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- In algebraic geometry, a '''sheaf of algebras''' on a [[ringed space]] ''X'' is a [[sheaf of commutative rings]] on ''X'' ...tics)|category]] of quasi-coherent (sheaves of) <math>\mathcal{O}_X</math>-algebras on ''X'' to the category of schemes that are '''affine''' over ''X'' (defin ...5 KB (820 words) - 17:03, 7 January 2025
- The '''classical Lie algebras''' are finite-dimensional [[Lie algebra]]s over a field which can be classi ...952065417|title=Introduction to finite and infinite dimensional lie (super)algebras|date=18 April 2016|publisher=Amsterdam Elsevie|isbn=9780128046753|location= ...2 KB (278 words) - 14:02, 9 January 2025
- ...ence G|last2=Pedersen|authorlink1=Lawrence G. Brown|first2=Gert K|title=C*-algebras of real rank zero|journal=Journal of Functional Analysis|date=July 1991|vol ...unori|last2=Osaka|first2=Hiroyuki|title=Real Rank of Tensor Products of C*-algebras|journal=Proceedings of the American Mathematical Society|date=July 1995|vol ...5 KB (726 words) - 00:03, 19 August 2023
- ...ators and the relations of the <math>A_i</math>'s. The free product of two algebras ''A'', ''B'' is denoted by ''A'' ∗ ''B''. The notion is a [[R ...(in that [[Category (mathematics)|category]]) is their [[tensor product of algebras|tensor product]]. ...2 KB (343 words) - 09:19, 20 July 2024
- ...the crowning achievements of the [[Lie theory|theory of Lie groups and Lie algebras]]. The theory was worked out mainly by [[E. Cartan]] and [[H. Weyl]] and be ...cover of such a group. Hence, the representation theory of semisimple Lie algebras marks the starting point for the general theory of [[representation of a co ...28 KB (4,267 words) - 05:38, 20 August 2024
- [[Category:C*-algebras]] ...17 KB (2,933 words) - 23:40, 28 March 2023
- ...is no root of unity, and the first examples of finite-dimensional Nichols algebras are the [[Borel subgroup#Lie algebra|Borel parts]] <math>u_q(\mathfrak{g})^ ...is generated by the support of <math>V</math>. For more details on Nichols algebras see [[Nichols algebra]]. ...26 KB (3,716 words) - 23:32, 26 January 2025
- ...ist provides Mubarakzyanov's '''classification of low-dimensional real Lie algebras''', published in Russian in 1963.<ref>{{harvnb|Mubarakzyanov|1963}}</ref> I Over the field <math>{\mathbb C}</math> algebras <math>{\mathfrak g}_{3.5}</math>, <math>{\mathfrak g}_{3.7}</math> are isom ...8 KB (1,032 words) - 13:42, 24 October 2023
- ...ebra]]s. For the topics in the representation theory of Lie groups and Lie algebras, see [[Glossary of representation theory]]. Because of the lack of other op ...Lie algebra <math>\mathfrak g</math> obtained by repeatedly taking derived algebras; i.e., <math>D^0 \mathfrak{g} = \mathfrak{g}, D^n \mathfrak{g} = D^{n-1}\ma ...23 KB (3,462 words) - 21:20, 10 January 2024
- {{Short description|Non-associative algebras with positive-definite quadratic form}} ...ere are no other possibilities. Such algebras, sometimes called '''Hurwitz algebras''', are examples of [[composition algebra]]s. ...28 KB (4,092 words) - 18:15, 8 February 2025
Page text matches
- ...escription|Identity valid in special Jordan algebras but not in all Jordan algebras}} ...</ref> which vanishes in all special Jordan algebras but not in all Jordan algebras. What is now known as Glennie's identity first appeared in his 1963 Yale Ph ...2 KB (286 words) - 11:54, 7 June 2024
- {{short description|Metric on the C*-algebras on a fixed Hilbert space}} ...[[Hausdorff distance]] between the [[unit ball]]s of the two C<sup>*</sup>-algebras, under the norm-induced metric on the space of all [[bounded operator]]s on ...2 KB (250 words) - 20:16, 12 August 2019
- ...they satisfy the flexible identity.<ref>Richard D. Schafer (1954) “On the algebras formed by the Cayley-Dickson process”, [[American Journal of Mathematics]] Besides [[associative algebra]]s, the following classes of nonassociative algebras are flexible: ...3 KB (339 words) - 17:37, 21 February 2025
- ...ere named after [[William of Ockham]] by {{harvtxt|Urquhart|1979}}. Ockham algebras form a [[variety (universal algebra)|variety]]. Examples of Ockham algebras include [[Boolean algebra]]s, [[De Morgan algebra]]s, [[Kleene algebra (wit ...2 KB (253 words) - 01:18, 21 May 2024
- The '''classical Lie algebras''' are finite-dimensional [[Lie algebra]]s over a field which can be classi ...952065417|title=Introduction to finite and infinite dimensional lie (super)algebras|date=18 April 2016|publisher=Amsterdam Elsevie|isbn=9780128046753|location= ...2 KB (278 words) - 14:02, 9 January 2025
- ...ators and the relations of the <math>A_i</math>'s. The free product of two algebras ''A'', ''B'' is denoted by ''A'' ∗ ''B''. The notion is a [[R ...(in that [[Category (mathematics)|category]]) is their [[tensor product of algebras|tensor product]]. ...2 KB (343 words) - 09:19, 20 July 2024
- A direct sum of simple Lie algebras is called a [[semisimple Lie algebra]]. == Complex simple Lie algebras == ...3 KB (531 words) - 03:00, 27 December 2024
- ==Linked quaternion algebras== ...lgebras over ''F''. The algebras ''A'' and ''B'' are '''linked quaternion algebras''' over ''F'' if there is ''x'' in ''F'' such that ''A'' is equivalent to ( ...3 KB (399 words) - 08:31, 25 June 2020
- Examples of such braided algebras (and even [[braided Hopf algebra|Hopf algebras]]) are the [[Nichols algebra]]s, that are by definition generated by a give ...uskiewitsch, Schneider: ''Pointed Hopf algebras'', New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, ...2 KB (371 words) - 03:12, 13 May 2024
- ...mov, M. T.|title=Unipotent and Nakayama automorphisms of quantum nilpotent algebras|date=1 Nov 2013|class=math.QA|eprint=1311.0278}}</ref> a concept related to ...title=Structure of Algebras|page=22|chapter=Chapt. 2: Ideals and Nilpotent Algebras|orig-year=1939|year=2003|series=Colloquium Publications, Col. 24|publisher= ...3 KB (409 words) - 10:02, 22 April 2021
- ...to a noncommutative setting by studying those ideas on general von Neumann algebras. ...ebras with a faithful normal tracial state, for example finite von Neumann algebras, the notion of conditional expectation is especially useful. ...3 KB (468 words) - 18:55, 23 January 2022
- ...are determined to a large extent by their projections. The idea behind AW*-algebras is to forgot the former, topological, condition, and use only the latter, a Hence an AW*-algebra is a C*-algebras that is at the same time a [[Baer *-ring]]. ...4 KB (632 words) - 12:23, 5 February 2025
- ...m/books?id=Yh1RHnYCDNsC&pg=PA77 p. 77]}}</ref> Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center. ...s by diagonalizable ones) and all splittings are conjugate; thus split Lie algebras are of most interest for non-algebraically closed fields. ...5 KB (790 words) - 19:44, 26 January 2024
- ...ons of C*-algebras. It implies that, for irreducible representations of C*-algebras, the only non-zero linear invariant subspace is the whole space. ...on, R. V.]]; [[Ringrose, J. R.]], ''Fundamentals of the Theory of Operator Algebras'', Vol. I : Elementary Theory, {{ISBN|978-0821808191}}</ref> If the C*-alge ...3 KB (374 words) - 17:50, 18 July 2024
- === Semisimple Lie algebras === ...|sl(n,R)]] and [[SU(N)#Lie algebra|su(n)]], as well as [[direct sum of Lie algebras|direct sum]]s of them. Let thus '''g''' be a semi-simple Lie algebra with ...3 KB (414 words) - 00:41, 19 May 2024
- ...]. In particular, the [[variety (universal algebra)|variety]] of all modal algebras is the equivalent algebraic semantics of the modal logic ''K'' in the sense ...dal algebra satisfying <math>\Box (-\Box x \lor x) = \Box x</math>. Magari algebras correspond to [[provability logic]]. ...2 KB (237 words) - 06:13, 14 January 2025
- ...or]] category. Lie conformal algebras are very closely related to [[vertex algebras]] and have many applications in other areas of algebra and integrable syste ==Definition and relation to Lie algebras== ...3 KB (449 words) - 19:11, 22 July 2022
- {{Lie groups |Algebras}} ===Reductive Lie algebras=== ...2 KB (345 words) - 21:07, 25 February 2025
- ...ariety of [[group (mathematics)|groups]], consist of congruence-permutable algebras, but some, like the variety of [[lattice (order)|lattices]], have members t ...> permute.{{r|Bergman|p=122}} A [[variety (universal algebra)|variety]] of algebras <math>\mathcal{V}</math> is referred to as '''congruence-permutable''' when ...3 KB (405 words) - 15:23, 17 December 2020
- ...square is the identity.<ref name=Schafer>{{cite news|title=On Structurable algebras|author=R.D. Schafer|journal=[[Journal of Algebra]]|year=1985|volume=92|page ...ructurable algebra'' if:<ref name=Garibaldi>{{cite news|title=Structurable Algebras and Groups of Type E_6 and E_7|author=Skip Garibaldi|author-link=Skip Garib ...2 KB (356 words) - 19:07, 29 December 2020