: 2007-11-06)
		In this book, based on a course given by the author at Wuhan University in 1999, all the concepts mentioned are introduced, and all the proofs are developed completely.
		Its main purpose is the proof of the existence and the uniqueness of the "hyperfocal subalgebra" in the source algebra.
		This result seems fundamental in block theory; for instance, the structure of the source algebra of a nilpotent block, an important fact in block theory, can be obtained as a corollary.
	bookweb.kinokuniya.co.jp /htmy/354043514X.html (215 words)

	Lie Algebra Notes
		Theorem: Suppose L is a Lie subalgebra of gl(V) consisting entirely of nilpotent endomorphisms, then there is some v in V so that xv=0 for all x in L. Proof: We proceed by induction on dim L. The statement is obviously true for dim L in {0,1}.
		Let K be a maximal proper Lie subalgebra of L. Via ad we get a representation of K on L. Since K is a subalgebra K is ad(K) invariant, thus we have a representation of K on L/K. Pick x in K and remember x is nilpotent.
		Corollary: Suppose L is a nilpotent subalgebra of gl(V).
	www.math.rutgers.edu /~nacin/Sahi4.html (1540 words)

	AlgebraicGroups - Cmat Wiki (Site not responding. Last check: 2007-11-06)
		If $\mathfrak h \subset \operatorname{\mathcal L}(G)$ is a Lie subalgebra with the property that $\mathfrak h \cdot V \subset U$, then $\mathfrak h^{+} \cdot V \subset U$.
		A Lie subalgebra $\mathfrak r$ of $\operatorname{\mathcal L}(G)$ is said to be {\em $G$-linearly reductive}\/ if every rational $G$--module is semisimple as an $\mathfrak r$--module.
		Among the $G$--linearly reductive Lie subalgebras of $\operatorname{\mathcal L}(G)$ that contain $\mathfrak s$, take a maximal one and denote it as $\mathfrak r$.
	www.cmat.edu.uy /moin/AlgebraicGroups (3010 words)

	GAP Manual: 38 Algebras (Site not responding. Last check: 2007-11-06)
		See Parent Algebras and Subalgebras, and the corresponding section More about Groups and Subgroups in the chapter about groups for details about the distinction between parent algebras and subalgebras.
		This is represented as a subalgebra of the common parent of
		The image of a subalgebra under a algebra homomorphism is computed by computing the images of a set of generators of the subalgebra, and the result is the subalgebra generated by those images.
	schmidt.nuigalway.ie /gap/CHAP038.htm (2936 words)

	The Mathematics of Boolean Algebra
		The cellularity c(A) of a BA is the supremum of the cardinalities of sets of pairwise disjoint elements of A.
		An important fact concerning cellularity is the Erdos-Tarski theorem: if the cellularity of a BA is a singular cardinal, then there actually is a set of disjoint elements of that size; for cellularity regular limit (inaccessible), there are counterexamples.
		On the other hand, the theory of a Boolean algebra with a distinguished subalgebra is undecidable.
	plato.stanford.edu /entries/boolalg-math (2064 words)

	CCR AND THE m DIMENSIONAL HEISENBERG ALGEBRAS
		and so, for every a, {a} is an invariant subalgebra of H. The complement of {a} in H is also a subalgebra of H which is also invariant in H, implying that for any m, H is a direct sum of m of the {a}.
		We have found a Lie algebra which contains H as an invariant subalgebra, and which is therefore an extension of H. The dimension of the algebra is reduced from 7 to 6, by the relation of linear dependence qp = pq + iI.
		The kinematic Heisenberg algebra is an invariant Lie subalgebra of the dynamical Lie algebra.
	graham.main.nc.us /~bhammel/PHYS/heisalg.html (4846 words)

	[No title]
		The Q-subalgebra * in the title is a polynomial subalgebra* generated by elements obtained by the acti* *on of maximum number of Milnor primitives, in this case n + 1, on the fundamental class n+1.
		The subalgebra in (6-20) has a very special property and it is of particular interest.
		The subalgebra Q is annihilated by the Milnor primi* *tive Qj for any j 1.
	hopf.math.purdue.edu /Tamanoi/Eilenberg-MacLane.txt (5576 words)

	Subalgebra -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		However, in more general situations, it's not safe to make analogous assumptions, and every operation must be checked.
		The term subalgebra is also used in the context of specific types of algebras such as (Click link for more info and facts about associative algebra) associative algebras and (Click link for more info and facts about Lie algebra) Lie algebras.
		In those contexts, you should think specifically of the algebraic structures relevant to them.
	www.absoluteastronomy.com /encyclopedia/s/su/subalgebra.htm (249 words)

	HKUST Institutional Repository: Item 1783.1/2013
		Let Z(k) and Z(g) be, respectively, the centers of the universal enveloping algebras of k and g.
		Kostant generalizes the result of Huang and Pandžić to the case where r is an arbitrary reductive subalgebra of g.
		In Kostant's work, the map Resh/t : S(h)W(g,h) → S(t)W(r,t), where t is a Cartan subalgebra of r contained in h, plays a similar role as the restriction ma...
	hdl.handle.net /1783.1/2013 (229 words)

	Canonical Subalgebra Bases in Non-commutative Polynomial Rings - Nordbeck (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Abstract: Canonical bases, also called SAGBI bases, for subalgebras of the non-commutative polynomial ring are investigated.
		Methods, including generalizations of the standard Grobner bases techniques, are developed for the test whether bases are canonical, and for the completion procedure of constructing canonical bases.
		The special case of homogeneous subalgebras is discussed.
	citeseer.ist.psu.edu /nordbeck98canonical.html (379 words)

	Cartan Subalgebra (Site not responding. Last check: 2007-11-06)
		Given a Lie algebra L, return a Cartan subalgebra of L. The algorithm works for Lie algebras L defined over a field F such that F > dim L and for restricted Lie algebras of characteristic p.
		If the Lie algebra does not fit into one of these classes then the correctness of the output is not guaranteed.
		We compute a Cartan subalgebra of the simple Lie algebra of type A_4 over the rational field.
	www.math.niu.edu /help/math/magmahelp/text907.html (75 words)

	Pioro (Site not responding. Last check: 2007-11-06)
		First, we show that for two arbitrary partial algebras, if their directed hypergraphs are isomorphic, then their weak, relative and strong subalgebra lattices are isomorphic.
		Secondly, we prove that two partial algebras have isomorphic weak subalgebra lattices iff their hypergraphs are isomorphic.
		Thirdly, for an arbitrary lattice $\mathbf{L}$ and a partial algebra $\mathbf{A}$ we describe (necessary and sufficient conditions) when the weak subalgebra lattice of $\mathbf{A}$ is isomorphic to $\mathbf{L}$.
	www.mat.ub.es /EMIS/journals/AM/00-1/pioro.html (130 words)

	Stone-Weierstrass theorem
		The set of all polynomial functions forms a subalgebra of C[a,b], and the content of the Weierstrass approximation theorem is that this subalgebra is dense in C[a,b].
		The approximation theorem is generalized in two directions: instead of the compact interval [a,b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, approximation with elements from other subalgebras of C(X) is investigated.
		The crucial property of the subalgebra is that it separates points: A subset A of C(X) is said to separate points, if for every two different points x and y in X and every two real numbers a and b there exists a function p in A with p(x) = a and p(y) = b.
	www.findword.org /st/stone-weierstrass-theorem.html (864 words)

	GluCat: Generic library of universal Clifford algebra templates
		It is the subalgebra defined by the frame.
		The result is, in general, contained in the subalgebra which spans the operands.
		The fold() and unfold() operations map an index set within a subalgebra to and from the isomorphic subalgebra defined by the corresponding contiguous index set.
	glucat.sourceforge.net (2306 words)

	A Completion Procedure for Computing a Canonical Basis for a k-Subalgebra - et, MADLENER (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		A canonical basis produced by the completion procedure shares many properties of a Grobner basis such as reducing an element of a k-subalgebra to 0 and generating unique normal forms for the...
		...approaches to solve the subalgebra problem in the commutative case using rewriting techniques can be found in the literature.
		and [RoSw90] is a generating set for a subalgebra such that the initial terms of the subalgebra are contained in the algebra generated...
	citeseer.ist.psu.edu /254395.html (570 words)

	Subalgebras
		But warranties are included in the Subalgebras of a product, while service contracts cost extra and are sold separately.
		Be aware that the Subalgebras rate on most adjustable rate mortgage loans (ARMs) can vary a great deal over the lifetime of the mortgage.
		The leasing payments may be lower because you don't own the Subalgebras at the end of the Subalgebras.
	subalgebra.ask.dyndns.dk /Subalgebras (539 words)

	ad-NILPOTENT b-IDEALS IN sl(n) HAVING A FIXED CLASS OF (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Abstract: We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n+1; C).
		We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence.
		4 Enumeration of ad-nilpotent ideals of a Borel subalgebra in..
	citeseer.lcs.mit.edu /605238.html (347 words)

	[ref] 61 Lie Algebras
		Creating and working with subalgebras goes exactly in the same way as for general algebras; so for that we refer to Chapter algebras.
		Here we describe functions that calculate well-known subalgebras and ideals of a Lie algebra (such as the centre, the centralizer of a subalgebra, etc.).
		is a semisimple subalgebra complementary to the radical of
	www.gap-system.org /Manuals/doc/htm/ref/CHAP061.htm (4974 words)

	Crossed products of operator algebras and Fell buindles over equivalence relations
		In particular, structure theorems were proved, that is, given algebra A and its subalgebra D, there are sufficient and necessary conditions for A to be isomorphic to the crossed product of D by some equivalence relation R.
		One can also define the ``diagonal subalgebra'' D as the set of functions supported on the diagonal of R.
		This theorem gives necessary and sufficient conditions for a C* -algebra to be isomorphic to the C* -algebra generated by some Fell bundle, such that its given subalgebra is isomorpfic to the diagonal subalgebra.
	math.la.asu.edu /~ifulman/report/report.html (986 words)

	Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 18, No. 1, pp. 1-6, 2002 (Site not responding. Last check: 2007-11-06)
		Recognizable properties of associative finitely presented algebras with the finite Gröbner basis were investigated by V. Latyshev, T. Gateva-Ivanova in \cite{gatlat}.
		While subalgebras may not be as important as ideals, they are the second major type of \emph{subobject} in ring theory.
		The paper considers recognizable properties of subalgebras with finite standard basis, or SAGBI-basis (Subalgebra Analogue to Gröbner Basis for Ideals).
	www.emis.ams.org /journals/AMAPN/vol18_1/1.html (114 words)

	Ete 2001
		Let F be a torsion free [^(G)]-module and P be the parabolic subalgebra of G generated by [^(G)], the Cartan subalgebra of G and all positive root vectors of G.
		N ([(L)\tilde] is a regular semisimple subalgebra of L, [(H)\tilde] is the complement of the Cartan subalgebra of [(L)\tilde] orthogonal to [(L)\tilde] in H and N is the nilpotent part of P).
		Here R is the root system of G, and this may be finite or only locally finite, in which case G is finite-dimensional respectively locally-finite.
	www.cms.math.ca /CMS/Events/summer01/abs/lt-f.html (1201 words)

	MATHEMATICA BOHEMICA, Vol. 126, No. 1, pp. 171-181, 2001 (Site not responding. Last check: 2007-11-06)
		On subalgebra lattices of a finite unary algebra II Konrad Pioro
		Next, we describe other pairs of subalgebra lattices (weak and relative, etc.) of a finite unary algebra.
		Finally, necessary and sufficient conditions are found for quadruples $\langle \mathbf {L}_{1},\mathbf {L}_{2}, \mathbf {L}_{3},\mathbf {L}_{4}\rangle$ of lattices for which there is a finite unary algebra having its weak, relative, strong subalgebra and initial segment lattices isomorphic to $\mathbf {L}_{1},\mathbf {L}_{2}, \mathbf {L}_{3},\mathbf {L}_{4}$, respectively.
	www.emis.de /journals/MB/126.1/14.html (175 words)

	MATHEMATICA BOHEMICA, Vol. 126, No. 1, pp. 161-170, 2001 (Site not responding. Last check: 2007-11-06)
		On subalgebra lattices of a finite unary algebra I
		Abstract: One of the main aims of the present and the next part [15] is to show that the theory of graphs (its language and results) can be very useful in algebraic investigations.
		We characterize, in terms of isomorphisms of some digraphs, all pairs $\langle\mathbf{A},\mathbf{L}\rangle$, where $\mathbf{A}$ is a finite unary algebra and $\Cal L$ a finite lattice such that the subalgebra lattice of $\mathbf{A}$ is isomorphic to $\mathbf{L}$.
	mb.math.cas.cz /mb126-1/13.html (232 words)

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