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Unitary Representation Of A Star Lie Superalgebra
In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra ''L'' on a Z2-graded vector space ''V'', such that if ''A'' and ''B'' are any two pure elements of ''L'' and ''X'' and ''Y'' are any two pure elements of ''V'', then :(c_1 A+c_2 B)\cdot X=c_1 A\cdot X + c_2 B\cdot X :A\cdot (c_1 X + c_2 Y)=c_1 A\cdot X + c_2 A\cdot Y :(-1)^=(-1)^A(-1)^X : ,Bcdot X=A\cdot (B\cdot X)-(-1)^B\cdot (A\cdot X). Equivalently, a representation of ''L'' is a Z2-graded representation of the universal enveloping algebra of ''L'' which respects the third equation above. Unitary representation of a star Lie superalgebra A * Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map * such that * respects the grading and : ,bsup>*= *,a* A unitary representation of such a Lie algebra is a Z2 graded Hilbert space which is a representation of a Lie superalgebra as above together with the requirement th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
