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Let G be a Lie group and let H be a closed subgroup ( not necessarily normal ).
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Let and G
Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.
Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀ x ∈ A ∀ g ∈ G ( g ( x ) ∈ ).
Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a ~ b ↔ ( ab < sup >− 1 </ sup > ∈ H ).
Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.
Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD.
Let F and G be a pair of adjoint functors with unit η and co-unit ε ( see the article on adjoint functors for the definitions ).
Let and be
Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.
Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.
Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.
Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.
Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.
Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.
Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.
Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.
Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.
Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.
Let and Lie
The film and its successors spawned countless imitators that borrowed elements instituted by Romero: Tombs of the Blind Dead, Let Sleeping Corpses Lie ( film ), Zombi 2, Hell of the Living Dead, Night of the Comet, Return of the Living Dead, Night of the Creeps, Children of the Living Dead, and the video game series Resident Evil ( later adapted as films in 2002, 2004, 2007 and 2010 ), Dead Rising, and House of the Dead.
# Let Sleeping Corpses Lie ( original title: Non si deve profanare il sonno dei morti ; also known as The Living Dead at Manchester Morgue, Don't Open the Window ) — passed with 2 minutes pre-cut in 1985 ; re-released uncut in 2002
Let be a Lie group and be its Lie algebra ( thought of as the tangent space to the identity element of ).
Let V be a representation of a Lie algebra g over a field F and let λ be a weight of g. Then the weight space of V with weight λ: g → F is the subspace
Let g be a Lie algebra, h a maximal commutative Lie subalgebra consisting of semi-simple elements ( sometimes called Cartan subalgebra ) and let V be a finite dimensional representation of g. If g is semisimple, then g = g and so all weights on g are trivial.
Let P → M be a principal bundle over a manifold M with structure Lie group G and a principal connection ω.
Let G be a Lie group and let be its Lie algebra ( which we identify with T < sub > e </ sub > G, the tangent space to the identity element in G ).
Let be the Cartan matrix of the Kac-Moody algebra, and let q be a nonzero complex number distinct from 1, then the quantum group, U < sub > q </ sub >( G ), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators ( where λ is an element of the weight lattice, i. e. for all i ), and and ( for simple roots, ), subject to the following relations:
Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → U < sub > L </ sub >, ( notation as above ) we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → A < sub > L </ sub > there exists a unique unital algebra homomorphism g: U → A such that: f (-) = g < sub > L </ sub > ( h (-)).
Let us extend to compact Lie group and consider " integrable " orbits for which the symplectic structure comes from a line bundle then quantization leads to the irreducible representations of.
Let π: P → M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P.
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